Some nonstandard problems for the Poisson equation
Authors:
L. E. Payne and P. W. Schaefer
Journal:
Quart. Appl. Math. 51 (1993), 81-90
MSC:
Primary 35J05; Secondary 35A05
DOI:
https://doi.org/10.1090/qam/1205938
MathSciNet review:
MR1205938
Full-text PDF Free Access
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- George E. Backus, Application of a non-linear boundary-value problem for Laplace’s equation to gravity and geomagnetic intensity surveys, Quart. J. Mech. Appl. Math. 21 (1968), 195–221. MR 227444, DOI https://doi.org/10.1093/qjmam/21.2.195
E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichung zweiter Ordnung von elliptischen Typus, Berlin Sber. Preuss. Akad. Wiss. 19, 147–152 (1927)
- Eberhard Hopf, A remark on linear elliptic differential equations of second order, Proc. Amer. Math. Soc. 3 (1952), 791–793. MR 50126, DOI https://doi.org/10.1090/S0002-9939-1952-0050126-X
- Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619
- Alan U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), no. 3, 687–704. MR 794582, DOI https://doi.org/10.1512/iumj.1985.34.34036
- Nicholas J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations 15 (1990), no. 4, 541–556. MR 1046708, DOI https://doi.org/10.1080/03605309908820698
- Rolando Magnanini, A fully nonlinear boundary value problem for the Laplace equation, Nonlinear analysis and applications (Arlington, Tex., 1986) Lecture Notes in Pure and Appl. Math., vol. 109, Dekker, New York, 1987, pp. 327–330. MR 912311
- Rolando Magnanini, A fully nonlinear boundary value problem for the Laplace equation in dimension two, Appl. Anal. 39 (1990), no. 2-3, 185–192. MR 1095632, DOI https://doi.org/10.1080/00036819008839979
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- L. E. Payne, Bounds for the maximum stress in the Saint Venant torsion problem, Indian J. Mech. Math. Special Issue Special Issue (1968/69), part I, 51–59. Special issue presented to Professor Bibhutibhusan Sen on the occasion of his seventieth birthday, Part I. MR 0351225
- L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature, Nonlinear Anal. 3 (1979), no. 2, 193–211. MR 525971, DOI https://doi.org/10.1016/0362-546X%2879%2990076-2
- L. E. Payne and G. A. Philippin, Some applications of the maximum principle in the problem of torsional creep, SIAM J. Appl. Math. 33 (1977), no. 3, 446–455. MR 455738, DOI https://doi.org/10.1137/0133028
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---, Nonlinear elliptic equations of second order, Lectures in Sympos. Partial Differential Equations, Berkeley, 1971 (mimeographed notes)
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G. E. Backus, Application of a non-linear boundary-value problem for Laplace’s equation to gravity and geomagnetic intensity surveys, Quart. J. Mech. Appl. Math. XXI, 195–221 (1968)
E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichung zweiter Ordnung von elliptischen Typus, Berlin Sber. Preuss. Akad. Wiss. 19, 147–152 (1927)
---, A remark on elliptic differential equations of the second order, Proc. Amer. Math. Soc. 3, 791–793 (1952)
B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin and New York, 1985
A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34, 687–704 (1985)
N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations 15, 541–556 (1990)
R. Magnanini, A fully nonlinear boundary value problem for the Laplace equation, Lecture Notes Pure Appl. Math., vol. 109, Marcel Dekker, 1987, pp. 327–330
---, A fully nonlinear boundary value problem for the Laplace equation in dimension two, Appl. Anal. 39, 185–192 (1990)
L. G. Makar-Limanov, Solutions of Dirichlet’s problem for the equation $\Delta u = - 1$ on a convex region, Math. Notes 9, 52–53 (1971)
L. E. Payne, Bounds for the maximum stress in the St. Venant torsion problem, Indian J. Mech. Math. (special issue) 51–59 (1968)
L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature, Nonlinear Anal. 3, 193–211 (1979)
---, Some applications of the maximum principle in the problem of torsional creep, SIAM J. Appl. Math. 33, 446–455 (1977)
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 304–318 (1971)
---, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London 264, 413–496 (1969)
---, Nonlinear elliptic equations of second order, Lectures in Sympos. Partial Differential Equations, Berkeley, 1971 (mimeographed notes)
R. P. Sperb, Extension of two theorems of Payne to some nonlinear Dirichlet problems, J. Appl. Math. Phys. (ZAMP) 26, 721–726 (1975)
---, Maximum principles and their applications, Math. in Science and Engineering, vol. 157, Academic Press, New York, 1981
H. F. Weinberger, Remarks on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43, 319–320 (1971)
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© Copyright 1993
American Mathematical Society