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I. Sobolev Type Inequalities |
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My Love Affair with the Sobolev Inequality |
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1 | (24) |
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4 | (3) |
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7 | (2) |
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A Morrey-Sobolev Inequality |
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9 | (3) |
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A Morrey-Besov Inequality |
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12 | (1) |
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Exponential Integrability |
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13 | (5) |
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Vanishing Exponential Integrability |
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18 | (2) |
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20 | (5) |
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21 | (4) |
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Maximal Functions in Sobolev Spaces |
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25 | (44) |
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25 | (2) |
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Maximal Function Defined on the Whole Space |
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27 | (6) |
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Boundedness in Sobolev spaces |
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27 | (5) |
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A capacitary weak type estimate |
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32 | (1) |
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Maximal Function Defined on a Subdomain |
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33 | (8) |
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Boundedness in Sobolev spaces |
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33 | (5) |
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38 | (3) |
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41 | (8) |
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Lusin type approximation of Sobolev functions |
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45 | (4) |
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49 | (5) |
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Maximal Functions on Metric Measure Spaces |
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54 | (15) |
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Sobolev spaces on metric measure spaces |
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55 | (2) |
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Maximal function defined on the whole space |
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57 | (5) |
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Maximal function defined on a subdomain |
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62 | (2) |
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Pointwise estimates and Lusin type approximation |
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64 | (1) |
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65 | (4) |
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Hardy Type Inequalities Via Riccati and Sturm-Liouville Equations |
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69 | (18) |
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69 | (2) |
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71 | (4) |
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Transition to Sturm-Liouville Equations |
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75 | (2) |
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Hardy Type Inequalities with Weights |
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77 | (6) |
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Poincare Type Inequalities |
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83 | (4) |
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85 | (2) |
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Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles |
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87 | (30) |
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87 | (2) |
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Symmetrization Inequalities |
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89 | (12) |
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Rearrangements of functions and function spaces |
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89 | (3) |
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The Hardy-Littlewood inequality |
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92 | (4) |
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The Polya-Szego inequality |
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96 | (5) |
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101 | (7) |
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Functions of Bounded Variation |
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101 | (2) |
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103 | (3) |
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106 | (2) |
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108 | (9) |
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108 | (3) |
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111 | (2) |
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113 | (4) |
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Inequalities of Hardy-Sobolev Type in Carnot-Caratheodory Spaces |
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117 | (36) |
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117 | (4) |
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121 | (6) |
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Pointwise Hardy Inequalities |
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127 | (12) |
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Hardy Inequalities on Bounded Domains |
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139 | (6) |
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Hardy Inequalities with Sharp Constants |
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145 | (8) |
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149 | (4) |
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Sobolev Embeddings and Hardy Operators |
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153 | (32) |
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153 | (1) |
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154 | (4) |
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The Poincare Inequality, α(E) and Hardy Type Operators |
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158 | (4) |
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Generalized Ridged Domains |
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162 | (8) |
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Approximation and Other s-Numbers of Hardy Type Operators |
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170 | (11) |
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Approximation Numbers of Embeddings on Generalized Ridged Domains |
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181 | (4) |
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182 | (3) |
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Sobolev Mappings between Manifolds and Metric Spaces |
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185 | (38) |
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185 | (2) |
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Sobolev Mappings between Manifolds |
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187 | (10) |
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Sobolev Mappings into Metric Spaces |
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197 | (8) |
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202 | (3) |
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Sobolev Spaces on Metric Measure Spaces |
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205 | (10) |
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Integration on rectifiable curves |
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205 | (2) |
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207 | (1) |
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208 | (1) |
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208 | (1) |
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209 | (2) |
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Other spaces of Sobolev type |
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211 | (3) |
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Spaces supporting the Poincare inequality |
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214 | (1) |
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Sobolev Mappings between Metric Spaces |
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215 | (8) |
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218 | (1) |
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219 | (4) |
