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Beteilige Person:	DiBenedetto, Emmanuele 1947-2021 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Boston [u.a.] Birkhäuser 2010
Ausgabe:	2. ed.
Schriftenreihe:	Cornerstones
Schlagwörter:	Differential equations, Partial Partielle Differentialgleichung Lehrbuch
Links:	http://deposit.dnb.de/cgi-bin/dokserv?id=2857128&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015678032&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XX, 389 S. graph. Darst.
ISBN:	9780817645519 0817645519 ebook 9780817645526

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Datensatz im Suchindex

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adam_text	Contents Preface to the Second Edition ................................. xvii Preface to the First Edition ................................... xix 0 Preliminaries .............................................. 1 1 Green s Theorem ........................................ 1 1.1 Differential Operators and Adjoints .................. 2 2 The Continuity Equation ................................. 3 3 The Heat Equation and the Laplace Equation ............... 5 3.1 Variable Coefficients ............................... 5 4 A Model for the Vibrating String .......................... 6 5 Small Vibrations of a Membrane .......................... 8 6 Transmission of Sound Waves ............................. 11 7 The Navier-Stokes System ................................ 13 8 The Euler Equations ..................................... 13 9 Isentropic Potential Flows ................................ 14 9.1 Steady Potential Isentropic Flows ................... 15 10 Partial Differential Equations ............................. 16 1 Quasi-Linear Equations and the Cauchy-Kowalewski Theorem .................................................. 17 1 Quasi-Linear Second-Order Equations in Two Variables ...... 17 2 Characteristics and Singularities ........................... 19 2.1 Coefficients Independent of ux and uy ................ 20 3 Quasi-Linear Second-Order Equations ...................... 21 3.1 Constant Coefficients .............................. 23 3.2 Variable Coefficients ............................... 23 4 Quasi-Linear Equations of Order m > 1.................... 24 4.1 Characteristic Surfaces ............................. 25 5 Analytic Data and the Cauchy-Kowalewski Theorem ........ 26 5.1 Reduction to Normal Form ([19]).................... 26 Contents 6 Proof of the Cauchy-Kowalewski Theorem .................. 27 6.1 Estimating the Derivatives of u at the Origin ......... 28 7 Auxiliary Inequalities .................................... 29 8 Auxiliary Estimations at the Origin ........................ 31 9 Proof of the Cauchy-Kowalewski Theorem (Concluded) ...... 32 9.1 Proof of Lemma 6.1 ............................... 33 Problems and Complements ............................... 33 lc Quasi-Linear Second-Order Equations in Two Variables ...... 33 5c Analytic Data and the Cauchy-Kowalewski Theorem ........ 34 6c Proof of the Cauchy-Kowalewski Theorem .................. 34 8c The Generalized Leibniz Rule ............................. 34 9c Proof of the Cauchy-Kowalewski Theorem (Concluded) ...... 35 The Laplace Equation ..................................... 37 1 Preliminaries ............................................ 37 1.1 The Dirichlet and Neumann Problems ............... 38 1.2 The Cauchy Problem .............................. 39 1.3 Well-Posedness and a Counterexample of Hadamard ..................................... 39 1.4 Radial Solutions ................................... 40 2 The Green and Stokes Identities ........................... 41 2.1 The Stokes Identities .............................. 41 3 Green s Function and the Dirichlet Problem for a Ball ........ 43 3.1 Green s Function for a Ball ......................... 45 4 Sub-Harmonic Functions and the Mean Value Property ....... 47 4.1 The Maximum Principle ........................... 50 4.2 Structure of Sub-Harmonic Functions ................ 50 5 Estimating Harmonic Functions and Their Derivatives ....... 52 5.1 The Harnack Inequality and the Liouville Theorem .... 52 5.2 Analyticity of Harmonic Functions ................... 53 6 The Dirichlet Problem ................................... 55 7 About the Exterior Sphere Condition ...................... 58 7.1 The Case N = 2 and dE Piecewise Smooth ........... 59 7.2 A Counterexample of Lebesgue for N — 3 ([101])...... 59 8 The Poisson Integral for the Half-Space .................... 60 9 Schauder Estimates of Newtonian Potentials ................ 62 10 Potential Estimates in LP(E) ............................. 65 11 Local Solutions .......................................... 68 11.1 Local Weak Solutions .............................. 69 12 Inhomogeneous Problems ................................. 70 12.1 On the Notion of Green s Function .................. 70 12.2 Inhomogeneous Problems ........................... 71 12.3 The Case ƒ є С™ (E).............................. 