Wentzell boundary conditions in the context of Dirichlet forms. (English) Zbl 1040.47032

For the formal elliptic expression \(L=\nabla a\nabla\) on \(\Omega\subset{\mathbb R}^d\) with Wentzell boundary condition \(-\alpha Au +n\cdot a\nabla u +\gamma u=0\) on \(\Sigma\subset\Omega\) and Dirichlet boundary condition on \(\Omega\backslash\Sigma\) (\(\alpha,\gamma\) are suitable functions, \(n\) is the outward normal), the authors prove that the \(C_0\)-semigroup \(e^{-tA}\) generated by the operator realization \(A\) of \(L\) defined by the form method is positivity preserving and contractive on \(L_p\). A more detailed analysis of the one-dimensional case is included. The proof is based on the context of the Dirichlet form. Mass conservation is also under discussion. The appendix to the paper contains a new result concerning the closability of the maximal form associated with \(-\nabla a\nabla\), where \(a\in W^1_{2,\text{loc}}(\Omega)\).


47D06 One-parameter semigroups and linear evolution equations
31C25 Dirichlet forms
34B05 Linear boundary value problems for ordinary differential equations
35J25 Boundary value problems for second-order elliptic equations