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A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions |
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223 | (26) |
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223 | (3) |
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Estimate for a Quadratic Form of the Gradient |
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226 | (4) |
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Weighted Garding Inequality for the Biharmonic Operator |
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230 | (3) |
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Dilation Invariant Hardy's Inequalities with Remainder Term |
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233 | (8) |
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Generalized Hardy-Sobolev Inequality with Sharp Constant |
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241 | (3) |
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Hardy's Inequality with Sharp Sobolev Remainder Term |
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244 | (5) |
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245 | (4) |
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Optimality of Function Spaces in Sobolev Embeddings |
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249 | (32) |
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249 | (7) |
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256 | (2) |
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258 | (3) |
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Optimal Range and Optimal Domain of Rearrangement-Invariant Spaces |
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261 | (3) |
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Formulas for Optimal Spaces Using the Functional f-f |
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264 | (3) |
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Explicit Formulas for Optimal Spaces in Sobolev Embeddings |
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267 | (3) |
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Compactness of Sobolev Embeddings |
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270 | (5) |
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275 | (1) |
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Gaussian Sobolev Embeddings |
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276 | (5) |
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278 | (3) |
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On the Hardy-Sobolev-Maz'ya Inequality and Its Generalizations |
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281 | (18) |
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281 | (3) |
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Generalization of the Hardy-Sobolev-Maz'ya Inequality |
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284 | (8) |
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The Space DV1,2(Ω) and Minimizers for the Hardy-Sobolev-Maz'ya Inequality |
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292 | (1) |
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Convexity Properties of the Functional Q for p>2 |
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293 | (6) |
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296 | (3) |
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Sobolev Inequalities in Familiar and Unfamiliar Settings |
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299 | (46) |
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299 | (1) |
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300 | (12) |
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300 | (2) |
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302 | (1) |
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Poincare, Sobolev, and the doubling property |
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303 | (7) |
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310 | (2) |
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Analysis and Geometry on Dirichlet Spaces |
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312 | (10) |
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312 | (1) |
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312 | (2) |
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Local weak solutions of the Laplace and heat equations |
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314 | (2) |
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Harnack type Dirichlet spaces |
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316 | (2) |
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Imaginary powers of---A and the wave equation |
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318 | (2) |
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320 | (2) |
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Flat Sobolev Inequalities |
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322 | (8) |
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How to prove a flat Sobolev inequality? |
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322 | (2) |
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Flat Sobolev inequalities and semigroups of operators |
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324 | (2) |
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The Rozenblum-Cwikel-Lieb inequality |
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326 | (3) |
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Flat Sobolev inequalities in the finite volume case |
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329 | (1) |
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Flat Sobolev inequalities and topology at infinity |
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330 | (1) |
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Sobolev Inequalities on Graphs |
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330 | (15) |
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331 | (1) |
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Sobolev inequalities and volume growth |
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332 | (1) |
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333 | (2) |
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335 | (4) |
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339 | (6) |
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A Universality Property of Sobolev Spaces in Metric Measure Spaces |
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345 | (16) |
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345 | (2) |
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347 | (2) |
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Dirichlet Forms and N1,2(X) |
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349 | (7) |
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Axiomatic Sobolev Spaces and N1, p(X) |
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356 | (5) |
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358 | (3) |
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Cocompact Imbeddings and Structure of Weakly Converget Sequences |
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361 | (16) |
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361 | (2) |
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Dislocation Space and Weak Convergence Decomposition |
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363 | (5) |
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Cocompactness and Minimizers |
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368 | (4) |
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372 | (1) |
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373 | (4) |
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375 | (2) |
| Index |
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377 | (10) |
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II. Applications in Analysis and Partial Differential Equations |
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On the Mathematical Works of S.L. Sobolev in the 1930s |
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1 | (10) |
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8 | (3) |
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11 | (8) |
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Boundary Harnack Principle and the Quasihyperbolic Boundary Condition |
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19 | (12) |
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19 | (4) |
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Boundary Harnack Principle and Carleson Estimate in Terms of the Green Function |
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23 | (1) |
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24 | (7) |
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30 | (1) |
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Sobolev Spaces and their Relatives: Local Polynomial Approximation Approach |
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31 | (38) |
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Topics in Local Polynomial Approximation Theory |
