Elements of the modern theory of partial differential equations:
Gespeichert in:
| Vorheriger Titel: | Encyclopaedia of mathematical sciences ; 31 |
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| Format: | Buch |
| Sprache: | English Russian |
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Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London
Springer
1999
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| Ausgabe: | 1. ed., 2. printing |
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Russ. übers. |
| Beschreibung: | 263 S. Ill. |
| ISBN: | 3540653775 |
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| 245 | 1 | 0 | |a Elements of the modern theory of partial differential equations |c Yu. V. Egorov ; A. I. Komech ; M. A. Shubin |
| 250 | |a 1. ed., 2. printing | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London |b Springer |c 1999 | |
| 300 | |a 263 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Aus dem Russ. übers. | ||
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| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Egorov, Jurij V. |d 1938- |e Sonstige |0 (DE-588)121177181 |4 oth | |
| 700 | 1 | |a Komeč, Aleksandr I. |d 1946- |e Sonstige |0 (DE-588)120962063 |4 oth | |
| 700 | 1 | |a Šubin, Michail A. |d 1944- |e Sonstige |0 (DE-588)121177211 |4 oth | |
| 780 | 0 | 0 | |i 1. Aufl. u.d.T. |t Encyclopaedia of mathematical sciences ; 31 |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-008560753 | ||
Datensatz im Suchindex
| _version_ | 1804127256787812352 |
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| adam_text | CONTENTS I. LINEAR PARTIAL DIFFERENTIAL EQUATIONS. ELEMENTS OF THE
MODERN THEORY YU.V. EGOROV AND M.A. SHUBIN 1 II. LINEAR PARTIAL
DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A.I. KOMECH 121 AUTHOR
INDEX 257 SUBJECT INDEX 261 I. LINEAR PARTIAL DIFFERENTIAL EQUATIONS.
ELEMENTS OF THE MODERN THEORY YU.V. EGOROV, M.A. SHUBIN TRANSLATED FROM
THE RUSSIAN BY P.C. SINHA CONTENTS PREFACE . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 4 NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 5 X 1.
PSEUDODIFFERENTIAL OPERATORS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 6 1.1. DEFINITION AND SIMPLEST PROPERTIES . . . .
. . . . . . . . . . . . . . . . . . . . 6 1.2. THE EXPRESSION FOR AN
OPERATOR IN TERMS OF AMPLITUDE. THE CONNECTION BETWEEN THE AMPLITUDE AND
THE SYMBOL. SYMBOLS OF TRANSPOSE AND ADJOINT OPERATORS . . . . . . . . .
. . . . . . 9 1.3. THE COMPOSITION THEOREM. THE PARAMETRIX OF AN
ELLIPTIC OPERATOR . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 14 1.4. ACTION OF PSEUDODIFFERENTIAL OPERATORS IN SOBOLEV
SPACES AND PRECISE REGULARITY THEOREMS FOR SOLUTIONS OF ELLIPTIC
EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 17 1.5. CHANGE OF VARIABLES AND PSEUDODIFFERENTIAL OPERATORS ON
A MANIFOLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 19 1.6. FORMULATION OF THE INDEX PROBLEM. THE
SIMPLEST INDEX FORMULAE . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 24 1.7. ELLIPTICITY WITH A PARAMETER. RESOLVENT AND COMPLEX
POWERS OF ELLIPTIC OPERATORS . . . . . . . . . . . . . . . . . 26 1.8.
PSEUDODIFFERENTIAL OPERATORS IN R N . . . . . . . . . . . . . . . . . .
. . . . . . 32 X 2. SINGULAR INTEGRAL OPERATORS AND THEIR APPLICATIONS.
CALDERON*S THEOREM. REDUCTION OF BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC
EQUATIONS TO PROBLEMS ON THE BOUNDARY . . . . . . . . . . . . . . . 36
2.1. DEFINITION AND BOUNDEDNESS THEOREMS . . . . . . . . . . . . . . . .
. . . . 36 2 YU.V. EGOROV, M.A. SHUBIN 2.2. SMOOTHNESS OF SOLUTIONS OF
SECOND-ORDER ELLIPTIC EQUATIONS . . . . . . . . . . . . . . . . . . . .
