On construction of Martin boundaries for second order elliptic equations
A generalized version of the classical Martin theorem says that any positive solution of a second order linear elliptic equation in a domain of a Riemannian manifold is represented uniquely by an integral of the Martin kernel over the Martin boundary with respect to a finite positive Borel measure which is zero at the non-minimal Martin boundary. This paper is concerned with positive solutions of a second order elliptic equation \[ (1)\quad \sum^{n}_{i,j=1}- \frac{1}{\sqrt{g}}\frac{\partial}{\partial x_ i}(\sqrt{g}a^{ij}\frac{\partial u}{\partial x_ j})+ \] \[ +\sum^{n}_{i=1}(b^ i\frac{\partial u}{\partial x_ i}- \frac{1}{\sqrt{g}}\frac{\partial}{\partial x_ i}(\sqrt{g}d^ iu))+cu=0 \] in a domain of a non-compact Riemannian manifold, where \(g=\det (g_{ij})\) with \((g_{ij})\) being the Riemannian metric, \((a^{ij})\) is positive definite and locally bounded, \((b^ i)\) and \((d^ i)\) are locally p-th integrable for some \(p>n\), and c is locally max(p/2,1)-th integrable. The aim of this paper is to establish two methods (decomposition methods) to construct the Martin boundary and Martin kernel for such an equation as (1), and to explicitly construct Martin boundaries and Martin kernels for equations in unbounded domains of \(R^ n\) by applying the decomposition methods to the equations. The first method (tensor product decomposition method) is concerned with decomposition of an operator and a domain into a sum of tensor products of operators and a direct product of domains; and the second one (direct sum decomposition method) is concerned with decomposition of a domain into the union of a finite number of ends and a relatively compact set.
scientific article; zbMATH DE number 4198435
scientific article