On the existence of Feller semigroups with boundary conditions. II
Let \(D\) be a bounded domain of \(\mathbb{R}^n\), with \(C^\infty\) 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., \(f\in C(\overline D)\), \(0\leq f\leq 1\Rightarrow 0\leq T_tf\leq 1\). The infinitesimal generator \({\mathbf A}\), defined by \({\mathbf A}u= \lim_{t\downarrow 0} t^{-1}(T_tu- u)\), is known to be a diffusion operator \(A\), i.e., second-order degenerate elliptic differential operator with real coefficients, \[ Au(x)= \sum^N_{i,j= 1}a^{ij}(x) {\partial^2u\over \partial x_i\partial x_j} (x)+ \sum^N_{i= 1}b^i(x) {\partial u\over\partial x_i} (x)+ c(x)u(x), \] satisfying a second-order Ventcel' boundary condition \(L\). \(L\) contains terms corresponding to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon and the viscosity phenomenon, respectively. This paper is concerned with the converse problem: construction of a Feller semigroup \(T_t\) whose infinitesimal generator is characterized by some given diffusion operator \(A\) and Ventcel' boundary condition. \textit{P. Cattiaux} [Ann. Inst. Fourier 42, No. 3, 541-624 (1992; Zbl 0780.35023)] considered the problem for the non-characteristic case that \(\sum^N_{i,j= 1}a^{ij}(x) n_in_j> 0\) for all \(x\in\partial D\), where \(n= (n_1,\dots, n_N)\) is the unit interior normal to \(\partial D\) at \(x\). This paper considers the construction problem for the characteristic case. The author constructs a Feller semigroup corresponding to a transversal Ventcel' boundary condition that one of the reflection and viscosity phenomenon occurs at each point of \(\partial D\).
second-order degenerate elliptic differential operator with real coefficients
second-order Ventcel' boundary condition
scientific article; zbMATH DE number 764020