. . . . 37 2.3. CONNECTION WITH PSEUDODIFFERENTIAL OPERATORS . . . . . .
. . . . . . . . . 37 2.4. DIAGONALIZATION OF HYPERBOLIC SYSTEM OF
EQUATIONS . . . . . . . . . 38 2.5. CALDERON*S THEOREM . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.6. REDUCTION OF
THE OBLIQUE DERIVATIVE PROBLEM TO A PROBLEM ON THE BOUNDARY . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 40 2.7. REDUCTION OF THE
BOUNDARY-VALUE PROBLEM FOR THE SECOND-ORDER EQUATION TO A PROBLEM ON THE
BOUNDARY . . 41 2.8. REDUCTION OF THE BOUNDARY-VALUE PROBLEM FOR AN
ELLIPTIC SYSTEM TO A PROBLEM ON THE BOUNDARY . . . . . . . . . 43 X 3.
WAVE FRONT OF A DISTRIBUTION AND SIMPLEST THEOREMS ON PROPAGATION OF
SINGULARITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 44 3.1. DEFINITION AND EXAMPLES . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 44 3.2. PROPERTIES OF THE WAVE FRONT SET .
. . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3. APPLICATIONS
TO DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . 47
3.4. SOME GENERALIZATIONS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 48 X 4. FOURIER INTEGRAL OPERATORS . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.
DEFINITION AND EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 48 4.2. SOME PROPERTIES OF FOURIER INTEGRAL OPERATORS .
. . . . . . . . . . . . . . 50 4.3. COMPOSITION OF FOURIER INTEGRAL
OPERATORS WITH PSEUDODIFFERENTIAL OPERATORS . . . . . . . . . . . . . .
. . . . . . . . . . 52 4.4. CANONICAL TRANSFORMATIONS . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 53 4.5. CONNECTION BETWEEN
CANONICAL TRANSFORMATIONS AND FOURIER INTEGRAL OPERATORS . . . . . . . .
. . . . . . . . . . . . . . . . . . . 55 4.6. LAGRANGIAN MANIFOLDS AND
PHASE FUNCTIONS . . . . . . . . . . . . . . . . . 57 4.7. LAGRANGIAN
MANIFOLDS AND FOURIER DISTRIBUTIONS . . . . . . . . . . . . . 59 4.8.
GLOBAL DEFINITION OF A FOURIER INTEGRAL OPERATOR . . . . . . . . . . . .
. 59 X 5. PSEUDODIFFERENTIAL OPERATORS OF PRINCIPAL TYPE . . . . . . . .
. . . . . . . . . . 60 5.1. DEFINITION AND EXAMPLES . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 60 5.2. OPERATORS WITH REAL
PRINCIPAL SYMBOL . . . . . . . . . . . . . . . . . . . . . 61 5.3.
SOLVABILITY OF EQUATIONS OF PRINCIPLE TYPE WITH REAL PRINCIPAL SYMBOL .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4.
SOLVABILITY OF OPERATORS OF PRINCIPAL TYPE WITH COMPLEX-VALUED PRINCIPAL
SYMBOL . . . . . . . . . . . . . . . . . . . . 64 X 6. MIXED PROBLEMS
FOR HYPERBOLIC EQUATIONS . . . . . . . . . . . . . . . . . . . . . . 65
6.1. FORMULATION OF THE PROBLEM . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 65 6.2. THE HERSH-KREISS CONDITION . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 66 6.3. THE SAKAMOTO CONDITIONS .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.4.
REFLECTION OF SINGULARITIES ON THE BOUNDARY . . . . . . . . . . . . . .
. . . 69 6.5. FRIEDLANDER*S EXAMPLE . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 71 I. LINEAR PARTIAL DIFFERENTIAL
EQUATIONS. ELEMENTS OF THE MODERN THEORY 3 6.6. APPLICATION OF CANONICAL
TRANSFORMATIONS . . . . . . . . . . . . . . . . . 73 6.7. CLASSIFICATION
OF BOUNDARY POINTS . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.8. TAYLOR*S EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 74 6.9. OBLIQUE DERIVATIVE PROBLEM . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 75 X 7. METHOD OF
STATIONARY PHASE AND SHORT-WAVE ASYMPTOTICS . . . . . . . . . . 78 7.1.
