Holdings: Elements of the modern theory of partial differential equations

Elements of the modern theory of partial differential equations:

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Bibliographic Details
Previous Title:	Encyclopaedia of mathematical sciences ; 31
Format:	Book
Language:	English Russian
Published:	Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London Springer 1999
Edition:	1. ed., 2. printing
Subjects:	Partielle Differentialgleichung
Online Access:	Inhaltsverzeichnis
Item Description:	Aus dem Russ. übers.
Physical Description:	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.
650	0	7	\|a Partielle Differentialgleichung \|0 (DE-588)4044779-0 \|2 gnd \|9 rswk-swf
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

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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. CALDERONS 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. CALDERONS 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. FRIEDLANDERS 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. TAYLORS 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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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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