Elliptic & parabolic equations:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2006
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 403-404) and index |
| Physical Description: | XV, 408 S. 24 cm |
| ISBN: | 9812700250 9812700269 |
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MARC
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| 100 | 1 | |a Wu, Zhuoqun |e Verfasser |0 (DE-588)1029136041 |4 aut | |
| 245 | 1 | 0 | |a Elliptic & parabolic equations |c Zhuoqun Wu, Jingxue Yin & Chunpeng Wang |
| 246 | 1 | 3 | |a Elliptic and parabolic equations |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2006 | |
| 300 | |a XV, 408 S. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (p. 403-404) and index | ||
| 650 | 4 | |a Équations différentielles elliptiques | |
| 650 | 4 | |a Équations différentielles paraboliques | |
| 650 | 4 | |a Differential equations, Elliptic | |
| 650 | 4 | |a Differential equations, Parabolic | |
| 650 | 0 | 7 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Yin, Jingxue |e Verfasser |4 aut | |
| 700 | 1 | |a Wang, Chunpeng |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804137959334608896 |
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| adam_text | ELLIPTIC & PARABOLIC EQUATIONS ZHUOQUN WU, JINGXUE YIN & CHUNPENG WANG
JILIN UNIVERSITY, CHINA WORLD SCIENTIFIC NEW JERSEY * LONDON * SINGAPORE
* BEIJING * SHANGHAI * HONG KONG * TAIPEI * CHENNAI CONTENTS PREFACE V
1. PRELIMINARY KNOWLEDGE 1 1.1 SOME FREQUENTLY APPLIED INEQUALITIES AND
BASIC TECHNIQUES 1 1.1.1 SOME FREQUENTLY APPLIED INEQUALITIES 1 1.1.2
SPACES C FC (Q) AND C#(F2) 2 1.1.3 SMOOTHING OPERATORS 3 1.1.4 CUT-OFF
FUNCTIONS 5 1.1.5 PARTITION OF UNITY 6 1.1.6 LOCAL FLATTING OF THE
BOUNDARY 6 1.2 HOLDER SPACES 7 1.2.1 SPACES C K A {TI) AND C K A
{SL) 7 1.2.2 INTERPOLATION INEQUALITIES 8 1.2.3 SPACES C 2K+A K+A / 2
(Q T ) 13 * - 1.3 ISOTROPIC SOBOLEV SPACES 14 1.3.1 WEAK DERIVATIVES 14
1.3.2 SOBOLEV SPACES W K P (Q) AND W P (TT) 15 1.3.3 OPERATION RULES
OF WEAK DERIVATIVES 17 1.3.4 INTERPOLATION INEQUALITY 17 1.3.5 EMBEDDING
THEOREM 19 1.3.6 POINCARE S INEQUALITY 21 1.4 -ANISOTROPIC SOBOLEV
SPACES 24 1.4.1 SPACES W 2K K (Q T ), W^ HQT), W 2PK K (Q T ), V 2
{QT) AND V(Q T ) 24 1.4.2 EMBEDDING THEOREM 26 1.4.3 POINCARE S
INEQUALITY 28 X ELLIPTIC AND PARABOLIC EQUATIONS 1.5 TRACE OF FUNCTIONS
IN H^Q) 29 1.5.1 SOME PROPOSITIONS ON FUNCTIONS IN H 1 (Q + ) 29 1.5.2
TRACE OF FUNCTIONS IN H 1 ^) 33 1.5.3 TRACE OF FUNCTIONS IN H Q T ) =
WL X {Q T ) 35 2. L 2 THEORY OF LINEAR ELLIPTIC EQUATIONS 39 2.1 WEAK
SOLUTIONS OF POISSON S EQUATION 39 2.1.1 DEFINITION OF WEAK SOLUTIONS 40
2.1.2 RIESZ S REPRESENTATION THEOREM AND ITS APPLICATION . . 41 2.1.3
TRANSFORMATION OF THE PROBLEM 43 2.1.4 EXISTENCE OF MINIMIZERS OF THE
CORRESPONDING FUNCTIONAL 44 2.2 REGULARITY OF WEAK SOLUTIONS OF
POISSON S EQUATION .... 47 2.2.1 DIFFERENCE OPERATORS 47 2.2.2 INTERIOR
REGULARITY 50 2.2.3 REGULARITY NEAR THE BOUNDARY 53 2.2.4 GLOBAL
REGULARITY 56 2.2.5 STUDY OF REGULARITY BY MEANS OF SMOOTHING OPERATORS
58 2.3 L 2 THEORY OF GENERAL ELLIPTIC EQUATIONS 60 2.3.1 WEAK SOLUTIONS
60 2.3.2 RIESZ S REPRESENTATION THEOREM AND ITS APPLICATION . . 61 2.3.3
VARIATIONAL METHOD 62 2.3.4 LAX-MILGRAM S THEOREM AND ITS APPLICATION 64
2.3.5 FREDHOLM S ALTERNATIVE THEOREM AND ITS APPLICATION . 67 3. L 2
THEORY OF LINEAR PARABOLIC EQUATIONS 71 3.1 ENERGY METHOD 71 3.1.1
DEFINITION OF WEAK SOLUTIONS 72 3.1.2 A MODIFIED LAX-MILGRAM S THEOREM
73 3.1.3 EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION . ... 75 3.2
ROTHE S METHOD 79 3.3 GALERKIN S METHOD 85 3.4 REGULARITY OF WEAK
SOLUTIONS 89 3.5 L 2 THEORY OF GENERAL PARABOLIC EQUATIONS 94 3.5.1
ENERGY METHOD 94 3.5.2 ROTHE S METHOD 96 3.5.3 GALERKIN S METHOD 97 4.
