Feller semigroups and degenerate elliptic operators with Wentzell boundary conditions (Q2717562)

Let \(D\) be a bounded domain in \(\mathbb{R}^N, \;N\geq 2\), with smooth boundary \(\partial D\). A strongly continuous semigroup \(\{T_t\}_{t\geq 0}\) of operators on \(C(\overline D)\) is called a Feller semigroup on \(\overline D\) if it is nonnegative and contractive on \(C(\overline D)\), i.e., \(f\in C(\overline D)\), \(0\leq f\leq 1\Rightarrow 0\leq T_tf\leq 1\). Let \(A\) be a degenerate elliptic differential operator NEWLINE\[NEWLINEAu(x)= \sum^N_{i,j= 1}a^{ij}(x) {\partial^2u\over \partial x_i\partial x_j} (x)+ \sum^N_{i= 1}b^i(x) {\partial u\over\partial x_i} (x)+ c(x)u(x),NEWLINE\]NEWLINE with real smooth coefficients, Ventsel' boundary condition and characteristic boundary. NEWLINENEWLINENEWLINEFor the operator \(A\) authors construct Feller semigroup corresponding to a diffusion phenomenon including absorption, reflection, viscosity, diffusion along the boundary and jump at each point of \(\partial D\). The construction is based on a functional analytic approach.