Stability of Unique Solvability in an Ill-Posed Dirichlet Problem

Suppose that Ω ? ∝ ⁿ is a compact domain with Lipschitz boundary ?Ω which is the closure of its interior Ω₀. Consider functions ? _i , τ _i : Ω→ ∝ belonging to the space L _q (Ω) for q ∈ (1, +∞] and a locally Holder mapping F: Ω × ∝ → ∝ such that F(·, 0) ≡ 0 on Ω. Consider two quasilinear inhomogeneous Dirichlet problems $$\left\{ {_{u = \tau _i \quad \quad \quad \quad \quad \quad \quad \,\operatorname{on} \partial \Omega ,}^{\Delta u_i = F(x,u_i ) + \varphi _i (x)\quad \operatorname{on} \Omega _0 ,} } \right.\quad \quad i = 1,2.$$ 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, τ_i, ?_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 Ω.