DE GIORGI ITERATION AND MOSER ITERATION 105 CONTENTS X I 4.1 GLOBAL
BOUNDEDNESS ESTIMATES OF WEAK SOLUTIONS OF POIS- SON S EQUATION 105 .:*.
. 4.1.1 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF LAPLACE S EQUATION 105
4.1.2 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF POISSON S EQUATION 107 4.2
GLOBAL BOUNDEDNESS ESTIMATES FOR WEAK SOLUTIONS OF THE HEAT EQUATION ILL
4.2.1 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF THE HOMO- GENEOUS HEAT
EQUATION ILL 4.2.2 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF THE NONHO-
MOGENEOUS HEAT EQUATION 112 4.3 LOCAL BOUNDEDNESS ESTIMATES FOR WEAK
SOLUTIONS OF POIS- SON S EQUATION 116 4.3.1 WEAK SUBSOLUTIONS
(SUPERSOLUTIONS) 116 4.3.2 LOCAL BOUNDEDNESS ESTIMATE FOR WEAK SOLUTIONS
OF LAPLACE S EQUATION 118 4.3.3 LOCAL BOUNDEDNESS ESTIMATE FOR SOLUTIONS
OF POISSON S EQUATION 120 4.3.4 ESTIMATE NEAR THE BOUNDARY FOR WEAK
SOLUTIONS OF POISSON S EQUATION 122 4.4 LOCAL BOUNDEDNESS ESTIMATES FOR
WEAK SOLUTIONS OF THE HEAT EQUATION 123 4.4.1 WEAK SUBSOLUTIONS
(SUPERSOLUTIONS) 123 4.4.2 LOCAL BOUNDEDNESS ESTIMATE FOR WEAK SOLUTIONS
OF THE HOMOGENEOUS HEAT EQUATION 123 4.4.3 LOCAL BOUNDEDNESS ESTIMATE
FOR WEAK SOLUTIONS OF THE NONHOMOGENEOUS HEAT EQUATION 126 5. HARNACK S
INEQUALITIES 131 5.1 HARNACK S INEQUALITIES FOR SOLUTIONS OF LAPLACE S
EQUATION . 131 5.1.1 MEAN VALUE FORMULA 131 5.1.2 CLASSICAL HARNACK S
INEQUALITY 133 5.1.3 ESTIMATE OF SUP U 133 BBR 5.1.4 ESTIMATE OF INF U
135 BBR 5.1.5 HARNACK S INEQUALITY 141 5.1.6 HOLDER S ESTIMATE 143 XII
ELLIPTIC AND PARABOLIC EQUATIONS 5.2 HARNACK S INEQUALITIES FOR
SOLUTIONS OF THE HOMOGENEOUS HEAT EQUATION 145 5.2.1 WEAK HARNACK S
INEQUALITY 146 5.2.2 HOLDER S ESTIMATE 155 5.2.3 HARNACK S INEQUALITY
156 6. SCHAUDER S ESTIMATES FOR LINEAR ELLIPTIC EQUATIONS 159 6.1
CAMPANATO SPACES 159 6.2 SCHAUDER S ESTIMATES FOR POISSON S EQUATION 165
6.2.1 ESTIMATES TO BE ESTABLISHED 165 6.2.2 CACCIOPPOLI S INEQUALITIES
168 6.2.3 INTERIOR ESTIMATE FOR LAPLACE S EQUATION 173 6.2.4 NEAR
BOUNDARY ESTIMATE FOR LAPLACE S EQUATION . . . 175 6.2.5 ITERATION LEMMA
177 6.2.6 INTERIOR ESTIMATE FOR POISSON S EQUATION 178 6.2.7 NEAR
BOUNDARY ESTIMATE FOR POISSON S EQUATION . . . 181 6.3 SCHAUDER S
ESTIMATES FOR GENERAL LINEAR ELLIPTIC EQUATIONS 187 6.3.1 SIMPLIFICATION
OF THE PROBLEM 188 6.3.2 INTERIOR ESTIMATE 188 6.3.3 NEAR BOUNDARY
ESTIMATE 191 6.3.4 GLOBAL ESTIMATE 193 7. SCHAUDER S ESTIMATES FOR
LINEAR PARABOLIC EQUATIONS 197 7.1 I-ANISOTROPIC CAMPANATO SPACES 197