METHOD OF STATIONARY PHASE . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 79 7.2. LOCAL ASYMPTOTIC SOLUTIONS OF HYPERBOLIC EQUATIONS .
. . . . . . . . 82 7.3. CAUCHY PROBLEM WITH RAPIDLY OSCILLATING INITIAL
DATA . . . . . . . . 86 7.4. LOCAL PARAMETRIX OF THE CAUCHY PROBLEM AND
PROPAGATION OF SINGULARITIES OF SOLUTIONS . . . . . . . . . . . . . . .
87 7.5. THE MASLOV CANONICAL OPERATOR AND GLOBAL ASYMPTOTIC SOLUTIONS OF
THE CAUCHY PROBLEM . . . . . 90 X 8. ASYMPTOTICS OF EIGENVALUES OF
SELF-ADJOINT DIFFERENTIAL AND PSEUDODIFFERENTIAL OPERATORS . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 96 8.1. VARIATIONAL
PRINCIPLES AND ESTIMATES FOR EIGENVALUES . . . . . . . . . 96 8.2.
ASYMPTOTICS OF THE EIGENVALUES OF THE LAPLACE OPERATOR IN A EUCLIDEAN
DOMAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99 8.3. GENERAL FORMULA OF WEYL ASYMPTOTICS AND THE METHOD OF
APPROXIMATE SPECTRAL PROJECTION . . . . . . . . . . . . . . . . . . . .
. . . 102 8.4. TAUBERIAN METHODS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 106 8.5. THE HYPERBOLIC EQUATION METHOD
. . . . . . . . . . . . . . . . . . . . . . . . 110 BIBLIOGRAPHICAL
COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 113 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 II. LINEAR
PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A.I. KOMECH
TRANSLATED FROM THE RUSSIAN BY P.C. SINHA CONTENTS PREFACE . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 125 CHAPTER 1. GENERALIZED FUNCTIONS AND
FUNDAMENTAL SOLUTIONS OF DIFFERENTIAL EQUATIONS . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 128 X 1. GENERALIZED FUNCTIONS AND
OPERATIONS ON THEM . . . . . . . . . . . . . . . . . . 128 1.1.
DIFFERENTIATION OF GENERALIZED FUNCTIONS . . . . . . . . . . . . . . . .
. . . 128 1.2. CHANGE OF VARIABLES IN GENERALIZED FUNCTIONS . . . . . .
. . . . . . . . 130 1.3. SUPPORT OF A GENERALIZED FUNCTION . . . . . . .
. . . . . . . . . . . . . . . . . 134 1.4. SINGULAR SUPPORT OF
GENERALIZED FUNCTIONS . . . . . . . . . . . . . . . . . 136 1.5. THE
CONVOLUTION OF GENERALIZED FUNCTIONS . . . . . . . . . . . . . . . . .
136 1.6. BOUNDARY VALUES OF ANALYTIC FUNCTIONS . . . . . . . . . . . . .
. . . . . . . 139 1.7. THE SPACE OF TEMPERED DISTRIBUTIONS . . . . . . .
. . . . . . . . . . . . . . . 141 X 2. FUNDAMENTAL SOLUTIONS OF
DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . 142 2.1. THE
FUNDAMENTAL SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 142 2.2. EXAMPLES OF FUNDAMENTAL SOLUTIONS . . . . . . . . . .
. . . . . . . . . . . . 143 2.3. THE PROPAGATION OF WAVES . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 146 2.4. THE
CONSTRUCTION OF FUNDAMENTAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
. . . . . . . . . . . . . . . . . . . . . . . . 147 2.5. A MEAN VALUE
THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148 CHAPTER 2. FOURIER TRANSFORMATION OF GENERALIZED FUNCTIONS . . . . .
. . . . . . 149 X 1. FOURIER TRANSFORMATION OF TEST FUNCTIONS . . . . .
. . . . . . . . . . . . . . . . . . 149 1.1. FOURIER TRANSFORMATION OF
RAPIDLY DECREASING FUNCTIONS . . . . . 149 1.2. PROPERTIES OF THE
FOURIER TRANSFORMATION . . . . . . . . . . . . . . . . . . . 149 1.3.
