Consider an elliptic (hyperbolic) equation $A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = 0$ in a bounded open plane set $D$, with real-valued functions $A$, $B$, and $C$. Is it true that at least one nontrivial solution $u$ can be continuously extended to the closure $\bar D$?
(“Nontrivial” means nonlinear, $u(x,y) \ne ax + by + c$.)
No boundary conditions are imposed. The set $D$ is supposed to be sufficiently regular; for example, is bounded by a smooth non self-intersecting closed curve (or a finite number of such curves). The functions $A$, $B$, and $C$ are sufficiently regular; for example, are analytic in a neighborhood of $\bar D$. It is assumed that ellipticity (hyperbolicity) may be lost on the boundary of $D$; that is, $AC – B^2 > 0$ in $D$ and $\ge 0$ in $\bar D$ (or $AC – B^2 < 0$ in $D$ and $\le 0$ in $\bar D$).