7.2 SCHAUDER S ESTIMATES FOR THE HEAT EQUATION 199 7.2.1 ESTIMATES TO BE
ESTABLISHED 199 7.2.2 INTERIOR ESTIMATE 200 7.2.3 NEAR BOTTOM ESTIMATE
208 7.2.4 NEAR LATERAL ESTIMATE 214 7.2.5 NEAR LATERAL-BOTTOM ESTIMATE
227 7.2.6 SCHAUDER S ESTIMATES FOR GENERAL LINEAR PARABOLIC EQUATIONS
231 8. EXISTENCE OF CLASSICAL SOLUTIONS FOR LINEAR EQUATIONS 233 8.1
MAXIMUM PRINCIPLE AND COMPARISON PRINCIPLE 233 8.1.1 THE CASE OF
ELLIPTIC EQUATIONS 233 8.1.2 THE CASE OF PARABOLIC EQUATIONS 236
CONTENTS XIII 8.2 EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR
LINEAR ELLIPTIC EQUATIONS 240 8.2.1 EXISTENCE AND UNIQUENESS OF THE
CLASSICAL SOLUTION FOR POISSON S EQUATION 240 8.2.2 THE METHOD OF
CONTINUITY 246 8.2.3 EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR
GENERAL LINEAR ELLIPTIC EQUATIONS 248 8.3 EXISTENCE AND UNIQUENESS OF
CLASSICAL SOLUTIONS FOR LINEAR PARABOLIC EQUATIONS 249 8.3.1 EXISTENCE
AND UNIQUENESS OF THE CLASSICAL SOLUTION FOR THE HEAT EQUATION 250 8.3.2
EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR GENERAL LINEAR
PARABOLIC EQUATIONS 251 9. L P ESTIMATES FOR LINEAR EQUATIONS AND
EXISTENCE OF STRONG SOLUTIONS 255 9.1 IP ESTIMATES FOR LINEAR ELLIPTIC
EQUATIONS AND EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS 255 9.1.1 L P
ESTIMATES FOR POISSON S EQUATION IN CUBES 255 9.1.2 L P ESTIMATES FOR
GENERAL LINEAR ELLIPTIC EQUATIONS . . . 260 9.1.3 EXISTENCE AND
UNIQUENESS OF STRONG SOLUTIONS FOR LINEAR ELLIPTIC EQUATIONS 264 9.2 V
ESTIMATES FOR LINEAR PARABOLIC EQUATIONS AND EXISTENCE AND UNIQUENESS OF
STRONG SOLUTIONS 266 9.2.1 L P ESTIMATES FOR THE HEAT EQUATION IN CUBES
266 9.2.2 L P ESTIMATES FOR GENERAL LINEAR PARABOLIC EQUATIONS . 271
9.2.3 EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR LINEAR PARABOLIC
EQUATIONS 272 10. FIXED POINT METHOD 277 10.1 FRAMEWORK OF SOLVING
QUASILINEAR EQUATIONS VIA FIXED POINT METHOD 277 10.1.1 LERAY-SCHAUDER S
FIXED POINT THEOREM 277 10.1.2 SOLVABILITY OF QUASILINEAR ELLIPTIC
EQUATIONS 277 10.1.3 SOLVABILITY OF QUASILINEAR PARABOLIC EQUATIONS 280
10.1.4 THE PROCEDURES OF THE A PRIORI ESTIMATES 282 10.2 MAXIMUM
ESTIMATE 282 10.3 INTERIOR HOLDER S ESTIMATE 284 XIV ELLIPTIC AND
PARABOLIC EQUATIONS 10.4 BOUNDARY HOLDER S ESTIMATE AND BOUNDARY
GRADIENT ESTI- MATE FOR SOLUTIONS OF POISSON S EQUATION 287 10.5
BOUNDARY HOLDER S ESTIMATE AND BOUNDARY GRADIENT ESTIMATE 289 10.6
GLOBAL GRADIENT ESTIMATE 296 10.7 HOLDER S ESTIMATE FOR A LINEAR
EQUATION 301 10.7.1 AN ITERATION LEMMA 301 10.7.2 MORREY S THEOREM 302
