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Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations. (English) Zbl 0635.35037
Author’s abstract: “We consider solutions (subsolutions and supersolutions) of the boundary value problem \[ a^{ij}(x,u,Du)D_{ij}u+a(x,u,Du)=0\quad in\quad \Omega;\quad \beta^ i(x)D_ iu+\gamma (x)u=g(x)\quad on\quad \partial \Omega \] 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.”
Reviewer: C.F.Wang

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B35 Stability in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] I. Ja. Bakel\(^{\prime}\)man, The Dirichlet problem for equations of Monge-Ampère type and their \?-dimensional analogues, Dokl. Akad. Nauk SSSR 126 (1959), 923 – 926 (Russian). · Zbl 0088.30402
[2] 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. · Zbl 0562.35001
[3] 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). · Zbl 0637.35028
[4] -, Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data (to appear). · Zbl 0648.35033
[5] 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. · Zbl 0619.35047
[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. · Zbl 0604.35027
[7] 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).
[8] 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). · Zbl 0563.35026
[9] N. S. Nadirashvili, On a problem with an oblique derivative, Mat. Sb. (N.S.) 127(169) (1985), no. 3, 398 – 416 (Russian).
[10] Neil S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61 (1980), no. 1, 67 – 79. · Zbl 0453.35028
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