Elliptic problems with unbounded drift coefficients and Neumann boundary condition.
The paper under review concerns an elliptic problem with Neumann boundary conditions of the form \[ \begin{cases} \lambda \varphi-\frac{1}{2}\Delta \varphi+\langle F, D\varphi\rangle =f & \text{in} \; K^\circ\\ \frac{\partial\varphi}{\partial n}=0 & \text{on} \; \partial K, \end{cases} \tag{1} \] where \(\lambda>0\), \(K\) is a closed convex subset of \({\mathbb R}^d\) with \(C^2\) boundary \(\partial K\) and interior \(K^\circ\). The vector field \(F\colon {\mathbb R}^d\to {\mathbb R}^d\) satisfies certain conditions, but is allowed to be unbounded. The main result of the paper implies that there is a function \(\rho\in L^1(\mathbb R^d)\), with \(\rho=0\) on \({\mathbb R}^d\setminus K\), such that for each \(f\in L^2({\mathbb R}^d, \rho\, \text{d}x)\), equation (1) has a unique strong solution (in the Friedrichs sense) \(\varphi \in W^{1,2}(K, \rho\,\text{d}x)\). The measure \(\rho\,\text{d}x\) is actually the invariant measure of a translation semigroup associated with a certain stochastic variational inequality.
scientific article; zbMATH DE number 2004928