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32 | (11) |
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Local Approximation Spaces |
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43 | (14) |
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48 | (3) |
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51 | (2) |
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53 | (1) |
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54 | (3) |
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57 | (12) |
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57 | (2) |
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59 | (3) |
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Pointwise differentiability |
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62 | (3) |
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65 | (2) |
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67 | (2) |
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Spectral Stability of Higher Order Uniformaly Elliptic Operators |
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69 | (34) |
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69 | (3) |
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Preliminaries and Notation |
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72 | (5) |
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Open Sets with Continuous Boundaries |
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77 | (2) |
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The Case of Diffeomorphic Open Sets |
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79 | (3) |
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Estimates for Dirichlet Eigenvalues via the Atlas Distance |
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82 | (4) |
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Estimates for Neumann Eigenvalues via the Atlas Distance |
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86 | (8) |
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Estimates via the Lower Hausdorff-Pompeiu Deviation |
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94 | (4) |
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98 | (5) |
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98 | (2) |
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Comparison of atlas distance, Hausdorff-Pompeiu distance, and lower Hausdorff-Pompeiu deviation |
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100 | (1) |
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101 | (2) |
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Conductor Inequalities and Criteria for Sobolev-Lorentz Two-Weight Inequalities |
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103 | (20) |
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103 | (2) |
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105 | (4) |
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Sobolev-Lorentz p, q-Capacitance |
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109 | (3) |
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112 | (3) |
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Necessary and Sufficient Conditions for Two-Weight Embeddings |
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115 | (8) |
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119 | (4) |
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Besov Regularity for the Poisson Equation in Smooth and Polyhedral Cones |
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123 | (24) |
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123 | (3) |
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Regularity Result for a Smooth Cone |
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126 | (8) |
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Besov Regularity for the Neumann Problem |
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134 | (5) |
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Appendix A. Regularity of Solutions of the Poisson Equation |
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139 | (2) |
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Appendix B. Function Spaces |
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141 | (6) |
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141 | (1) |
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Sobolev spaces on domains |
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141 | (1) |
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Besov spaces and wavelets |
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142 | (2) |
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144 | (3) |
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Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution Partial Differential Equations |
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147 | (52) |
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Introduction. Higher-Order Models and Blow-up or Compacton Solutions |
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148 | (6) |
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Three types of nonlinear PDEs under consideration |
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148 | (1) |
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Combustion type models: regional blow-up, global stability, main goals, and first discussion |
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149 | (2) |
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Regional blow-up in quasilinear hyperbolic equations |
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151 | (1) |
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Nonlinear dispersion equations and compactons |
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152 | (2) |
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Blow-up Problem: General Blow-up Analysis of Parabolic and Hyperbolic PDEs |
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154 | (8) |
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Global existence and blow-up in higher order parabolic equations |
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154 | (6) |
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Blow-up data for higher order parabolic and hyperbolic PDEs |
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160 | (1) |
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Blow-up rescaled equation as a gradient system: towards the generic blow-up behavior for parabolic PDEs |
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161 | (1) |
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Existence Problem: Variational Approach and Countable Families of Solutions by Lusternik-Schnirelman Category and Fibering Theory |
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162 | (11) |
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Variational setting and compactly supported solutions |
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162 | (1) |
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The Lusternik-Schnirelman theory and direct application of fibering method |
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163 | (2) |
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On a model with an explicit description of the Lusternik-Schnirelman sequence |
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165 | (1) |
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Preliminary analysis of geometric shapes of patterns |
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166 | (7) |
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Oscillation Problem: Local Oscillatory Structure of Solutions Close to Interfaces and Periodic Connections with Singularities |
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173 | (8) |
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Autonomous ODEs for oscillatory components |
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174 | (1) |
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Periodic oscillatory components |
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175 | (1) |
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Numerical construction of periodic orbits; m = 2 |
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176 | (1) |
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Numerical construction of periodic orbits; m = 3 |
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177 | (4) |
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Numeric Problem: Numerical Construction and First Classification of Basic Types of Localized Blow-up or Compacton Patterns |
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181 | (18) |
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Fourth order equation: m = 2 |
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182 | (3) |
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Countable family of {F0, F0}-interactions |
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185 | (3) |
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Countable family of {---F0, F0}-interactions |
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188 | (1) |
