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

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.