FOURIER TRANSFORMATION OF FUNCTIONS WITH COMPACT SUPPORT . . . 150 122
A.I. KOMECH X 2. FOURIER TRANSFORMATION OF TEMPERED GENERALIZED
FUNCTIONS . . . . . . . . 151 2.1. CLOSURE OF THE FOURIER TRANSFORMATION
WITH RESPECT TO CONTINUITY . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 151 2.2. PROPERTIES OF THE FOURIER TRANSFORMATION . . . .
. . . . . . . . . . . . . . . 151 2.3. METHODS FOR COMPUTING FOURIER
TRANSFORMS . . . . . . . . . . . . . . . . 153 2.4. EXAMPLES OF THE
COMPUTATION OF FOURIER TRANSFORMS . . . . . . . . . 154 X 3. THE SOBOLEV
FUNCTION SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 155 X 4. FOURIER TRANSFORMATION OF RAPIDLY GROWING GENERALIZED
FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . 156 4.1. FUNCTIONS
ON THE SPACE Z . C N / . . . . . . . . . . . . . . . . . . . . . . . . .
. . 156 4.2. FOURIER TRANSFORMATION ON THE SPACE D 0 . R N / . . . . . .
. . . . . . . . . 157 4.3. OPERATIONS ON THE SPACE Z 0 . C N / . . . . .
. . . . . . . . . . . . . . . . . . . . . 158 4.4. PROPERTIES OF THE
FOURIER TRANSFORMATION . . . . . . . . . . . . . . . . . . . 158 4.5.
ANALYTIC FUNCTIONALS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 158 X 5. THE PALEY-WIENER THEORY . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 160 5.1. FOURIER
TRANSFORM OF GENERALIZED FUNCTIONS WITH COMPACT SUPPORTS . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 160 5.2. TEMPERED
DISTRIBUTIONS WITH SUPPORT IN A CONE . . . . . . . . . . . . . 160 5.3.
EXPONENTIALLY GROWING DISTRIBUTIONS HAVING SUPPORT IN A CONE . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 161 X 6. CONVOLUTION
AND FOURIER TRANSFORM . . . . . . . . . . . . . . . . . . . . . . . . .
. . 163 CHAPTER 3. EXISTENCE AND UNIQUENESS OF SOLUTIONS OF DIFFERENTIAL
EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164 X 1. THE PROBLEM OF DIVISION . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 164 1.1. THE PROBLEM OF DIVISION IN
CLASSES OF RAPIDLY GROWING DISTRIBUTIONS . . . . . . . . . . . . . . . .
. . . . . . . . . 164 1.2. THE PROBLEM OF DIVISION IN CLASSES OF
EXPONENTIALLY GROWING GENERALIZED FUNCTIONS. THE H¨ ORMANDER STAIRCASE .
. . . . . . . . . . . 166 1.3. THE PROBLEM OF DIVISION IN CLASSES OF
TEMPERED DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 167 X 2. REGULARIZATION. THE METHODS OF *SUBTRACTION* AND
EXIT TO THE COMPLEX DOMAIN AND THE RIESZ POWER METHOD . . . . . . . . .
. . . . 168 2.1. THE METHOD OF SUBTRACTION . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 169 2.2. THE METHOD OF EXIT TO THE COMPLEX
DOMAIN . . . . . . . . . . . . . . . . 171 2.3. THE RIESZ METHOD OF
COMPLEX POWERS . . . . . . . . . . . . . . . . . . . . 172 X 3.
EQUATIONS IN A CONVEX CONE. AN OPERATIONAL CALCULUS . . . . . . . . . .
. . 173 3.1. EQUATIONS IN A CONE . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 173 3.2. AN OPERATIONAL CALCULUS . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.3.
DIFFERENTIAL-DIFFERENCE EQUATIONS ON A SEMI-AXIS . . . . . . . . . . . .
177 X 4. PROPAGATION OF SINGULARITIES AND SMOOTHNESS OF SOLUTIONS . . .
. . . . . . 178 4.1. CHARACTERISTICS OF DIFFERENTIAL EQUATIONS . . . . .