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189 | (2) |
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191 | (1) |
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More complicated patterns: towards chaotic structures |
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191 | (4) |
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195 | (4) |
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Lq, p-Cohomology of Riemannian Manifolds with Negative Curvature |
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199 | (10) |
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199 | (4) |
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Lq, p-cohomology and Sobolev inequalities |
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199 | (2) |
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Statement of the main result |
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201 | (2) |
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Manifolds with Contraction onto the Closed Unit Ball |
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203 | (2) |
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205 | (4) |
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207 | (2) |
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Volume Growth and Escape Rate of Brownian Motion on a Cartan-Hadamard Manifold |
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209 | (18) |
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209 | (4) |
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Heat Equation Solution Estimates |
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213 | (3) |
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Escape Rate of Brownian Motion |
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216 | (4) |
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Escape Rate on Model Manifolds |
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220 | (7) |
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220 | (1) |
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220 | (4) |
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224 | (3) |
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Sobolev Estimates for the Green Potential Associated with the Robin-Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition |
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227 | (34) |
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227 | (5) |
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232 | (2) |
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Smoothness Spaces on Lipschitz Boundaries and Lipschitz Domains |
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234 | (9) |
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243 | (3) |
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246 | (3) |
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249 | (7) |
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256 | (2) |
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258 | (3) |
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259 | (2) |
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Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains |
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261 | (50) |
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Statement of Problems and Preliminary Description of Results |
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261 | (9) |
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261 | (1) |
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Spectral boundary value problem |
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262 | (3) |
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Polynomial property and the Korn inequality |
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265 | (2) |
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Formulation of the problem in the operator form |
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267 | (1) |
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268 | (2) |
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The Model Problem and the Operator Pencil |
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270 | (15) |
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Model problem in the quasicylinder |
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270 | (2) |
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The Fredholm property of the problem operator |
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272 | (4) |
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Exponential decay and finite dimension of the kernel |
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276 | (3) |
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279 | (5) |
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On the positive threshold |
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284 | (1) |
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Specific Properties of Spectra in Particular Situations |
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285 | (26) |
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The absence of the point spectrum |
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285 | (2) |
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Concentration of the discrete spectrum |
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287 | (3) |
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290 | (1) |
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Artificial boundary conditions |
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291 | (4) |
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Opening gaps in the continuous spectrum |
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295 | (4) |
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Variational methods for searching trapped modes below the cut-off |
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299 | (4) |
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Remarks on cracks and edges |
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303 | (1) |
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304 | (2) |
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306 | (5) |
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Estimates for Completeley Integrable Systems of Differential Operators and Applications |
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311 | (18) |
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311 | (1) |
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Notation and Preliminaries |
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312 | (2) |
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Remarks on Completely Integrable Linear Systems of Differential Equations |
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314 | (4) |
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Estimates for Operators Satisfying the Complete Integrability Condition |
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318 | (4) |
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Stability of Solutions of Completely Linear Integrable Systems |
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322 | (3) |
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Applications to Differential Geometry |
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325 | (4) |
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327 | (2) |
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Counting Schrodinger Boundstates: Semiclassics and Beyond |
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329 | (26) |
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329 | (3) |
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332 | (1) |
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The Rozenblum-Lieb-Cwikel estimate |
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332 | (1) |
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The General Rozenblum-Lieb-Cwikel Inequality |
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333 | (3) |
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The approach by Li and Yau |
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333 | (1) |
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334 | (2) |
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Operators on Rd, d≥3: Non-Semiclassical Behavior of N_(0;HαV) |
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336 | (2) |
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Operators on the Semi-Axis |
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338 | (3) |
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338 | (2) |
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Non-semiclassical behavior of N_(0;HαV) |
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340 | (1) |
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341 | (2) |
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341 | (2) |
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Non-semiclassical behavior |
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343 | (1) |
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Schrodinger Operator on Manifolds |
