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Partial Differential Equations
http://dx.doi.org/10.1090/pspum/004
Proceedings of Symposia in
PURE MATHEMATICS
Volume 4
Partial Differential Equations Charles B. Morrey, Jr. Editor
Yfe American Mathematical Society £ i i i u | | ii it ji ^ Mm <*
%\ II11IIII / ? Providence, Rhode Island
PROCEEDINGS OF THE FOURTH SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY
HELD AT THE UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA APRIL 21-22, 1960
with the Support of the UNITED STATES AIR FORCE
Under Contract Number AF 49 (638) - 746
Library of Congress Catalog Number 50-1183 International Standard Book Number 0-8218-1404-4
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Copyright © 1961 by the American Mathematical Society. All rights reserved. Printed in the United States of America.
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10 9 8 7 6 5 4 3 96 95 94 93 92
CONTENTS
Extensions and Applications of the De Giorgi-Nash Results. By CHARLES B. MORREY, JR 1
Dirichlefs Principle in the Calculus of Variations. By JAMES SERRIN 17
Associated Spaces, Interpolation Theorems and the Regularity of Solutions of Differential Problems.
By N. ARONSZAJN 23
Lebesgue Spaces of Differentiate Functions and Distributions. By A. P. CALDERON 33
The Majorant Method. By P. C. ROSENBLOOM . 5 1
A Priori Estimates for Elliptic and Parabolic Equations. By FELIX E. BROWDER 73
Differential Equations in Hilbert Spaces. By FRANCOIS TREVES 83
A Maximum Property of Cauchy's Problem in Three-Dimensional Space-Time.
By H. F. WEINBERGER 91
Comments on Elliptic Partial Differential Equations. By Louis NIRENBERG 101
Some Unusual Boundary Value Problems. By MARTIN SCHECHTER 109
A New Proof and Generalizations of the Cauchy-Kowalewski Theorem to Nonanalytic and to Non-normal Systems.
By AVNER FRIEDMAN . . . . . 1 1 5
Regularity of Continuations of Solutions. By FRITZ JOHN 121
Some Local Properties of Elliptic Equations. By DAVID GILBARG 127
Estimates at Infinity for Steady State Solutions of the Navier-Stokes Equations.
By ROBERT FINN 143
VI CONTENTS
Interior Estimates for Solutions of Elliptic Monge-Ampere Equations. By ERHARD HEINZ 149
Zero Order A Priori Estimates for Solutions of Elliptic Differential Equations. By H. O. CORDES . . . 157
INDEX 167
INDEX
A priori estimates, 73-74, 138 for elliptic and parabolic boundary value
problems, 73 for mixed initial-boundary value prob
lems for general parabolic equations, 73
A priori Holder estimate, 127, 132 Adjoint
boundary problem, 76 pairing, 26
Admissible (norm), 26 Associated
norms,^23, 26-27 space F , 23 spaces, 23, 26-27
Asymptotic behavior, 133, 139 expansions, 133, 139
Bernstein's inequality, 58 Bessel potentials, 23, 29, 33
class Pa of of order a in B", 29 Boundary conditions, 111
overdetermined, 112 underdetermined, 112
Boundary value problems, 109 adjoint of, 76 estimates for, 103 exterior, 144, 147 Leray's, 144 mixed initial-, 78 regular, 74 (See also Elliptic)
Calculus of variations, existence theorems in, 20
Canonical pairing [£*, B, <&*, &>], 28 self-pairing, 28
Cauchy initial value problem, 101 -Kowalewski system, 115 problem, 51
Characteristics, 102
Class
Pa of Bessel potentials of order a in B", 29
Comparison arguments, 128 function, 130
Compatible (norms), 25 Compensating function, 31 Compensation method, 24, 31 Conditionally coercive (quadratic forms),