. . . . . . . . . . . . . . 178 4.2. WAVE FRONTS BICHARACTERISTICS AND
PROPAGATION OF SINGULARITIES . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 180 II. LINEAR PARTIAL DIFFERENTIAL
EQUATIONS WITH CONSTANT COEFFICIENTS 123 X 5. SMOOTHNESS OF SOLUTIONS OF
ELLIPTIC EQUATIONS. HYPOELLIPTICITY . . . . . 183 5.1. SMOOTHNESS OF
GENERALIZED SOLUTIONS OF ELLIPTIC EQUATIONS . . . . 183 5.2.
HYPOELLIPTIC OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 184 CHAPTER 4. THE FUNCTION P * C FOR POLYNOMIALS OF
SECOND-DEGREE AND ITS APPLICATION IN THE CONSTRUCTION OF FUNDAMENTAL
SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186 X 1. THE FUNCTION P * C FOR THE CASE WHEN P IS A REAL LINEAR
FUNCTION . . . . 186 1.1. ANALYTIC CONTINUATION WITH RESPECT TO * . . .
. . . . . . . . . . . . . . . . 186 1.2. AN APPLICATION TO BESSEL
FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . 188 X 2. THE
FUNCTION P * C FOR THE CASE WHEN P . X / IS A QUADRATIC FORM OF THE TYPE
( M , N * M ) WITH REAL COEFFICIENTS . . . . . . . . . . . . . . . . . .
. 188 2.1. THE CASE M D N . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 189 2.2. APPLICATION TO DECOMPOSITION OF
* -FUNCTION INTO PLANE WAVES . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 190 2.3. THE CASE 1 6 M 6 N * 1 . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.4.
APPLICATION TO BESSEL FUNCTIONS . . . . . . . . . . . . . . . . . . . .
. . . . . . 193 X 3. INVARIANT FUNDAMENTAL SOLUTIONS OF SECOND-ORDER
EQUATIONS WITH REAL COEFFICIENTS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 196 3.1. ANALYSIS OF INVARIANCE
PROPERTIES OF THE EQUATION . . . . . . . . . . . . 197 3.2.
DETERMINATION OF THE REGULAR PART OF AN INVARIANT FUNDAMENTAL SOLUTION .
. . . . . . . . . . . . . . . . . . . . . 198 X 4. REGULARIZATION OF THE
FORMAL FUNDAMENTAL SOLUTION FOR THE CASE Q D 0 . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.1. THE
CASE M D 0 OR M D N . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 200 4.2. THE CASE 1 6 M 6 N * 1 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 201 X 5. REGULARIZATION OF THE FUNDAMENTAL
SOLUTION FOR THE CASE Q 6D 0 . . . . . 204 5.1. THE CASE 1 6 M 6 N * 1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.2. THE
CASE M D 0 OR M D N . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 207 X 6. ON SINGULARITIES OF FUNDAMENTAL SOLUTIONS OF
SECOND-ORDER EQUATIONS WITH REAL COEFFICIENTS AND WITH NON-DEGENERATE
QUADRATIC FORM . . . . . . . . . . . . . . . . . . . . . . . 211 CHAPTER
5. BOUNDARY-VALUE PROBLEMS IN HALF-SPACE . . . . . . . . . . . . . . . .
. . 212 X 1. EQUATIONS WITH CONSTANT COEFFICIENTS IN A HALF-SPACE . . .
. . . . . . . . . . 213 1.1. GENERAL SOLUTION OF EQUATION (0.1) IN A
HALF-SPACE . . . . . . . . . . . 213 1.2. CLASSIFICATION OF EQUATIONS IN
HALF-SPACE . . . . . . . . . . . . . . . . . . . 215 X 2. REGULAR
BOUNDARY-VALUE PROBLEMS IN A HALF-SPACE IN CLASSES OF BOUNDED FUNCTIONS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 220 2.1. REGULAR BOUNDARY-VALUE PROBLEMS . . . . . . . . . . . . .
. . . . . . . . . . . 221 2.2. EXAMPLES OF REGULAR BOUNDARY-VALUE
PROBLEMS . . . . . . . . . . . . . 224 X 3. REGULAR BOUNDARY-VALUE
PROBLEMS IN CLASSES OF EXPONENTIALLY GROWING FUNCTIONS . . . . . . . . .
