Dual semigroups and second order linear elliptic boundary value problems (Q791027)

It is shown that second order elliptic boundary value problems on a bounded domain \(\Omega\) of \({\mathbb{R}}^ n\) generate analytic semigroups in \(L_ 1(\Omega)\). The proof is based on Phillip's theory of dual semigroups, which allows to consider \(L_ 1(\Omega)\) as a ''reflexive'' space with ''dual'' \(C({\bar \Omega})\). There are also given several sharp growth estimates for the corresponding semigroups in \(L_ p\), \(1\leq p<\infty\), as well as necessary and sufficient conditions that the semigroups to be contraction semigroups in every \(L_ p\), \(1\leq p<\infty\). Moreover the questions of positive resolvents (''maximum principle'') and the existence of eigenvalues with positive eigenfunctions are studied in great generality.