Dirichlet problems with discontinuous coefficients and Feller semigroups
Let \(\Omega \subset {\mathbb R}^n\), \(n \geq 3\), be a bounded domain with \(C^{1,1}\) boundary. The author considers a second-order, elliptic integro-differential operator \(W\) with real discontinuous coefficients of the form \[Wu(x)=Au(x)+Su(x),\ x\in \Omega, \] where \(A\) is uniformly elliptic differential operator \[Au(x) = \sum\limits_{i,j=1}^n a^{ij}(x)\frac{\partial^2 u}{\partial x_i \partial x_j} + \sum\limits_{i=1}^n b^i(x)\frac{\partial u}{\partial x_i} + c(x)u(x) \] and \(S\) is the integro-differential operator \[Su(x) = \int\limits_{{\mathbb R}^n\backslash \{0\}} \left(u(x+z)-u(x)-\sum\limits_{k=1}^n z_k\frac{\partial u}{\partial x_k}(x) \right) K(x,z)\,\mu(dz).\] It is assumed that \(a^{ij}=a^{ji}\), \(a^{ij}\in \mathrm{VMO} \cap L^\infty(\Omega)\), \(b^i\in L^\infty(\Omega)\), \(c\in L^\infty(\Omega)\), \(c \leq 0 \), \(0 \leq K\in L^\infty({\mathbb R}^n \times {\mathbb R}^n)\), and \(K(x,z) = 0\) if \(x\in \Omega\) and \(x+z \notin \overline{\Omega}\). \(\mu(dz)\) is a Radon measure which satisfies the moment condition \[\int\limits_{\{0 < |z| \leq 1\}} |z|^2 \mu(dz) + \int\limits_{\{|z| > 1\}} |z| \mu(dz) < \infty. \] For the non-homogeneous Dirichlet problem \(Wu=f\) in \(\Omega\) with boundary condition \(\gamma_0 u = \varphi\) on \(\partial\Omega\) it is proved an existence and uniqueness theorem in the framework of Sobolev and Besov spaces of \(L^p\) type. The second important result is the existence of a Feller semigroup on \(C_0(\overline{\Omega})\) corresponding to the operator \(W\) with Dirichlet boundary conditions. These results are generalizations of Bony-Courrège-Priouret theorems to the \(\mathrm{VMO}\) case. The proofs are based on real analysis techniques (Hille-Yosida theorem, maximum principle, etc.).
scientific article; zbMATH DE number 7217845
scientific article