Explicit \(H_ 2\)-estimates and pointwise bounds for solutions of second- order elliptic boundary value problems (Q1193187)

Let \(L\) be a second-order linear elliptic operator, \(B\) a linear boundary operator. If some conditions are fulfilled, then the \(H_ 2\)-norm and \(C_ 0\)-norm of a function \(u\in H_ 2(\Omega)\) can be estimated in terms of \(Lu\), \(Bu\) and some positive constants. The main goal of the present article is the computation of these constants when the dimension of the space is 2 or 3. Using the established estimates for linear operators, the author studies nonlinear boundary-value problems of the form \(-\Delta u+F(\cdot,u)=0\) in \(\Omega\), \(Bu=s\) on \(\partial\Omega\), provided that approximate solutions \(\omega\in H_ 2(\Omega)\) (with some given properties) can be computed.