| Preface |
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1 | (38) |
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Some Frequently Applied Inequalities and Basic Techniques |
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1 | (6) |
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Some frequently applied inequalities |
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1 | (1) |
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Spaces Ck (Ω) and Ck0 (Ω) |
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2 | (1) |
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3 | (2) |
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5 | (1) |
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6 | (1) |
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Local flatting of the boundary |
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6 | (1) |
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7 | (7) |
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Spaces Ck, α (Ω) and Ck, α (Ω) |
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7 | (1) |
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Interpolation inequalities |
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8 | (5) |
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13 | (1) |
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14 | (10) |
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14 | (1) |
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Sobolev spaces Wk, p (Ω) and Wk, p0 (Ω) |
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15 | (2) |
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Operation rules of weak derivatives |
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17 | (1) |
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17 | (2) |
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19 | (2) |
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21 | (3) |
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t-Anisotropic Sobolev Spaces |
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24 | (5) |
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Spaces W2k,k (QT), W2k,k p (QT), W2k,k p (QT), V2 (QT) and V (QT) |
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24 | (2) |
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26 | (2) |
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28 | (1) |
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Trace of Functions in H1 (Ω) |
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29 | (10) |
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Some propositions on functions in H1 (Q+) |
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29 | (4) |
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Trace of functions in H1 (Ω) |
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33 | (2) |
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Trace of functions in H1 (QT) = W1,1 2 (QT) |
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35 | (4) |
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L2 Theory of Linear Elliptic Equations |
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39 | (32) |
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Weak Solutions of Poisson's Equation |
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39 | (8) |
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Definition of weak solutions |
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40 | (1) |
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Riesz's representation theorem and its application |
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41 | (2) |
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Transformation of the problem |
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43 | (1) |
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Existence of minimizers of the corresponding functional |
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44 | (3) |
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Regularity of Weak Solutions of Poisson's Equation |
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47 | (13) |
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47 | (3) |
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50 | (3) |
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Regularity near the boundary |
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53 | (3) |
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56 | (2) |
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Study of regularity by means of smoothing operators |
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58 | (2) |
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L2 Theory of General Elliptic Equations |
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60 | (11) |
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60 | (1) |
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Riesz's representation theorem and its application |
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61 | (1) |
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62 | (2) |
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Lax-Milgram's theorem and its application |
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64 | (3) |
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Fredholm's alternative theorem and its application |
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67 | (4) |
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L2 Theory of Linear Parabolic Equations |
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71 | (34) |
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71 | (8) |
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Definition of weak solutions |
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72 | (1) |
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A modified Lax-Milgram's theorem |
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73 | (2) |
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Existence and uniqueness of the weak solution |
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75 | (4) |
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79 | (6) |
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85 | (4) |
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Regularity of Weak Solutions |
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89 | (5) |
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L2 Theory of General Parabolic Equations |
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94 | (11) |
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94 | (2) |
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96 | (1) |
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97 | (8) |
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De Giorgi Iteration and Moser Iteration |
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105 | (26) |
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Global Boundedness Estimates of Weak Solutions of Poisson's Equation |
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105 | (6) |
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Weak maximum principle for solutions of Laplace's equation |
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105 | (2) |
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Weak maximum principle for solutions of Poisson's equation |
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107 | (4) |
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Global Boundedness Estimates for Weak Solutions of the Heat Equation |
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111 | (5) |
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Weak maximum principle for solutions of the homogeneous heat equation |
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111 | (1) |
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Weak maximum principle for solutions of the nonhomogeneous heat equation |
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112 | (4) |
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Local Boundedness Estimates for Weak Solutions of Poisson's Equation |
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116 | (7) |
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Weak subsolutions (supersolutions) |
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116 | (2) |
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Local boundedness estimate for weak solutions of Laplace's equation |
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118 | (2) |
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Local boundedness estimate for solutions of Poisson's equation |
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120 | (2) |
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Estimate near the boundary for weak solutions of Poisson's equation |
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122 | (1) |
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Local Boundedness Estimates for Weak Solutions of the Heat Equation |
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123 | (8) |
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Weak subsolutions (supersolutions) |
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123 | (1) |
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Local boundedness estimate for weak solutions of the homogeneous heat equation |
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123 | (3) |
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Local boundedness estimate for weak solutions of the nonhomogeneous heat equation |
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126 | (5) |
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131 | (28) |
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Harnack's Inequalities for Solutions of Laplace's Equation |
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131 | (14) |
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131 | (2) |
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Classical Harnack's inequality |
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133 | (1) |
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133 | (2) |
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Estimate of inf BθR BθR u |
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135 | (6) |
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141 | (2) |
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143 | (2) |
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Harnack's Inequalities for Solutions of the Homogeneous Heat Equation |
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145 | (14) |
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Weak Harnack's inequality |
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146 | (9) |
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155 | (1) |
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156 | (3) |
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Schauder's Estimates for Linear Elliptic Equations |
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159 | (38) |
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159 | (6) |
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Schauder's Estimates for Poisson's Equation |
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165 | (22) |
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Estimates to be established |
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165 | (3) |
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Caccioppoli's inequalities |
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168 | (5) |
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Interior estimate for Laplace's equation |
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173 | (2) |
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Near boundary estimate for Laplace's equation |
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175 | (2) |
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177 | (1) |
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Interior estimate for Poisson's equation |
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178 | (3) |
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Near boundary estimate for Poisson's equation |
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181 | (6) |
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Schauder's Estimates for General Linear Elliptic Equations |
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187 | (10) |
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Simplification of the problem |
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188 | (1) |
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188 | (3) |
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191 | (2) |
