Feller semigroups generated by degenerate elliptic operators (Q1971723)

A \(C_0\)-semigroup \(e^{tA}\) on a space \(C(\overline D)\) is called a Feller semigroup if it is nonnegative and \(f\in C(\overline D)\), \(0\leq f(x)\leq 1\) and \(\overline D\Rightarrow 0\leq e^{tA}f(x)\leq 1\) on \(\overline D\). Let \(D\subseteq \mathbb{R}^N\), \(N\geq 2\), with smooth boundary \(\partial D\). Consider the degenerate elliptic operator \[ Au(x)= \sum^N_{i,j=1} a^{ij}(x) {\partial^2u\over\partial x_i\partial x_j}(x)+ \sum^N_{i=1} b^i(x){\partial u\over\partial x_i}(x)+ C(x) u(x) \] and define the sets \[ \Sigma_2= \Biggl\{x'\in \partial D: \sum^N_{i,j=1} a^{ij}(x') n_i n_j= 0,\;b(x')= 0\Biggr\}, \] \[ \Sigma_3= \Biggl\{x'\in \partial D: \sum^N_{i,j=1} a^{ij}(x') n_in_j> 0\Biggr\}, \] where \(\vec n= (n_1,n_2,\dots)\) is the unit interior normal to the \(\partial D\) at \(x'\) and \[ b(x')= \sum^N_{i=1} \Biggl(b^i(x')- \sum^N_{j=1} {\partial a^{ij}(x')\over\partial x_j}\Biggr) n_i,\quad x'\in \partial D. \] Theorem 1. Assume that \(\partial D= \Sigma_2\cup\Sigma_3\) and each \(\Sigma_2\) and \(\Sigma_3\) consists of finite number of connected hypersurfaces. Then there exists a Feller semigroup \(e^{tS}\) on \(C(\overline D)\) such that \(Su= Au\), \(u\in D(S)= \{u\in C(\overline D): Au\in C(\overline D)\), \(Au= 0\) on \(\partial D\}\). Theorem 2. Under conditions of Theorem 1 there exists a Feller semigroup \(e^{tS}\) on \(C_\pi(\overline D)= \{u\in C(\overline D): u\) is constant on \(\partial D\}\) such that \[ Su= Au,\quad u\in D(S)= \{u\in C_\pi(\overline D): Au\in C_\pi(\overline D),\;Au= 0\text{ on }\partial D\}. \] Theorem 3. Under conditions of Theorem 1 there exists a Feller semigroup \(e^{tS}\) on \(C_0(\overline D)= \{u\in C(\overline D): u= 0\) on \(\partial D\}\) such that \[ Su= Au,\quad u\in D(S)= \{u\in C_0(\overline D): Au\in C_0(\overline D)\}. \]