10.7.3 HOLDER S ESTIMATE 303 10.8 HOLDER S ESTIMATE FOR GRADIENTS 307
10.8.1 INTERIOR HOLDER S ESTIMATE FOR GRADIENTS OF SOLUTIONS . 307
10.8.2 BOUNDARY HOLDER S ESTIMATE FOR GRADIENTS OF SOLUTIONS 308 10.8.3
GLOBAL HOLDER S ESTIMATE FOR GRADIENTS OF SOLUTIONS . 310 10.9
SOLVABILITY OF MORE GENERAL QUASILINEAR EQUATIONS 310 10.9.1 SOLVABILITY
OF MORE GENERAL QUASILINEAR ELLIPTIC EQUATIONS 310 10.9.2 SOLVABILITY OF
MORE GENERAL QUASILINEAR PARABOLIC EQUATIONS 311 11. TOPOLOGICAL DEGREE
METHOD 313 11.1 TOPOLOGICAL DEGREE 313 11.1.1 BROUWER DEGREE 313 11.1.2
LERAY-SCHAUDER DEGREE 315 11.2 EXISTENCE OF A HEAT EQUATION WITH STRONG
NONLINEAR SOURCE 317 12. MONOTONE METHOD 323 12.1 MONOTONE METHOD FOR
PARABOLIC PROBLEMS 323 12.1.1 DEFINITION OF SUPERSOLUTIONS AND
SUBSOLUTIONS 324 12.1.2 ITERATION AND MONOTONE PROPERTY 324 12.1.3
EXISTENCE RESULTS 327 12.1.4 APPLICATION TO MORE GENERAL PARABOLIC
EQUATIONS . . . 330 12.1.5 NONUNIQUENESS OF SOLUTIONS 332 12.2 MONOTONE
METHOD FOR COUPLED PARABOLIC SYSTEMS 336 12.2.1 QUASIMONOTONE REACTION
FUNCTIONS 337 12.2.2 DEFINITION OF SUPERSOLUTIONS AND SUBSOLUTIONS 337
12.2.3 MONOTONE SEQUENCES 339 12.2.4 EXISTENCE RESULTS 350 12.2.5
EXTENSION 353 CONTENTS XV 13. DEGENERATE EQUATIONS 355 13.1 LINEAR
EQUATIONS 355 13.1.1 FORMULATION OF THE FIRST BOUNDARY VALUE PROBLEM . .
356 13.1.2 SOLVABILITY OF THE PROBLEM IN A SPACE SIMILAR TO H 1 . 361
13.1.3 SOLVABILITY OF THE PROBLEM IN L P (CL) . . . 362 13.1.4 METHOD OF
ELLIPTIC REGULARIZATION 365 13.1.5 UNIQUENESS OF WEAK SOLUTIONS IN L P
(FL) AND REGULARITY 366 13.2 A CLASS OF SPECIAL QUASILINEAR DEGENERATE
PARABOLIC EQUA- TIONS - FILTRATION EQUATIONS 368 13.2.1 DEFINITION OF
WEAK SOLUTIONS 369 13.2.2 UNIQUENESS OF WEAK SOLUTIONS FOR ONE
DIMENSIONAL EQUATIONS 371 13.2.3 EXISTENCE OF WEAK SOLUTIONS FOR ONE
DIMENSIONAL EQUA- TIONS 373 13.2.4 UNIQUENESS OF WEAK SOLUTIONS FOR
HIGHER DIMENSIONAL EQUATIONS 378 13.2.5 EXISTENCE OF WEAK SOLUTIONS FOR
HIGHER DIMENSIONAL EQUATIONS 381 13.3 GENERAL QUASILINEAR DEGENERATE
PARABOLIC EQUATIONS . . . . 384 13.3.1 UNIQUENESS OF WEAK SOLUTIONS FOR
WEAKLY DEGENERATE EQUATIONS 385 13.3.2 EXISTENCE OF WEAK SOLUTIONS FOR
WEAKLY DEGENERATE EQUATIONS 393 13.3.3 A REMARK ON QUASILINEAR PARABOLIC
EQUATIONS WITH STRONG DEGENERACY 399 BIBLIOGRAPHY 403 INDEX 405
|
| adam_txt |
ELLIPTIC & PARABOLIC EQUATIONS ZHUOQUN WU, JINGXUE YIN & CHUNPENG WANG
JILIN UNIVERSITY, CHINA WORLD SCIENTIFIC NEW JERSEY * LONDON * SINGAPORE
