Two-dimensional nonlinear boundary value problems for elliptic equations

Lieberman, Gary M.

doi:10.1090/S0002-9947-1987-0871676-8

Two-dimensional nonlinear boundary value problems for elliptic equations
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Trans. Amer. Math. Soc. 300 (1987), 287-295 Request permission

Abstract:

Boundary regularity of solutions of the fully nonlinear boundary value problem \[ F(x,u,Du,{D^2}u) = 0\quad {\text {in}}\;\Omega ,\qquad G(x,u,Du) = 0\quad {\text {on}}\;\partial \Omega \] is discussed for two-dimensional domains $\Omega$. The function $F$ is assumed uniformly elliptic and $G$ is assumed to depend (in a nonvacuous manner) on $Du$. Continuity estimates are proved for first and second derivatives of $u$ under weak hypotheses for smoothness of $F$, $G$, and $\Omega$.

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