The probabilistic solution of the third boundary value problem for second order elliptic equations

Abstract

Using standard reflected Brownian motion (SRBM) and martingales we define (in the spirit of Stroock and Varadhan-see [S-V]) the probabilistic solution of the boundary value problem $$\begin{gathered} \frac{1}{2}\Delta u + qu = 0, in D; \hfill \\ \frac{{\partial u}}{{\partial n}} + cu = - f,on\partial D, \hfill \\ \end{gathered} $$ whereD is a bounded domain withC 3 boundary andn is the inward unit oormal vector on ∂D. The assumptions forq, c andf are quite general. The corresponding Dirichlet problem was studied by Chung, Rao, Zhao and others (see [C-R1] and [Z-M]) and the corresponding Neumann by Pei Hsu in [H2]. Here we show that the probabilistic solution of our problem exists, is unique (unless we hit an eigenvalue), continuous on $$\bar D$$ and equivalent to the weak analytic solution. The method we use is to reduce the problem to an integral equation inD that involves the associated semigroup and, hence, to the study of the properties of this semigroup. In this way we do not have to assume that the spectrum is negative (almost every previous work on these probabilistic solutions makes this assumption). We construct the kernel of this semigroup and we prove certain estimates for it which help us to establish many other results, including the gauge theorem. We also show that, if the boundary functionc is continuous, our semigroup is a uniform limit of Neumann semigroups and, furthermore, that the Dirichlet semigroup is a uniform limit of semigroups of our type. Therefore the Dirichlet spectrum is a “monotone” limit of spectra of mixed problems (see Sect. 5B), a fact which is mentioned without proof in Vol 1, Ch. IV, Sect. 2 of theMethods of Mathematical Physics by Courant and Hilbert. This establishes the interrelation of the three boundary value problems. Finally, we add a drift term to our differential equation, which becomes $$\frac{1}{2}\Delta u + b \cdot \nabla u + qu = 0$$ and we solve the third boundary value problem for this equation probabilistically, with the help of Girsanov's transformation.