* BEIJING * SHANGHAI * HONG KONG * TAIPEI * CHENNAI CONTENTS PREFACE V
1. PRELIMINARY KNOWLEDGE 1 1.1 SOME FREQUENTLY APPLIED INEQUALITIES AND
BASIC TECHNIQUES 1 1.1.1 SOME FREQUENTLY APPLIED INEQUALITIES 1 1.1.2
SPACES C FC (Q) AND C#(F2) 2 1.1.3 SMOOTHING OPERATORS 3 1.1.4 CUT-OFF
FUNCTIONS 5 1.1.5 PARTITION OF UNITY 6 1.1.6 LOCAL FLATTING OF THE
BOUNDARY 6 1.2 HOLDER SPACES 7 1.2.1 SPACES C K A {TI) AND C K ' A
{SL) 7 1.2.2 INTERPOLATION INEQUALITIES 8 1.2.3 SPACES C 2K+A ' K+A / 2
(Q T ) 13 * - 1.3 ISOTROPIC SOBOLEV SPACES 14 1.3.1 WEAK DERIVATIVES 14
1.3.2 SOBOLEV SPACES W K ' P (Q) AND W' P (TT) 15 1.3.3 OPERATION RULES
OF WEAK DERIVATIVES 17 1.3.4 INTERPOLATION INEQUALITY 17 1.3.5 EMBEDDING
THEOREM 19 1.3.6 POINCARE'S INEQUALITY 21 1.4 -ANISOTROPIC SOBOLEV
SPACES 24 1.4.1 SPACES W 2K ' K (Q T ), W^'HQT), W 2PK ' K (Q T ), V 2
{QT) AND V(Q T ) 24 1.4.2 EMBEDDING THEOREM 26 1.4.3 POINCARE'S
INEQUALITY 28 X ELLIPTIC AND PARABOLIC EQUATIONS 1.5 TRACE OF FUNCTIONS
IN H^Q) 29 1.5.1 SOME PROPOSITIONS ON FUNCTIONS IN H 1 (Q + ) 29 1.5.2
TRACE OF FUNCTIONS IN H 1 ^) 33 1.5.3 TRACE OF FUNCTIONS IN H\Q T ) =
WL' X {Q T ) 35 2. L 2 THEORY OF LINEAR ELLIPTIC EQUATIONS 39 2.1 WEAK
SOLUTIONS OF POISSON'S EQUATION 39 2.1.1 DEFINITION OF WEAK SOLUTIONS 40
2.1.2 RIESZ'S REPRESENTATION THEOREM AND ITS APPLICATION . . 41 2.1.3
TRANSFORMATION OF THE PROBLEM 43 2.1.4 EXISTENCE OF MINIMIZERS OF THE
CORRESPONDING FUNCTIONAL 44 2.2 REGULARITY OF WEAK SOLUTIONS OF
POISSON'S EQUATION . 47 2.2.1 DIFFERENCE OPERATORS 47 2.2.2 INTERIOR
REGULARITY 50 2.2.3 REGULARITY NEAR THE BOUNDARY 53 2.2.4 GLOBAL
REGULARITY 56 2.2.5 STUDY OF REGULARITY BY MEANS OF SMOOTHING OPERATORS
58 2.3 L 2 THEORY OF GENERAL ELLIPTIC EQUATIONS 60 2.3.1 WEAK SOLUTIONS
60 2.3.2 RIESZ'S REPRESENTATION THEOREM AND ITS APPLICATION . . 61 2.3.3
VARIATIONAL METHOD 62 2.3.4 LAX-MILGRAM'S THEOREM AND ITS APPLICATION 64
2.3.5 FREDHOLM'S ALTERNATIVE THEOREM AND ITS APPLICATION . 67 3. L 2
THEORY OF LINEAR PARABOLIC EQUATIONS 71 3.1 ENERGY METHOD 71 3.1.1
DEFINITION OF WEAK SOLUTIONS 72 3.1.2 A MODIFIED LAX-MILGRAM'S THEOREM
73 3.1.3 EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION . . 75 3.2
ROTHE'S METHOD 79 3.3 GALERKIN'S METHOD 85 3.4 REGULARITY OF WEAK
SOLUTIONS 89 3.5 L 2 THEORY OF GENERAL PARABOLIC EQUATIONS 94 3.5.1
ENERGY METHOD 94 3.5.2 ROTHE'S METHOD 96 3.5.3 GALERKIN'S METHOD 97 4.
DE GIORGI ITERATION AND MOSER ITERATION 105 CONTENTS X I 4.1 GLOBAL
BOUNDEDNESS ESTIMATES OF WEAK SOLUTIONS OF POIS- SON'S EQUATION 105 .:*.