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343 | (3) |
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343 | (2) |
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345 | (1) |
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Operators on Manifolds: Beyond Theorem 3.4 |
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346 | (3) |
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Schrodinger Operator on a Lattice |
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349 | (3) |
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352 | (3) |
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352 | (3) |
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Function Spaces on Cellular Domains |
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355 | (32) |
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Introduction and Preliminaries |
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355 | (13) |
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355 | (1) |
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356 | (3) |
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Wavelet systems and sequence spaces |
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359 | (2) |
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361 | (1) |
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362 | (3) |
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365 | (2) |
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367 | (1) |
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368 | (8) |
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368 | (4) |
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Approximation, density, decomposition |
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372 | (4) |
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Spaces on Cubes and Polyhedrons |
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376 | (4) |
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380 | (7) |
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380 | (2) |
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382 | (1) |
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382 | (2) |
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384 | (3) |
| Index |
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387 | |
| Preface |
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1 | (4) |
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III. Applications in Mathematical Physics |
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Geometrization of Rings as a Method for Solving Inverse Problems |
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5 | (20) |
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5 | (2) |
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7 | (2) |
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9 | (5) |
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Hyperbolic Inverse Problem |
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14 | (11) |
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23 | (2) |
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The Ginzburg-Landau Equations for Superconductivity with Random Fluctions |
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25 | (110) |
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25 | (6) |
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The Ginzburg-Landau Equation and Its Finite Difference Approximation |
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31 | (7) |
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Boundary value problem for the Ginzburg-Landau equation |
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31 | (1) |
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Approximation by the method of lines |
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32 | (6) |
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The stochastic Ginzburg-Landau Equation |
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38 | (5) |
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38 | (3) |
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The stochastic problem for the Ginzburg-Landau equation |
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41 | (2) |
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Discrete Approximation of the Stochastic Problem |
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43 | (9) |
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Definition of a projector Ph in L2(G) |
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43 | (2) |
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Approximation of Wiener processes |
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45 | (3) |
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48 | (1) |
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The discrete stochastic system |
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49 | (1) |
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50 | (2) |
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52 | (17) |
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Application of the Ito formula |
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53 | (2) |
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A priori estimate for p = 1 |
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55 | (1) |
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A priori estimate for p = 2 |
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56 | (3) |
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59 | (6) |
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A priori estimates for Δhψk |
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65 | (4) |
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Existence Theorem for Approximations |
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69 | (6) |
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69 | (1) |
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70 | (2) |
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Solvability of the discrete stochastic system |
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72 | (3) |
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Smoothness of the Strong Solution with respect to t |
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75 | (6) |
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Estimate of the mean maximum |
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75 | (1) |
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Estimate of the auxiliary random process |
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76 | (4) |
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Estimate of the mean modulus of continuity |
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80 | (1) |
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81 | (4) |
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81 | (2) |
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Compact sets in the space of time-dependent functions |
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83 | (2) |
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Weak Solution of the Discrete Stochastic Problem |
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85 | (3) |
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Definition of the weak solution for the discrete problem |
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85 | (2) |
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The equation for the weak solution of the discrete problem |
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87 | (1) |
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Passage to the Limit in a Family of vhn |
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88 | (2) |
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Compactness of the family of measures Vhn |
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88 | (2) |
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90 | (1) |
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Estimates for the Weak Solution |
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90 | (7) |
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91 | (1) |
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92 | (3) |
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Estimates for the measure V |
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95 | (2) |
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The Equation for the Weak Solution of the Stochastic Ginzburg-Landau Problem |
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97 | (13) |
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Definition of the weak solution |
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97 | (1) |
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The first steps of the proof for V to satisfy (12.6) |
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98 | (4) |
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Investigation of f2, h(ψ) |
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102 | (1) |
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Subspaces of piecewise linear functions |
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103 | (1) |
