Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems
A computer-assisted method for proving the existence of a solution of the problem \(-\Delta U(x)+F(x,U(x),U_ x(x))=0\) \((x\in\Omega)\), \(B[U](x)=0\) \((x\in\partial\Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) \((n=2\) or 3) and the nonlinearity \(F\) is defined on \(\overline\Omega\times\mathbb{R}\times\mathbb{R}^ n\) with values \(F(x,y,z)\in\mathbb{R}\), is presented. It is assumed that \(F\) grows at most quadratically with respect to the gradient \(U_ x\) and with a mixed-type linear boundary operator \(B\). All kinds of monotonicity conditions on the nonlinearity \(F\), or inverse-positivity assumptions are avoided. Some numerical examples are reported finally.
nonlinear elliptic boundary value problems
scientific article; zbMATH DE number 86032
scientific article