. 4.1.1 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF LAPLACE'S EQUATION 105
4.1.2 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF POISSON'S EQUATION 107 4.2
GLOBAL BOUNDEDNESS ESTIMATES FOR WEAK SOLUTIONS OF THE HEAT EQUATION ILL
4.2.1 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF THE HOMO- GENEOUS HEAT
EQUATION ILL 4.2.2 WEAK MAXIMUM PRINCIPLE FOR SOLUTIONS OF THE NONHO-
MOGENEOUS HEAT EQUATION 112 4.3 LOCAL BOUNDEDNESS ESTIMATES FOR WEAK
SOLUTIONS OF POIS- SON'S EQUATION 116 4.3.1 WEAK SUBSOLUTIONS
(SUPERSOLUTIONS) 116 4.3.2 LOCAL BOUNDEDNESS ESTIMATE FOR WEAK SOLUTIONS
OF LAPLACE'S EQUATION 118 4.3.3 LOCAL BOUNDEDNESS ESTIMATE FOR SOLUTIONS
OF POISSON'S EQUATION 120 4.3.4 ESTIMATE NEAR THE BOUNDARY FOR WEAK
SOLUTIONS OF POISSON'S EQUATION 122 4.4 LOCAL BOUNDEDNESS ESTIMATES FOR
WEAK SOLUTIONS OF THE HEAT EQUATION 123 4.4.1 WEAK SUBSOLUTIONS
(SUPERSOLUTIONS) 123 4.4.2 LOCAL BOUNDEDNESS ESTIMATE FOR WEAK SOLUTIONS
OF THE HOMOGENEOUS HEAT EQUATION 123 4.4.3 LOCAL BOUNDEDNESS ESTIMATE
FOR WEAK SOLUTIONS OF THE NONHOMOGENEOUS HEAT EQUATION 126 5. HARNACK'S
INEQUALITIES 131 5.1 HARNACK'S INEQUALITIES FOR SOLUTIONS OF LAPLACE'S
EQUATION . 131 5.1.1 MEAN VALUE FORMULA 131 5.1.2 CLASSICAL HARNACK'S
INEQUALITY 133 5.1.3 ESTIMATE OF SUP U 133 BBR 5.1.4 ESTIMATE OF INF U
135 BBR 5.1.5 HARNACK'S INEQUALITY 141 5.1.6 HOLDER'S ESTIMATE 143 XII
ELLIPTIC AND PARABOLIC EQUATIONS 5.2 HARNACK'S INEQUALITIES FOR
SOLUTIONS OF THE HOMOGENEOUS HEAT EQUATION 145 5.2.1 WEAK HARNACK'S
INEQUALITY 146 5.2.2 HOLDER'S ESTIMATE 155 5.2.3 HARNACK'S INEQUALITY
156 6. SCHAUDER'S ESTIMATES FOR LINEAR ELLIPTIC EQUATIONS 159 6.1
CAMPANATO SPACES 159 6.2 SCHAUDER'S ESTIMATES FOR POISSON'S EQUATION 165
6.2.1 ESTIMATES TO BE ESTABLISHED 165 6.2.2 CACCIOPPOLI'S INEQUALITIES
168 6.2.3 INTERIOR ESTIMATE FOR LAPLACE'S EQUATION 173 6.2.4 NEAR
BOUNDARY ESTIMATE FOR LAPLACE'S EQUATION . . . 175 6.2.5 ITERATION LEMMA
177 6.2.6 INTERIOR ESTIMATE FOR POISSON'S EQUATION 178 6.2.7 NEAR
BOUNDARY ESTIMATE FOR POISSON'S EQUATION . . . 181 6.3 SCHAUDER'S
ESTIMATES FOR GENERAL LINEAR ELLIPTIC EQUATIONS 187 6.3.1 SIMPLIFICATION
OF THE PROBLEM 188 6.3.2 INTERIOR ESTIMATE 188 6.3.3 NEAR BOUNDARY
ESTIMATE 191 6.3.4 GLOBAL ESTIMATE 193 7. SCHAUDER'S ESTIMATES FOR
LINEAR PARABOLIC EQUATIONS 197 7.1 I-ANISOTROPIC CAMPANATO SPACES 197
7.2 SCHAUDER'S ESTIMATES FOR THE HEAT EQUATION 199 7.2.1 ESTIMATES TO BE
ESTABLISHED 199 7.2.2 INTERIOR ESTIMATE 200 7.2.3 NEAR BOTTOM ESTIMATE
208 7.2.4 NEAR LATERAL ESTIMATE 214 7.2.5 NEAR LATERAL-BOTTOM ESTIMATE
227 7.2.6 SCHAUDER'S ESTIMATES FOR GENERAL LINEAR PARABOLIC EQUATIONS
231 8. EXISTENCE OF CLASSICAL SOLUTIONS FOR LINEAR EQUATIONS 233 8.1
MAXIMUM PRINCIPLE AND COMPARISON PRINCIPLE 233 8.1.1 THE CASE OF
ELLIPTIC EQUATIONS 233 8.1.2 THE CASE OF PARABOLIC EQUATIONS 236
CONTENTS XIII 8.2 EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR
LINEAR ELLIPTIC EQUATIONS 240 8.2.1 EXISTENCE AND UNIQUENESS OF THE
CLASSICAL SOLUTION FOR POISSON'S EQUATION 240 8.2.2 THE METHOD OF
CONTINUITY 246 8.2.3 EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR