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The measures νhn and their weak compactness |
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104 | (2) |
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The final steps for passage to the limit |
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106 | (3) |
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Proof of the equality (12.7) |
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109 | (1) |
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Certain Properties of the Weak Statistical Solution ν |
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110 | (3) |
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110 | (2) |
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Solvability for almost all data |
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112 | (1) |
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Uniqueness of the Weak Statistical Solution |
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113 | (11) |
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Reduction of uniqueness for statistical solution ν to uniqueness of the solution for (12.1) |
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113 | (1) |
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Proof of the uniqueness of the solution of (12.1) and (2.2): the first step |
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114 | (2) |
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Estimation of the terms T2 to T5, T7, and T9 |
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116 | (3) |
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119 | (3) |
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122 | (2) |
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The Strong Statistical Solution of the Stochastic Ginzburg-Landau Equation |
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124 | (11) |
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Existence and uniqueness of a strong statistical solution |
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124 | (1) |
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On one family of scalar Wiener processes |
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125 | (2) |
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Equation for a strong statistical solution |
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127 | (4) |
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131 | (4) |
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Carleman Estimates with Second Large Parameter for Second Order Operators |
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135 | (26) |
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135 | (4) |
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Pseudoconvexity Condition |
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139 | (3) |
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Proof of Carleman Estimates for Scalar Operators |
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142 | (9) |
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Proof of Carleman Estimates for Elasticity System |
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151 | (2) |
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Holder Type Stability in the Cauchy Problem |
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153 | (2) |
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Lipschitz Stability in the Cauchy Problem |
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155 | (2) |
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157 | (4) |
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158 | (3) |
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Sharp Spectral Asymptotics for Dirac Energy |
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161 | (26) |
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161 | (3) |
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164 | (4) |
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164 | (2) |
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166 | (1) |
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Singular homogeneous case |
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166 | (2) |
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168 | (19) |
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Constant coefficients case |
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169 | (1) |
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General microhyperbolic case |
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170 | (1) |
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171 | (4) |
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175 | (2) |
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177 | (6) |
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General microhyperbolic case II |
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183 | (1) |
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184 | (3) |
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Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates |
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187 | (90) |
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Open-Loop and Closed-Loop Abstract Setting for Hyperbolic/Petrowski Type PDEs with Boundary Control |
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187 | (16) |
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A key open-loop boundary control-boundary observation map: orientation |
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188 | (2) |
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An historical overview on regularity, exact controllability, and uniform stabilization of hyperbolic and Petrowski type PDEs under boundary control |
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190 | (6) |
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Abstract setting encompassing the second order and first order (in time) hyperbolic and Petrowski type PDEs of the present paper |
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196 | (7) |
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Open Loop Problem (1.2.1): From B*L Bounded to L Bounded, Equivalently B*eA* Bounded |
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203 | (5) |
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Closed-Loop Nonlinear Feedback System: Uniform Stabilization with Optimal Decay Rates |
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208 | (5) |
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A Second Order in Time Hyperbolic Illustration: The Wave Equation with Dirichlet Boundary Control and Suitably Lifted Velocity Boundary Observation |
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213 | (19) |
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From the Dirichlet boundary control g for the wave solution {ν, νt} to the boundary observation ∂z/∂ν|Γ, via the Poisson equation lifting z = A-1νt |
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213 | (8) |
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221 | (4) |
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The half-space problem: A direct computation |
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225 | (4) |
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Implication on the uniform feedback stabilization of the boundary nonlinear dissipative feedback system ω in (4.1.1a-c) |
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229 | (2) |
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Implication on exact controllability of the (linear) dissipative system under boundary control |
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231 | (1) |
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Corollary of Section 4: The Multidimensional Kirchhoff Equation with `Moments' Boundary Control and Normal Derivatives of the Velocity as Boundary Observation |
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232 | (7) |
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Preliminaries. The operator B*L |
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232 | (4) |
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Implication on the uniform feedback stabilization of the boundary nonlinear dissipative feedback system ω in (5.1.1a-c) |
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236 | (2) |
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Implication on exact controllability of the (linear) dissipative system under boundary control |
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238 | (1) |
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A First Order in Time PDE Illustration: The Schrodinger Equation under Dirichlet Boundary Control and Suitably Lifted Solution as Boundary Observation |
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239 | (8) |
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From the Dirichlet boundary control u for the Schrodinger equation solution y to the boundary observation ∂z/∂ν|Γ, via the Poisson equation lifting z = A-1y |
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239 | (5) |
