Asymptotic behavior of solutions of oblique derivative boundary value problems (Q583684)

We study the asymptotic properties of the oblique boundary value problem of the equation \((\epsilon^ 2L+V)u^{\epsilon}-qu^{\epsilon}=0\) on a bounded domain D; here L is a second order, uniformly elliptic operator, and the dynamic system determined by the vector field V has a unique equilibrium point inside D. The boundary condition is given by \(\partial \gamma /\partial n=f\) for a nontangential, outward pointing vector field \(\gamma\) and a nonnegative function f on the boundary \(\partial D\). We use ideas of large deviations from probability theory to prove that the limit \(\lim_{\epsilon \to 0}\epsilon^ 2u^{\epsilon}\) exists and describe the limit in terms of large deviation rate functions.