72 12.4 The Case ƒ Є CT>(Ë) ............................... 72 Contents vii Problems and Complements ............................... 73 le Preliminaries ............................................ 73 1.1c Newtonian Potentials on Ellipsoids .................. 73 1.2c Invariance Properties .............................. 74 2c The Green and Stokes Identities ........................... 74 3c Green s Function and the Dirichlet Problem for the Ball ............................................. 74 3.1c Separation of Variables ............................. 75 4c Sub-Harmonic Functions and the Mean Value Property ....... 76 4.1c Reflection and Harmonic Extension .................. 77 4.2c The Weak Maximum Principle ...................... 77 4.3c Sub-Harmonic Functions ........................... 78 5c Estimating Harmonic Functions ........................... 79 5.1c Harnack-Type Estimates ........................... 80 5.2c Ill-Posed Problems: An Example of Hadamard ........ 80 5.3c Removable Singularities ............................ 81 7c About the Exterior Sphere Condition ...................... 82 8c Problems in Unbounded Domains ......................... 83 8.1c The Dirichlet Problem Exterior to a Ball ............. 83 9c Schauder Estimates up to the Boundary ([135, 136])......... 84 10c Potential Estimates in LP(E) ............................. 84 10.1c Integrability of Riesz Potentials ..................... 85 10.2c Second Derivatives of Potentials ..................... 85 Boundary Value Problems by Double-Layer Potentials ..... 87 1 The Double-Layer Potential ............................... 87 2 On the Integral Defining the Double-Layer Potential ......... 89 3 The Jump Condition of W(dE, жо; υ) Across дЕ............. 91 4 More on the Jump Condition Across dE .................... 93 5 The Dirichlet Problem by Integral Equations ([111]) ......... 94 6 The Neumann Problem by Integral Equations ([111]) ........ 95 7 The Green Function for the Neumann Problem .............. 97 7.1 Finding £(·; ■)..................................... 98 8 Eigenvalue Problems for the Laplacian ..................... 99 8.1 Compact Kernels Generated by Green s Function ......100 9 Compactness of AF in U>{E) for 1 < ρ < oc ................100 10 Compactness of ΑΦ in LP{E) for 1 < ρ < oo ................102 11 Compactness of ΑΦ in L°°{E) .............................102 Problems and Complements ...............................104 2c On the Integral Defining the Double-Layer Potential .........104 5c The Dirichlet Problem by Integral Equations ................105 6c The Neumann Problem by Integral Equations ...............106 viii Contents 7c Green s Function for the Neumann Problem ................106 7.1c Constructing £(■; ·) for a Ball in R2 and K3 ...........106 8c Eigenvalue Problems .....................................107 4 Integral Equations and Eigenvalue Problems ...............109 1 Kernels in L2(E) ........................................109 1.1 Examples of Kernels in L2(E) .......................110 2 Integral Equations in L2(E) .............................. Ill 2.1 Existence of Solutions for Small \|A\| .................. Ill 3 Separable Kernels .......................................112 3.1 Solving the Homogeneous Equations .................113 3.2 Solving the Inhomogeneous Equation ................113 4 Small Perturbations of Separable Kernels ...................114 4.1 Existence and Uniqueness of Solutions ...............115 5 Almost Separable Kernels and Compactness ................116 5.1 Solving Integral Equations for Almost Separable Kernels .......................117 5.2 Potential Kernels Are Almost Separable ..............117 6 Applications to the Neumann Problem .....................118 7 The Eigenvalue Problem .................................119 8 Finding a First Eigenvalue and Its Eigenfunctions ...........121 9 The Sequence of Eigenvalues ..............................122 9.1 An Alternative Construction Procedure of the Sequence of Eigenvalues ............................123 10 Questions of Completeness and the ffilbert-Schmidt Theorem . 124 10.1 The Case of K(x; ■) є L2(E) Uniformly in ж ..........125 11 The Eigenvalue Problem for the Laplacean ..................126 11.1 An Expansion of Green s Function ...................127 Problems and Complements ...............................128 2c Integral Equations .......................................128 2.1c Integral Equations of the First Kind .................128 2.2c Abel Equations ([2, 3]).............................128 2.3c Solving Abel Integral Equations .....................129 2.4c The Cycloid ([3])..................................130 2.5c Volterra Integral Equations ([158, 159]) ..............130 3c Separable Kernels .......................................131 3.1c Hammerstein Integral Equations ([64])...............131 6c Applications to the Neumann Problem .....................132 9c The Sequence of Eigenvalues ..............................132 10c Questions of Completeness ................................132 10.1c Periodic Functions in RN ...........................133 10.2c The Poisson Equation with Periodic Boundary Conditions .......................................134 lie The Eigenvalue Problem for the Lapiacian ..................134 Contents ix The Heat Equation ........................................135 1 Preliminaries ............................................135 1.1 The Dirichlet Problem .............................136 1.2 The Neumann Problem ............................136 1.3 The Characteristic Cauchy Problem .................136 2 The Cauchy Problem by Similarity Solutions ................136 2.1 The Backward Cauchy Problem .....................140 3 The Maximum Principle and Uniqueness (Bounded Domains) . 