Stability of Unique Solvability in an Ill-Posed Dirichlet Problem

Suppose that $\Omega\subset\mathbb R^n$ is a compact domain with Lipschitz boundary $\partial\Omega$ which is the closure of its interior $\Omega_0$. Consider functions $\phi_i,\tau_i\colon\Omega\to\mathbb R$ belonging to the space $L_q(\Omega)$ for $q\in(1,+\infty]$ and a locally Holder mapping $F\colon\Omega\times\mathbb R\to\mathbb R$ such that $F(\,\cdot\,,0)\equiv0$ on $\Omega$. Consider two quasilinear inhomogeneous Dirichlet problems
$$ \begin{cases} \Delta u_i=F(x,u_i)+\phi_i(x) & \text{on $\Omega_0$}, \\ u=\tau_i & \text{on $\partial\Omega$}, \end{cases} \qquad i=1,2. $$
In this paper, we study the following problem: Under certain conditions on the function $F$ generally not ensuring either the uniqueness or the existence of solutions in these problems, estimate the deviation of the solutions $u_i$ (assuming that they exist) from each other in the uniform metric, using additional information about the solutions $u_i$ . Here we assume that the solutions are continuous, although their continuity is a consequence of the constraints imposed on $F$, $\tau_i$, $\phi_i$. For the additional information on the solutions $u_i$, $i=1,2$ we take their values on the grid; in particular, we show that if their values are identical on some finite grid, then these functions coincide on $\Omega$.