Twodimensional nonlinear boundary value problems for elliptic equations
Author:
Gary M. Lieberman
Journal:
Trans. Amer. Math. Soc. 300 (1987), 287295
MSC:
Primary 35J65; Secondary 35B65
MathSciNet review:
871676
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Abstract: Boundary regularity of solutions of the fully nonlinear boundary value problem is discussed for twodimensional domains . The function is assumed uniformly elliptic and is assumed to depend (in a nonvacuous manner) on . Continuity estimates are proved for first and second derivatives of under weak hypotheses for smoothness of , , and .
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 G. E. Backus, Application of a nonlinear boundaryvalue problem for Laplace's equation to gravity and geomagnetic intensity surveys, Quart. J. Mech. Appl. Math. 21 (1968), 195221. MR 0227444 (37:3028)
 [2]
 D. Gilbarg and L. Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal. 74 (1980), 297318. MR 588031 (82a:35038)
 [3]
 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren Math. Wiss., Band 224, SpringerVerlag, Berlin and New York, 1983. MR 737190 (86c:35035)
 [4]
 E. Giusti, Boundary behavior of nonparametric minimal surfaces, Indiana Univ. Math. J. 22 (1972), 435444. MR 0305253 (46:4383)
 [5]
 G. M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary, Comm. Partial Differential Equations 6 (1981), 437497. MR 612553 (83g:35032)
 [6]
 , The Dirichlet problem for quasilinear elliptic equations with Holder continuous boundary values, Arch. Rational Mech. Anal. 79 (1982), 3053[ill]3. MR 656797 (83j:35048)
 [7]
 , Regularized distance and its applications, Pacific J. Math. 117 (1985), 329352. MR 779924 (87j:35101)
 [8]
 , The Dirichlet problem for quasilinear elliptic equations with continuously differentiate boundary data, Comm. Partial Differential Equations 11 (1986), 167229. MR 818099 (87f:35073)
 [9]
 G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509546. MR 833695 (87h:35114)
 [10]
 J. D. Madjarova, On the Neumann problem for a fully nonlinear convex, elliptic equation, C. R. Acad. Bulgare Sci. 38 (1985), 183186. MR 789129 (86g:35078)
 [11]
 M. V. Safonov, On the classical solution of Bellman's elliptic equation, Soviet Math. Dokl. 30 (1984), 482485.
 [12]
 J. Taylor, Boundary regularity for solutions to various capillarity and free boundary problems, Comm. Partial Differential Equations 2 (1977), 323357. MR 0487721 (58:7336)
 [13]
 N. S. Trudinger, On an interpolation inequality and its application to nonlinear elliptic equations, Proc. Amer. Math. Soc. 95 (1985), 7378. MR 796449 (87a:35077)
 [14]
 G. C. Wen, Some nonlinear boundary value problems for nonlinear elliptic equations of second order in the plane, Complex Variables 4 (1985), 189204. MR 801637 (86i:35053)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708716768
PII:
S 00029947(1987)08716768
Keywords:
Elliptic equations,
boundary value problems,
twodimensional,
boundary regularity
Article copyright:
© Copyright 1987
American Mathematical Society