140 3.1 A Priori Estimates ................................141 3.2 Ill-Posed Problems ................................141 3.3 Uniqueness (Bounded Domains) .....................142 4 The Maximum Principle in RN ............................142 4.1 A Priori Estimates ................................144 4.2 About the Growth Conditions (4.3) and (4.4).........145 5 Uniqueness of Solutions to the Cauchy Problem .............145 5.1 A Counterexample of Tychonov ([155])...............145 6 Initial Data in L¡OC(RN) ..................................147 6.1 Initial Data in the Sense of L¡OC(RN) ................149 7 Remarks on the Cauchy Problem ..........................149 7.1 About Regularity ..................................149 7.2 Instability of the Backward Problem .................150 8 Estimates Near t = 0 ....................................151 9 The Inhomogeneous Cauchy Problem ......................152 10 Problems in Bounded Domains ............................154 10.1 The Strong Solution ...............................155 10.2 The Weak Solution and Energy Inequalities ...........156 11 Energy and Logarithmic Convexity ........................157 11.1 Uniqueness for Some Ill-Posed Problems ..............158 12 Local Solutions ..........................................158 12.1 Variable Cylinders .................................162 12.2 The Case a = 0..................................162 13 The Harnack Inequality ..................................163 13.1 Compactly Supported Sub-Solutions .................164 13.2 Proof of Theorem 13.1.............................165 14 Positive Solutions in 5χ ..................................167 14.1 Non-Negative Solutions ............................169 Problems and Complements ...............................171 2c Similarity Methods ......................................171 2.1c The Heat Kernel Has Unit Mass .....................171 2.2c The Porous Media Equation ........................172 2.3c The p-Laplacean Equation ..........................172 2.4c The Error Function ................................173 2.5c The Appell Transformation ([7]).....................173 2.6c The Heat Kernel by Fourier Transform ...............173 Contents 2.7c Rapidly Decreasing Functions .......................174 2.8c The Fourier Transform of the Heat Kernel ............174 2.9c The Inversion Formula .............................175 3c The Maximum Principle in Bounded Domains ...............176 3.1c The Blow-Up Phenomenon for Super-Linear Equations . 177 3.2c The Maximum Principle for General Parabolic Equations ........................................178 4c The Maximum Principle in Rw ............................178 4.1c A Counterexample of the Tychonov Type ............180 7c Remarks on the Cauchy Problem ..........................180 12c On the Local Behavior of Solutions ........................180 The Wave Equation ........................................183 1 The One-Dimensional Wave Equation ......................183 1.1 A Property of Solutions ............................184 2 The Cauchy Problem ....................................185 3 Inhomogeneous Problems .................................186 4 A Boundary Value Problem (Vibrating String) ..............188 4.1 Separation of Variables .............................189 4.2 Odd Reflection ....................................190 4.3 Energy and Uniqueness ............................190 4.4 Inhomogeneous Problems ...........................191 5 The Initial Value Problem in N Dimensions .................191 5.1 Spherical Means ...................................192 5.2 The Darboux Formula .............................192 5.3 An Equivalent Formulation of the Cauchy Problem .... 