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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations

Author: Gary M. Lieberman
Journal: Trans. Amer. Math. Soc. 304 (1987), 343-353
MSC: Primary 35J65; Secondary 35B45
MathSciNet review: 906819
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Abstract: We consider solutions (and subsolutions and supersolutions) of the boundary value problem

\begin{displaymath}\begin{array}{*{20}{c}} {{a^{ij}}(x,\,u,\,Du){D_{ij}}u + a(x,... ...(x)u = g(x)\quad {\text{on}}\;\partial \Omega } \\ \end{array} \end{displaymath}

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.

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Article copyright: © Copyright 1987 American Mathematical Society

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