24 Conjugate norm, 26 Correct operators, 86 Curvature, 96
V<p, 5 Derivative of an operator, 87 Dini continuous, 128 Dirichlet
problem, 84, 104, 137 system, 112
Dirichlet Integral, 144, 147 of order j3 in D, 30
Distribution, 109 Domain
G of class C1, 4 of polyhedral type and class Cy**\ 30
Elliptic equations, 109 operator, 102, 109 systems of second order, 149 uniformly, 128
Elliptic boundary value problems existence of solutions to — in
operator-theoretic terms, 76 regularity of solutions of, 73
Ellipticity, non-uniform, 134 Energy inequalities, 86 Equation
of the type d/dt + A(t), 89 of waves, 121 second order of the type d2/dfi +
B(t)d/dt + A{t), 89
168 INDEX
Equivalent norms (J ||i ~ || ||2>» 2 9
Estimates for boundary value problems, 103 of Di Giorgi-Nash type, 73 zero order, 158 (See also A priori)
Existence of solutions to elliptic boundary value
problems in operator-theoretic terms, 76
theorems in the calculus of variations, 20
Extended maximum principle, 128 Extension theorem, 24, 30 Exterior boundary value problem, 144,
147 Extremal, 8
Force, 146 Frechet derivative, 57 Fundamental solution tensor, 144, 146
General restriction theorem, 31 Generalized solutions, 109 Goursat problem, 51, 71, 116
Harnack inequality, 132, 134 Hyperbolic, 101
Initial value problems, 85 Integral
equation, 93 singular operators, 102
Interpolated norms, 25 Interpolation of norms, 23 Isolated singular points, 131 Isomorphism {St T) of the pairing
[F, W, (v, wy] into (or onto) [V, W, <v', «;'>'], 28
K€- [K'€-] condition, 159 with respect to the symmetric positive
definite n x n matrix C = ((ctVfc)), 161 Kowalewski-Cauchy system, 115
Leray exterior boundary value problem of, 144 -Schauder fixed point theorem, 138
Limit theorem, 132 Liouville, 147
theorem, 134-135
Majorant method, 51 Majorants, 62 Maximum
principle, 21, 106, 128 property, 91, 95
Minimal surface equation, 136, 140 Mixed
initial-boundary value problem, 78 problems in cylindrical domains, 83
Monge-Ampere equations, 149
Navier-Stokes equations, 143 Neighborhood of infinity, 143 Neumann problem, 84 Newton's method, 54 Nonlinear equations—removable singu
larities, 135 Non-normal systems, 116 Non-uniform ellipticity, 134 Norms
associated, 23, 26 conjugate, 26 equivalent, 29 interpolated, 23, 25 standard IMU,D» 30
Normal (boundary conditions), 109
Operator correct, 86 derivative of, 87 elliptic, 109 singular integral, 102
Pairing, 23, 26 adjoint, 26 of vector spaces, 23 proper, 26 (See also Canonical and Isomorphism)
Perturbation methods, 54 Properly elliptic, 109
Quadratic interpolation, 24-26 theorem, 26
Quasi-linear parabolic equation, 79
INDEX 169
Regular boundary value problem, 74 Regularity
of continuations (of solutions), 121 of solutions of elliptic boundary \
problems, 73 Removable singularities, 130 Restriction theorem, converse to, 31
Singular integral operators, 102 Space
associated, 23, 26-27 associated V, 23 G\(G)y 1 Cj>h(G)y 74 @l(E), 89 H\ 85 Hl(D), 2
LUEn), 35 AS, 37 Lipc (G), 4 P * ^ " ) , 30 P&(^n), 30 W'*[G), 74
Standard norm ||w||afz>, 30
Stokes, Navier- equations, 143 Stress tensor, 145 Stronger (norm) (>-), 28 Strongly hyperbolic, 101 Subsonic flows, 139 Surface area, 19
Telescopic series, 125
Variational integral definition of, 17-18 lower semicontinuity of, 18
Variational problems, 17 Velocity field, 143
Wake, 146 Wave equation, 92, 121 W7eak solution, 128 Weaker (norm) (<), 28 Well posed problem, 121
Zero order estimate, 158