193 6 The Cauchy Problem in Ш3 ...............................193 7 The Cauchy Problem in Ж2 ...............................196 8 The Inhomogeneous Cauchy Problem ......................198 9 The Cauchy Problem for Inhomogeneous Surfaces ...........199 9.1 Reduction to Homogeneous Data on t = Φ............ 200 9.2 The Problem with Homogeneous Data ...............200 10 Solutions in Half-Space. The Reflection Technique ...........201 10.1 An Auxiliary Problem .............................202 10.2 Homogeneous Data on the Hyperplane £3 = 0.........202 11 A Boundary Value Problem ...............................203 12 Hyperbolic Equations in Two Variables ....................204 13 The Characteristic Goursat Problem .......................205 13.1 Proof of Theorem 13.1: Existence ....................205 13.2 Proof of Theorem 13.1: Uniqueness ..................207 13.3 Goursat Problems in Rectangles .....................207 14 The Non-Characteristic Cauchy Problem and the Riemann Function ...............................................208 15 Symmetry of the Riemann Function .......................210 Contents xi Problems and Complements ...............................211 2c The d Alembert Formula .................................211 3c Inhomogeneous Problems.................................211 3.1c The Duhamel Principle ([38]).......................211 4c Solutions for the Vibrating String .........................212 6c Cauchy Problems in E3 ..................................214 6.1c Asymptotic Behavior ..............................214 6.2c Radial Solutions ...................................214 6.3c Solving the Cauchy Problem by Fourier Transform .... 216 7c Cauchy Problems in R2 and the Method of Descent ..........217 7.1c The Cauchy Problem for N = 4,5...................218 8c Inhomogeneous Cauchy Problems ..........................218 8.1c The Wave Equation for the N and (N + 1)-Laplacian .. 218 8.2c Miscellaneous Problems ............................219 10c The Reflection Technique .................................221 lie Problems in Bounded Domains ............................221 11.1c Uniqueness .......................................221 11.2c Separation of Variables .............................222 12c Hyperbolic Equations in Two Variables ....................222 12.1c The General Telegraph Equation ....................222 14c Goursat Problems .......................................223 14.1c The Riemann Function and the Fundamental Solution of the Heat Equation ..............................223 Quasi-Linear Equations of First-Order .....................225 1 Quasi-Linear Equations ..................................225 2 The Cauchy Problem ....................................226 2.1 The Case of Two Independent Variables ..............226 2.2 The Case of N Independent Variables ................227 3 Solving the Cauchy Problem ..............................227 3.1 Constant Coefficients ..............................228 3.2 Solutions in Implicit Form ..........................229 4 Equations in Divergence Form and Weak Solutions ..........230 4.1 Surfaces of Discontinuity ...........................231 4.2 The Shock Line ...................................231 5 The Initial Value Problem ................................232 5.1 Conservation Laws ................................233 6 Conservation Laws in One Space Dimension ................234 6.1 Weak Solutions and Shocks .........................235 6.2 Lack of Uniqueness ................................236 7 Hopf Solution of The Burgers Equation ....................236 8 Weak Solutions to (6.4) When α(·) is Strictly Increasing ......238 8.1 Lax VariationaJ Solution ...........................239 9 Constructing Variational Solutions I .......................240 9.1 Proof of Lemma 9.1 ...............................241 Contents 10 Constructing Variational Solutions II ......................242 11 The Theorems of Existence and Stability ...................244 11.1 Existence of Variational Solutions ...................244 11.2 Stability of Variational Solutions ....................245 12 Proof of Theorem 11.1...................................246 12.1 The Representation Formula (11.4)..................246 12.2 Initial Datum in the Sense of ^(R) ................247 12.3 Weak Forms of the PDE ...........................248 13 The Entropy Condition ..................................248 13.1 Entropy Solutions .................................249 13.2 Variational Solutions of (6.4) are Entropy Solutions -----249 13.3 Remarks on the Shock and the Entropy Conditions .... 251 14 The Kruzhkov Uniqueness Theorem ........................253 14.1 Proof of the Uniqueness Theorem I ..................253 14.2 Proof of the Uniqueness Theorem II .................254 14.3 Stability in Ll(RN) ................................256 15 The Maximum Principle for Entropy Solutions ..............256 Problems and Complements ...............................257 3c Solving the Cauchy Problem ..............................257 6c Explicit Solutions to the Burgers Equation .................259 6.2c Invariance of Burgers Equations by Some Transformation of Variables ........................259 6.3c The Generalized Riemann Problem ..................260 13c The Entropy Condition ..................................261 14c The Kruzhkov Uniqueness Theorem ........................262 Non-Linear Equations of First-Order ......................265 1 Integral Surfaces and Monge s Cones .......................265 1.1 Constructing Monge s Cones ........................266 1.2 The Symmetric Equation of Monge s Cones ...........266 2 Characteristic Curves and Characteristic Strips .............267 2.1 Characteristic Strips ...............................268 3 The Cauchy Problem ....................................269 3.1 Identifying the Initial Data p(0, s) ...................269 3.2 Constructing the Characteristic Strips ...............270 4 Solving the Cauchy Problem ..............................270 4.1 Verifying (4.3)....................................271 4.2 A Quasi-Linear Example in E2 ......................272 5 The Cauchy Problem for the Equation of Geometrical Optics ....................................273 5.1 Wave Fronts, Light Rays, Local Solutions, and Caustics ......................................274 6 The Initial Value Problem for Hamilton-Jacobi Equations .... 