Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations
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- by Gary M. Lieberman PDF
- Trans. Amer. Math. Soc. 304 (1987), 343-353 Request permission
Abstract:
We consider solutions (and subsolutions and supersolutions) of the boundary value problem \[ \begin {array}{*{20}{c}} {{a^{ij}}(x, u, Du){D_{ij}}u + a(x, u, Du) = 0\quad {\text {in}}\;\Omega ,} \\ {{\beta ^i}(x){D_i}u + \gamma (x)u = g(x)\quad {\text {on}}\;\partial \Omega } \\ \end {array} \] for a Lipschitz domain $\Omega$, a positive-definite matrix-valued function $[{a^{ij}}]$, and a vector field $\beta$ which points uniformly into $\Omega$. Without making any continuity assumptions on the known functions, we prove Harnack and Hölder estimates for $u$ near $\partial \Omega$. In addition we bound the ${L^\infty }$ norm of $u$ near $\partial \Omega$ in terms of an appropriate ${L^p}$ norm and the known functions. Our approach is based on that for the corresponding interior estimates of Trudinger.References
- I. Ja. Bakel′man, The Dirichlet problem for equations of Monge-Ampère type and their $n$-dimensional analogues, Dokl. Akad. Nauk SSSR 126 (1959), 923–926 (Russian). MR 0118947
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Gary M. Lieberman, Oblique derivative problems in Lipschitz domains. I. Continuous boundary data, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 4, 1185–1210 (English, with Italian summary). MR 923448 —, Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data (to appear).
- Gary M. Lieberman and Neil S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), no. 2, 509–546. MR 833695, DOI 10.1090/S0002-9947-1986-0833695-6
- P.-L. Lions, N. S. Trudinger, and J. I. E. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math. 39 (1986), no. 4, 539–563. MR 840340, DOI 10.1002/cpa.3160390405
- N. S. Nadirashvili, Lemma on the interior derivative and uniqueness of the solution of the second boundary value problem for second-order elliptic equations, Dokl. Akad. Nauk SSSR 261 (1981), no. 4, 804–808 (Russian). MR 638068
- N. S. Nadirashvili, On the question of the uniqueness of the solution of the second boundary value problem for second-order elliptic equations, Mat. Sb. (N.S.) 122(164) (1983), no. 3, 341–359 (Russian). MR 721393
- N. S. Nadirashvili, On a problem with an oblique derivative, Mat. Sb. (N.S.) 127(169) (1985), no. 3, 398–416 (Russian). MR 798384
- Neil S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61 (1980), no. 1, 67–79. MR 587334, DOI 10.1007/BF01389895
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 304 (1987), 343-353
- MSC: Primary 35J65; Secondary 35B45
- DOI: https://doi.org/10.1090/S0002-9947-1987-0906819-0
- MathSciNet review: 906819