. . . . . . . . . . . . . . . . . . 226 3.1. DEFINITION AND EXAMPLES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 124 A.I.
KOMECH 3.2. THE CAUCHY PROBLEM . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 228 3.3. THE DIRICHLET PROBLEM FOR ELLIPTIC
EQUATIONS . . . . . . . . . . . . . . . . 229 X 4. REGULAR
BOUNDARY-VALUE PROBLEMS IN THE CLASS OF FUNCTIONS OF ARBITRARY GROWTH .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 229 X 5. WELL-POSED AND CONTINUOUS BOUNDARY-VALUE PROBLEMS IN A
HALF-SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 231 5.1. WELL-POSED BOUNDARY VALUE PROBLEMS
. . . . . . . . . . . . . . . . . . . . . 231 5.2. CONTINUOUS WELL-POSED
BOUNDARY-VALUE PROBLEMS . . . . . . . . . . . 232 X 6. THE POISSON
KERNEL FOR THE BOUNDARY-VALUE PROBLEM IN A HALF-SPACE . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234 6.1. THE POISSON KERNEL AND THE FUNDAMENTAL SOLUTION OF THE
BOUNDARY-VALUE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . .
. 234 6.2. THE CONNECTION BETWEEN THE FUNDAMENTAL SOLUTION OF THE CAUCHY
PROBLEM AND THE RETARDED FUNDAMENTAL SOLUTION OF THE OPERATOR P .@ X / .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
X 7. BOUNDARY-VALUE PROBLEMS IN A HALF-SPACE FOR NON-HOMOGENEOUS
EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
238 7.1. NON-HOMOGENEOUS EQUATIONS IN A HALF-SPACE . . . . . . . . . . .
. . . . 238 7.2. BOUNDARY-VALUE PROBLEMS FOR NON-HOMOGENEOUS EQUATIONS .
. . . 240 CHAPTER 6. SHARP AND DIFFUSION FRONTS OF HYPERBOLIC EQUATIONS
. . . . . . . . 240 X 1. BASIC NOTIONS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 X 2. THE
PETROVSKIJ CRITERION . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 244 X 3. THE LOCAL PETROVSKIJ CRITERION . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 246 X 4. GEOMETRY
OF LACUNAE NEAR CONCRETE SINGULARITIES OF FRONTS . . . . . . . . 247 X
5. EQUATIONS WITH VARIABLE COEFFICIENTS . . . . . . . . . . . . . . . .
. . . . . . . . . . . 250 BIBLIOGRAPHICAL COMMENTS . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 250 REFERENCES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 251
|
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| id | DE-604.BV012605146 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:30:28Z |
| institution | BVB |
| isbn | 3540653775 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008560753 |
| oclc_num | 611846440 |
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| physical | 263 S. Ill. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| spelling | Elements of the modern theory of partial differential equations Yu. V. Egorov ; A. I. Komech ; M. A. Shubin 1. ed., 2. printing Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London Springer 1999 263 S. Ill. txt rdacontent n rdamedia nc rdacarrier Aus dem Russ. übers. Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Egorov, Jurij V. 1938- Sonstige (DE-588)121177181 oth Komeč, Aleksandr I. 1946- Sonstige (DE-588)120962063 oth Šubin, Michail A. 1944- Sonstige (DE-588)121177211 oth 1. Aufl. u.d.T. Encyclopaedia of mathematical sciences ; 31 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008560753&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Elements of the modern theory of partial differential equations Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 |
| title | Elements of the modern theory of partial differential equations |
| title_auth | Elements of the modern theory of partial differential equations |
| title_exact_search | Elements of the modern theory of partial differential equations |
| title_full | Elements of the modern theory of partial differential equations Yu. V. Egorov ; A. I. Komech ; M. A. Shubin |
| title_fullStr | Elements of the modern theory of partial differential equations Yu. V. Egorov ; A. I. Komech ; M. A. Shubin |
| title_full_unstemmed | Elements of the modern theory of partial differential equations Yu. V. Egorov ; A. I. Komech ; M. A. Shubin |
| title_old | Encyclopaedia of mathematical sciences ; 31 |
| title_short | Elements of the modern theory of partial differential equations |
| title_sort | elements of the modern theory of partial differential equations |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008560753&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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