274 7 The Cauchy Problem in Terms of the Lagrangian ............276 Contents xiii 8 The Hopf Variational Solution ............................ 277 8.1 The First Hopf Variational Formula .................278 8.2 The Second Hopf Variational Formula ................278 9 Semigroup Property of Hopf Variational Solutions ...........279 10 Regularity of Hopf Variational Solutions ....................280 11 Hopf Variational Solutions (8.3) are Weak Solutions of the Cauchy Problem (6.4)....................................281 12 Some Examples .........................................283 12.1 Example I ........................................283 12.2 Example II .......................................284 12.3 Example III ......................................284 13 Uniqueness .............................................285 14 More on Uniqueness and Stability .........................287 14.1 Stability in ľp{Rn) for All ρ > 1....................287 14.2 Comparison Principle ..............................288 15 Semi-Concave Solutions of the Cauchy Problem .............288 15.1 Uniqueness of Semi-Concave Solutions ...............288 16 A Weak Notion of Semi-Concavity .........................289 17 Semi-Concavity of Hopf Variational Solutions ...............290 17.1 Weak Semi-Concavity of Hopf Variational Solutions Induced by the Initial Datum uo ....................290 17.2 Strictly Convex Hamiltonian ........................291 18 Uniqueness of Weakly Semi-Concave Variational Hopf Solutions ................................293 Linear Elliptic Equations with Measurable Coefficients .... 297 1 Weak Formulations and Weak Derivatives ..................297 1.1 Weak Derivatives ..................................298 2 Embeddings of Wl^{E) ..................................299 2.1 Compact Embeddings of W^P(E) ...................300 3 Multiplicative Embeddings of W^p(E) and Wl{E) .........300 3.1 Some Consequences of the Multiplicative Embedding Inequalities .......................................301 4 The Homogeneous Dirichlet Problem .......................302 5 Solving the Homogeneous Dirichlet Problem (4.1) by the Riesz Representation Theorem ............................302 6 Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods .....................................303 6.1 The Case N = 2...................................304 6.2 Gâteaux Derivative and The Euler Equation of J(·) .... 305 7 Solving the Homogeneous Dirichlet Problem (4.1) by Galerkin Approximations .................................305 7.1 On the Selection of an Orthonormal System in W¿(E) .......................................306 xiv Contents 7.2 Conditions on f and ƒ for the Solvability of the Dirichlet Problem (4.1) ............................307 8 Traces on dE of Functions in Wl{E) .....................307 8.1 The Segment Property .............................307 8.2 Defining Traces ...................................308 8.3 Characterizing the Traces on dE of Functions in W1 P(E) .........................................309 9 The Inhomogeneous Dirichlet Problem .....................309 10 The Neumann Problem ..................................310 10.1 A Variant of (10.1)................................311 11 The Eigenvalue Problem .................................312 12 Constructing the Eigenvalues of (11.1) .....................313 13 The Sequence of Eigenvalues and Eigenfunctions ............315 14 A Priori L°°(E) Estimates for Solutions of the Dirichlet Problem (9.1)...........................................317 15 Proof of Propositions 14.1-14.2............................318 15.1 An Auxiliary Lemma on Fast Geometric Convergence .. 319 15.2 Proof of Proposition 14.1 for N > 2..................319 15.3 Proof of Proposition 14.1 for N = 2..................320 16 A Priori L°°(E) Estimates for Solutions of the Neumann Problem (10.1)..........................................320 17 Proof of Propositions 16.1-16.2............................322 17.1 Proof of Proposition 16.1 for N > 2..................324 17.2 Proof of Proposition 16.1 for N = 2..................325 18 Miscellaneous Remarks on Further Regularity ...............325 Problems and Complements ...............................326 lc Weak Formulations and Weak Derivatives ..................326 1.1c The Chain Rule in W^(E) ........................326 2c Embeddings of Wl(E) ..................................327 2.1c Proof of (2.4).....................................327 2.2c Compact Embeddings of W1-P{E) ...................328 3c Multiplicative Embeddings of W} P(E) and Whp{E) .........329 3.1c Proof of Theorem 3.1 for 1 < ρ < N.................329 3.2c Proof of Theorem 3.1 for ρ > N > 1.................331 3.3c Proof of Theorem 3.2 for 1 < ρ < N and E Convex ___332 5c Solving the Homogeneous Dirichlet Problem (4.1) by the Riesz Representation Theorem ............................333 6c Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods .....................................334 6.1c More General Variational Problems ..................334 6.8c Gâteaux Derivatives, Euler Equations, and Quasi-Linear Elliptic Equations .....................336 Contents xv 8c Traces on дЕ of Functions in W1 P(E) .....................337 8.1c Extending Functions in W1 P(E) ....................337 8.2c The Trace Inequality ...............................338 8.3c Characterizing the Traces on dE of Functions in Wl*{E) .........................................339 9c The Inhomogeneous Dirichlet Problem .....................341 9.1c The Lebesgue Spike ...............................341 9.2c Variational Integrals and Quasi-Linear Equations ......341 10c The Neumann Problem ..................................342 lie The Eigenvalue Problem .................................343 12c Constructing the Eigenvalues .............................343 13c The Sequence of Eigenvalues and Eigenfunctions ............343 14c A Priori L°°(E) Estimates for Solutions of the Dirichlet Problem (9.1)...........................................343 15c A Priori L°°(E) Estimates for Solutions of the Neumann Problem (10.1)..........................................344 15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c) .....344 10 DeGiorgi Classes ..........................................347 1 Quasi-Linear Equations and DeGiorgi Classes ...............347 1.1 DeGiorgi Classes ..................................349 2 Local Boundedness of Functions in the DeGiorgi Classes ......350 2.1 Proof of Theorem 2.1 for 1< ρ < N.................351 2.2 Proof of Theorem 2.1 for ρ = N.....................352 3 Holder Continuity of Functions in the DG Classes ...........353 3.1 On the Proof of Theorem 3.1 .......................354 4 Estimating the Values of и by the Measure of the Set where и is Either Near μ+ or Near μ~ ...........................354 5 Reducing the Measure of the Set where и is Either Near μ+ or Near μ~ .............................................355 5.1 The Discrete Isoperimetric Inequality ................356 5.2 Proof of Proposition 5.1............................357 6 Proof of Theorem 3.1....................................358 7 Boundary DeGiorgi Classes: Dirichlet Data .................359 7.1 Continuity up to dE of Functions in the Boundary DG Classes (Dirichlet Data) ........................360 8 Boundary DeGiorgi Classes: Neumann Data ................361 8.1 Continuity up to dE of Functions in the Boundary DG Classes (Neumann Data) .......................363 9 The Harnack Inequality ..................................364 9.1 Proof of Theorem 9.1 (Preliminaries) ................364 9.2 Proof of Theorem 9.1. Expansion of Positivity .........365 9.3 Proof of Theorem 9.1..............................365 10 Harnack Inequality and Holder Continuity ..................367 xvi Contents 11 Local Clustering of the Positivity Set of Functions in W^l{E) .............................................368 12 A Proof of the Harnack Inequality Independent of Holder Continuity ..............................................370 References .....................................................373 Index ..........................................................381
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genre_facet	Lehrbuch
id	DE-604.BV022470541
illustrated	Illustrated
indexdate	2024-12-20T12:56:07Z
institution	BVB
isbn	9780817645519 0817645519 ebook 9780817645526
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-015678032
oclc_num	171111213
open_access_boolean
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physical	XX, 389 S. graph. Darst.
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publishDateSort	2010
publisher	Birkhäuser
record_format	marc
series2	Cornerstones
spellingShingle	DiBenedetto, Emmanuele 1947-2021 Partial differential equations Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd
subject_GND	(DE-588)4044779-0 (DE-588)4123623-3
title	Partial differential equations
title_auth	Partial differential equations
title_exact_search	Partial differential equations
title_full	Partial differential equations Emmanuele DiBenedetto
title_fullStr	Partial differential equations Emmanuele DiBenedetto
title_full_unstemmed	Partial differential equations Emmanuele DiBenedetto
title_short	Partial differential equations
title_sort	partial differential equations
topic	Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd
topic_facet	Differential equations, Partial Partielle Differentialgleichung Lehrbuch
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