Consider a solution $f\in C^{ 2}(\Omega)$ of a prescribed mean curvature equation \[ { \rm div}\left(\frac{ \nabla f}{ \sqrt{ 1+|\nabla f|^{ 2}}}\right)=2H(x,f) \ \ \ \ { \rm in} \ \ \Omega, \] where $\Omega\subset \Real^{ 2}$ is a domain whose boundary has a corner at ${ \cal O}=(0,0)\in\partial\Omega.$ If $\sup_{ x\in\Omega} |f(x)|$ and $\sup_{ x\in\Omega} |H(x,f(x))|$ are both finite and $\Omega$ has a reentrant corner at ${ \cal O},$ then the radial limits of $f$ at ${ \cal O},$ \[ Rf(\theta) \myeq \lim_{ r\downarrow 0} f(r\cos(\theta),r\sin(\theta)), \] are shown to exist and to have a specific type of behavior, independent of the boundary behavior of $f$ on $\partial\Omega.$ If $\sup_{ x\in\Omega} |f(x)|$ and $\sup_{ x\in\Omega} |H(x,f(x))|$ are both finite and the trace of $f$ on one side has a limit at ${ \cal O},$ then the radial limits of $f$ at ${ \cal O}$ exist and have a specific type of behavior.
Categories: math.AP