Stability properties in some classes of second order partial differential equations (Q480879)

A unique solution of the problem \[ -\Delta u_1 =0 \text{ in } \Omega,\qquad u_1=g \text{ on }\partial \Omega, \] is given through the Green's formula such that \(\Omega\) is a Euclidean bounded set with a smooth boundary, \(\Delta\) is the Laplacian operator and \(g\) is a continuous function on \(\partial \Omega\), the boundary of \(\Omega\). For \(u\in C^2(\Omega)\cap C(\overline{\Omega})\) such that \(u=g\) on \(\partial{\Omega}\), the author shows that whenever \(\Delta u\) is bounded by \(h\), a positive function on \(\Omega\), then \(|u(x)-u_1(x)|\leq K_{\Omega,h}\) a constant depending on \(\Omega\) and on \(h\) (Theorem 3). Next, the author prolongs its result to \(\mathbb R^n\).