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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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##### References:
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