Stability of Unique Solvability in an Ill-Posed Dirichlet Problem

Abstract

Suppose that Ω ⊂ ∝n is a compact domain with Lipschitz boundary ∂Ω which is the closure of its interior Ω0. Consider functions ϕi, τi: Ω→ ∝ belonging to the space Lq(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 ui (assuming that they exist) from each other in the uniform metric, using additional information about the solutions ui. Here we assume that the solutions are continuous, although their continuity is a consequence of the constraints imposed on F, τi, ϕi. For the additional information on the solutions ui, 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.