Geodesic
Stability of Unique Solvability in an Ill-Posed Dirichlet Problem
Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 294-308.

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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 $\Omega$. 
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     author = {I. G. Tsar'kov},
     title = {Stability of {Unique} {Solvability} in an {Ill-Posed} {Dirichlet} {Problem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {294--308},
     publisher = {mathdoc},
     volume = {79},
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     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a12/}
}
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I. G. Tsar'kov. Stability of Unique Solvability in an Ill-Posed Dirichlet Problem. Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 294-308. http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a12/

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