## 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 $$a\in W^1_{2,\text{loc}}(\Omega)$$.

### MSC:

 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