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Oblique derivative problems and invariant measures. (English) Zbl 0635.35020
The present paper is devoted to the study of some second-order oblique derivative problems for an operator A not in the divergence form and for a boundary operator B with Hölder continuous coefficients.
One major purpose is to describe the limiting behaviour of the solutions \(u_{\lambda}\) of the unilateral problem: \[ Max[u_{\lambda}- \psi;\quad Au_{\lambda}+\lambda u_{\lambda}-f]=0\quad in\quad \Omega;\quad Bu_{\lambda}=0\quad on\quad \Gamma, \] as the positive parameter \(\lambda\) tends to zero. This is motivated by the control theory of stochastic processes. The above boundary value problem actually comprises t0) of the equation \(-\Delta u+u^ p=0\), \(u\geq 0\) on \(B_ R\setminus 0\), the dimension of the underlying space being N. When \(1<p<N/(N-2)\) there are solutions with a singularity at 0. The presented paper shows that if non-negative smooth Dirichlet boundary data are prescribed there is a unique solution which is \(C^ 2\) on the closure of the domain. Moreover a unique solution can be specified satisfying the boundary conditions by giving an asymptotic form for the singularity at 0. The uniqueness statement as a consequence of the imposition of boundary data extends work of Veron. In addition the present paper removes the restriction to a ball and obtains the analogous results for any smooth domain.
Reviewer: J.F.Toland

MSC:
35J25 Boundary value problems for second-order elliptic equations
93E20 Optimal stochastic control
60G40 Stopping times; optimal stopping problems; gambling theory
35B65 Smoothness and regularity of solutions to PDEs
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