GENERAL LINEAR ELLIPTIC EQUATIONS 248 8.3 EXISTENCE AND UNIQUENESS OF
CLASSICAL SOLUTIONS FOR LINEAR PARABOLIC EQUATIONS 249 8.3.1 EXISTENCE
AND UNIQUENESS OF THE CLASSICAL SOLUTION FOR THE HEAT EQUATION 250 8.3.2
EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR GENERAL LINEAR
PARABOLIC EQUATIONS 251 9. L P ESTIMATES FOR LINEAR EQUATIONS AND
EXISTENCE OF STRONG SOLUTIONS 255 9.1 IP ESTIMATES FOR LINEAR ELLIPTIC
EQUATIONS AND EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS 255 9.1.1 L P
ESTIMATES FOR POISSON'S EQUATION IN CUBES 255 9.1.2 L P ESTIMATES FOR
GENERAL LINEAR ELLIPTIC EQUATIONS . . . 260 9.1.3 EXISTENCE AND
UNIQUENESS OF STRONG SOLUTIONS FOR LINEAR ELLIPTIC EQUATIONS 264 9.2 V
ESTIMATES FOR LINEAR PARABOLIC EQUATIONS AND EXISTENCE AND UNIQUENESS OF
STRONG SOLUTIONS 266 9.2.1 L P ESTIMATES FOR THE HEAT EQUATION IN CUBES
266 9.2.2 L P ESTIMATES FOR GENERAL LINEAR PARABOLIC EQUATIONS . 271
9.2.3 EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR LINEAR PARABOLIC
EQUATIONS 272 10. FIXED POINT METHOD 277 10.1 FRAMEWORK OF SOLVING
QUASILINEAR EQUATIONS VIA FIXED POINT METHOD 277 10.1.1 LERAY-SCHAUDER'S
FIXED POINT THEOREM 277 10.1.2 SOLVABILITY OF QUASILINEAR ELLIPTIC
EQUATIONS 277 10.1.3 SOLVABILITY OF QUASILINEAR PARABOLIC EQUATIONS 280
10.1.4 THE PROCEDURES OF THE A PRIORI ESTIMATES 282 10.2 MAXIMUM
ESTIMATE 282 10.3 INTERIOR HOLDER'S ESTIMATE 284 XIV ELLIPTIC AND
PARABOLIC EQUATIONS 10.4 BOUNDARY HOLDER'S ESTIMATE AND BOUNDARY
GRADIENT ESTI- MATE FOR SOLUTIONS OF POISSON'S EQUATION 287 10.5
BOUNDARY HOLDER'S ESTIMATE AND BOUNDARY GRADIENT ESTIMATE 289 10.6
GLOBAL GRADIENT ESTIMATE 296 10.7 HOLDER'S ESTIMATE FOR A LINEAR
EQUATION 301 10.7.1 AN ITERATION LEMMA 301 10.7.2 MORREY'S THEOREM 302
10.7.3 HOLDER'S ESTIMATE 303 10.8 HOLDER'S ESTIMATE FOR GRADIENTS 307
10.8.1 INTERIOR HOLDER'S ESTIMATE FOR GRADIENTS OF SOLUTIONS . 307
10.8.2 BOUNDARY HOLDER'S ESTIMATE FOR GRADIENTS OF SOLUTIONS 308 10.8.3
GLOBAL HOLDER'S ESTIMATE FOR GRADIENTS OF SOLUTIONS . 310 10.9
SOLVABILITY OF MORE GENERAL QUASILINEAR EQUATIONS 310 10.9.1 SOLVABILITY
OF MORE GENERAL QUASILINEAR ELLIPTIC EQUATIONS 310 10.9.2 SOLVABILITY OF
MORE GENERAL QUASILINEAR PARABOLIC EQUATIONS 311 11. TOPOLOGICAL DEGREE
METHOD 313 11.1 TOPOLOGICAL DEGREE 313 11.1.1 BROUWER DEGREE 313 11.1.2
LERAY-SCHAUDER DEGREE 315 11.2 EXISTENCE OF A HEAT EQUATION WITH STRONG
NONLINEAR SOURCE 317 12. MONOTONE METHOD 323 12.1 MONOTONE METHOD FOR
PARABOLIC PROBLEMS 323 12.1.1 DEFINITION OF SUPERSOLUTIONS AND
SUBSOLUTIONS 324 12.1.2 ITERATION AND MONOTONE PROPERTY 324 12.1.3
EXISTENCE RESULTS 327 12.1.4 APPLICATION TO MORE GENERAL PARABOLIC
EQUATIONS . . . 330 12.1.5 NONUNIQUENESS OF SOLUTIONS 332 12.2 MONOTONE
METHOD FOR COUPLED PARABOLIC SYSTEMS 336 12.2.1 QUASIMONOTONE REACTION
FUNCTIONS 337 12.2.2 DEFINITION OF SUPERSOLUTIONS AND SUBSOLUTIONS 337
12.2.3 MONOTONE SEQUENCES 339 12.2.4 EXISTENCE RESULTS 350 12.2.5
EXTENSION 353 CONTENTS XV 13. DEGENERATE EQUATIONS 355 13.1 LINEAR
EQUATIONS 355 13.1.1 FORMULATION OF THE FIRST BOUNDARY VALUE PROBLEM . .