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Implication on the uniform feedback stabilization of the boundary nonlinear dissipative feedback system ω in (6.1.1a-c) |
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244 | (1) |
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Implication on exact controllability of the (linear) dissipative system under boundary control |
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244 | (1) |
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Asymptotic behavior of the transfer function: (B*L)(λ) = O(λ-(1/2-ε)), as positive λ → +∞. A direct, independent proof |
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245 | (2) |
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Euler-Bernoulli Plate with Clamped Boundary Controls. Neumann Boundary Control and Velocity Boundary Observation |
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247 | (9) |
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From the Neumann boundary control of the Euler-Bernoulli plate to the boundary observation -Az|Γ, via the Poisson lifting z = A-1νt |
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247 | (7) |
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Implication on the uniform feedback stabilization of the boundary nonlinear dissipative feedback system w in (7.1.1a-d) |
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254 | (1) |
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Implication on exact controllability of the (linear) dissipative system under boundary control |
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255 | (1) |
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Euler-Bernoulli Plate with Hinged Boundary Controls. Boundary Control in the `Moment' Boundary Condition and Suitably Lifted Velocity Boundary Observation |
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256 | (9) |
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From the `moment' boundary control of the Euler-Bernoulli plate to the boundary observation ∂zt/∂ν|Γ, via an elliptic lifting zt = A-1νt |
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256 | (6) |
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Implication on the uniform feedback stabilization of the boundary nonlinear dissipative feedback system w in (8.1.1a-d) |
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262 | (1) |
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Implication on exact controllability of the (linear) dissipative system under boundary control |
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262 | (1) |
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Asymptotic behavior of the transfer function (B*L)(λ) = O(λ-(1/2+ε)), as positive λ → +∞. A direct, independent proof |
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263 | (2) |
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The Multidimensional Schrodinger Equation with Neumann Boundary Control on the State Space H1(Ω) and on the State Space L2(Ω) |
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265 | (12) |
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Exact controllability/uniform stabilization in H1(Ω), dim Ω≥1 |
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265 | (1) |
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Exact controllability/uniform stabilization in L2(Ω), dim Ω≥1 |
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266 | (1) |
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Counterexample for the multidimensional Schrodinger equation with Neumann boundary control: L ¢ L(L2(0, T;L2(Γ); L2(0, T;Hε(Ω)), ε >
0. A fortiori: B*L ¢ L(L2(0, T;U)), with B* related to the state space Hε(Ω) and control space U = L2(Γ) |
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267 | (3) |
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The operator B*L, with U = L2(Γ) and state space L2(Ω) of the open-loop y-problem (9.1.1a-d) |
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270 | (1) |
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271 | (6) |
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Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions |
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277 | (40) |
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277 | (3) |
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Green's Kernel for a Mixed Boundary Value Problem in a Planar Domain with a Small Hole or a Crack |
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280 | (13) |
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Special solutions of model problems |
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280 | (2) |
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282 | (1) |
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Pointwise estimate of a solution to the exterior Neumann problem |
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283 | (3) |
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Asymptotic properties of the regular part of the Neumann function in R2\F |
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286 | (2) |
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Maximum modulus estimate for solutions to the mixed problem in Ωε with the Neumann data on ∂Fε |
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288 | (2) |
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Approximation of Green's function G(N)ε |
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290 | (2) |
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Simpler asymptotic formulas for Green's function G(N)ε |
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292 | (1) |
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Mixed Boundary Value Problem with the Dirichlet Condition on ∂Fε |
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293 | (10) |
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Special solutions of model problems |
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294 | (2) |
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Asymptotic property of the regular part of Green's function in R2\F |
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296 | (1) |
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Maximum modulus estimate for solutions to the mixed problem in Ωε with the Dirichlet data on ∂Fε |
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297 | (2) |
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Approximation of Green's function G(D)ε |
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299 | (2) |
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Simpler asymptotic representation of Green's function G(D)ε |
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301 | (2) |
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The Neumann Function for a Planar Domain with a Small Hole or Crack |
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303 | (6) |
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Special solutions of model problems |
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303 | (1) |
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Maximum modulus estimate for solutions to the Neumann problem in Ωε |
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304 | (2) |
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Asymptotic approximation of Nε |
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306 | (2) |
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Simpler asymptotic representation of the Neumann function Nε |
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308 | (1) |
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Asymptotic approximations of Green's kernels for mixed and Neumann problems in three dimensions |
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309 | (8) |
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Special solutions of model problems in limit domains |
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309 | (3) |
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Approximations of Green's kernels |
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312 | (3) |
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315 | (2) |
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Finsler Structures and Wave Propagation |
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317 | (18) |
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317 | (2) |
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Finsler Metrics and Finsler Symbols |
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319 | (2) |
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Finsler Symbols, Pseudodifferential Operators, and Hyperbolic PDEs |
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321 | (4) |
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Katok's Construction and Its Harmonic Analysis Counterpart |
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325 | (8) |
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Appendix. Randers-Randers Duality |
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333 | (2) |
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334 | (1) |
| Index |
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335 | |