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193 | (4) |
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Schauder's Estimates for Linear Parabolic Equations |
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197 | (36) |
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t-Anisotropic Campanato Spaces |
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197 | (2) |
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Schauder's Estimates for the Heat Equation |
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199 | (34) |
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Estimates to be established |
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199 | (1) |
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200 | (8) |
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208 | (6) |
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214 | (13) |
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Near lateral-bottom estimate |
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227 | (4) |
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Schauder's estimates for general linear parabolic equations |
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231 | (2) |
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Existence of Classical Solutions for Linear Equations |
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233 | (22) |
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Maximum Principle and Comparison Principle |
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233 | (7) |
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The case of elliptic equations |
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233 | (3) |
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The case of parabolic equations |
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236 | (4) |
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Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations |
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240 | (9) |
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Existence and uniqueness of the classical solution for Poisson's equation |
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240 | (6) |
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246 | (2) |
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Existence and uniqueness of classical solutions for general linear elliptic equations |
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248 | (1) |
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Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations |
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249 | (6) |
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Existence and uniqueness of the classical solution for the heat equation |
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250 | (1) |
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Existence and uniqueness of classical solutions for general linear parabolic equations |
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251 | (4) |
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Lp Estimates for Linear Equations and Existence of Strong Solutions |
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255 | (22) |
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Lp Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions |
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255 | (11) |
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Lp estimates for Poisson's equation in cubes |
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255 | (5) |
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Lp estimates for general linear elliptic equations |
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260 | (4) |
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Existence and uniqueness of strong solutions for linear elliptic equations |
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264 | (2) |
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Lp Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions |
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266 | (11) |
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Lp estimates for the heat equation in cubes |
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266 | (5) |
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Lp estimates for general linear parabolic equations |
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271 | (1) |
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Existence and uniqueness of strong solutions for linear parabolic equations |
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272 | (5) |
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277 | (36) |
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Framework of Solving Quasilinear Equations via Fixed Point Method |
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277 | (5) |
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Leray-Schauder's fixed point theorem |
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277 | (1) |
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Solvability of quasilinear elliptic equations |
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277 | (3) |
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Solvability of quasilinear parabolic equations |
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280 | (2) |
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The procedures of the a priori estimates |
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282 | (1) |
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282 | (2) |
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Interior Holder's Estimate |
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284 | (3) |
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Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation |
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287 | (2) |
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Boundary Holder's Estimate and Boundary Gradient Estimate |
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289 | (7) |
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296 | (5) |
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Holder's Estimate for a Linear Equation |
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301 | (6) |
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301 | (1) |
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302 | (1) |
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303 | (4) |
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Holder's Estimate for Gradients |
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307 | (3) |
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Interior Holder's estimate for gradients of solutions |
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307 | (1) |
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Boundary Holder's estimate for gradients of solutions |
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308 | (2) |
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Global Holder's estimate for gradients of solutions |
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310 | (1) |
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Solvability of More General Quasilinear Equations |
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310 | (3) |
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Solvability of more general quasilinear elliptic equations |
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310 | (1) |
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Solvability of more general quasilinear parabolic equations |
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311 | (2) |
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Topological Degree Method |
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313 | (10) |
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313 | (4) |
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313 | (2) |
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315 | (2) |
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Existence of a Heat Equation with Strong Nonlinear Source |
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317 | (6) |
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323 | (32) |
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Monotone Method for Parabolic Problems |
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323 | (13) |
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Definition of supersolutions and subsolutions |
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324 | (1) |
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Iteration and monotone property |
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324 | (3) |
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327 | (3) |
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Application to more general parabolic equations |
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330 | (2) |
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Nonuniqueness of solutions |
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332 | (4) |
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Monotone Method for Coupled Parabolic Systems |
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336 | (19) |
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Quasimonotone reaction functions |
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337 | (1) |
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Definition of supersolutions and subsolutions |
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337 | (2) |
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339 | (11) |
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350 | (3) |
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353 | (2) |
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355 | (48) |
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355 | (13) |
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Formulation of the first boundary value problem |
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356 | (5) |
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Solvability of the problem in a space similar to H1 |
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361 | (1) |
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Solvability of the problem in Lp(Ω) |
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362 | (3) |
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Method of elliptic regularization |
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365 | (1) |
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Uniqueness of weak solutions in Lp (Ω) and regularity |
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366 | (2) |
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A Class of Special Quasilinear Degenerate Parabolic Equations -- Filtration Equations |
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368 | (16) |
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Definition of weak solutions |
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369 | (2) |
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Uniqueness of weak solutions for one dimensional equations |
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371 | (2) |
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Existence of weak solutions for one dimensional equations |
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373 | (5) |
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Uniqueness of weak solutions for higher dimensional equations |
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378 | (3) |
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Existence of weak solutions for higher dimensional equations |
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381 | (3) |
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General Quasilinear Degenerate Parabolic Equations |
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384 | (19) |
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Uniqueness of weak solutions for weakly degenerate equations |
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385 | (8) |
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Existence of weak solutions for weakly degenerate equations |
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393 | (6) |
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A remark on quasilinear parabolic equations with strong degeneracy |
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399 | (4) |
| Bibliography |
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403 | (2) |
| Index |
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405 | |