356 13.1.2 SOLVABILITY OF THE PROBLEM IN A SPACE SIMILAR TO H 1 . 361
13.1.3 SOLVABILITY OF THE PROBLEM IN L P (CL) . . . 362 13.1.4 METHOD OF
ELLIPTIC REGULARIZATION 365 13.1.5 UNIQUENESS OF WEAK SOLUTIONS IN L P
(FL) AND REGULARITY 366 13.2 A CLASS OF SPECIAL QUASILINEAR DEGENERATE
PARABOLIC EQUA- TIONS - FILTRATION EQUATIONS 368 13.2.1 DEFINITION OF
WEAK SOLUTIONS 369 13.2.2 UNIQUENESS OF WEAK SOLUTIONS FOR ONE
DIMENSIONAL EQUATIONS 371 13.2.3 EXISTENCE OF WEAK SOLUTIONS FOR ONE
DIMENSIONAL EQUA- TIONS 373 13.2.4 UNIQUENESS OF WEAK SOLUTIONS FOR
HIGHER DIMENSIONAL EQUATIONS 378 13.2.5 EXISTENCE OF WEAK SOLUTIONS FOR
HIGHER DIMENSIONAL EQUATIONS 381 13.3 GENERAL QUASILINEAR DEGENERATE
PARABOLIC EQUATIONS . . . . 384 13.3.1 UNIQUENESS OF WEAK SOLUTIONS FOR
WEAKLY DEGENERATE EQUATIONS 385 13.3.2 EXISTENCE OF WEAK SOLUTIONS FOR
WEAKLY DEGENERATE EQUATIONS 393 13.3.3 A REMARK ON QUASILINEAR PARABOLIC
EQUATIONS WITH STRONG DEGENERACY 399 BIBLIOGRAPHY 403 INDEX 405 |
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| author | Wu, Zhuoqun Yin, Jingxue Wang, Chunpeng |
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| building | Verbundindex |
| bvnumber | BV035029416 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 560 |
| ctrlnum | (OCoLC)85264136 (DE-599)BVBBV035029416 |
| dewey-full | 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035029416 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:49:17Z |
| indexdate | 2024-07-09T21:20:35Z |
| institution | BVB |
| isbn | 9812700250 9812700269 |
| language | English |
| lccn | 2007295412 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016698418 |
| oclc_num | 85264136 |
| open_access_boolean | |
| owner | DE-29T DE-11 DE-20 |
| owner_facet | DE-29T DE-11 DE-20 |
| physical | XV, 408 S. 24 cm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Wu, Zhuoqun Verfasser (DE-588)1029136041 aut Elliptic & parabolic equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang Elliptic and parabolic equations Singapore [u.a.] World Scientific 2006 XV, 408 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 403-404) and index Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s DE-604 Parabolische Differentialgleichung (DE-588)4173245-5 s Yin, Jingxue Verfasser aut Wang, Chunpeng Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698418&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wu, Zhuoqun Yin, Jingxue Wang, Chunpeng Elliptic & parabolic equations Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Parabolische Differentialgleichung (DE-588)4173245-5 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4173245-5 (DE-588)4014485-9 |
| title | Elliptic & parabolic equations |
| title_alt | Elliptic and parabolic equations |
| title_auth | Elliptic & parabolic equations |
| title_exact_search | Elliptic & parabolic equations |
| title_exact_search_txtP | Elliptic & parabolic equations |
| title_full | Elliptic & parabolic equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang |
| title_fullStr | Elliptic & parabolic equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang |
| title_full_unstemmed | Elliptic & parabolic equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang |
| title_short | Elliptic & parabolic equations |
| title_sort | elliptic parabolic equations |
| topic | Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Parabolische Differentialgleichung (DE-588)4173245-5 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Parabolische Differentialgleichung Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698418&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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