Dirichlet Boundary Value Problems For Elliptic Partial Differential Equations With Singular Coefficients:a Probabilistic Approach

In this thesis,we prove that there exist unique weak solutions to the Dirichlet boundary value problems for two classes of second order elliptic operators with sin-gular coefficients.The first class is semilinear second order elliptic operators,which do not necessarily have the maximum principle and are non-symmetric in general.The second is linear second order elliptic operators whose coefficients are signed measures.And we also give a representation of the solution in this case.Our method is probabilis-tic.As far as we know,this is the first time to solve the boundary value problem with such a generality that the operator has a signed measure drift.To begin with,we consider the Dirichlet boundary value problems to semilinear second order elliptic differential equations.When the elliptic operator does not have singular coefficient,using the Dirichlet theory,we know that there exists a diffusion process associated with the elliptic operator.To solve the semilinear boundary value problem,also as an independent interest,we first need to consider a class of backward stochastic differential equations generated by the non-linear term in the boundary value problem.More specifically,the backward stochastic differential equations are driven by the martingale part of the diffusion process.And we will solve the backward stochastic differential equations with very singular coefficients and random terminal times.So we will produce a candidate to the solution by appealing to the theory of backward stochas-tic differential equations.It turns out that the classical L2 setting of backward stochastic differential equations is not suitable here.We have to work in the framework of L1 and deal with class(D)stochastic processes.When the elliptic operator has singular coef-ficients,we will use the W1,p-estimates of divergence form and h-transform method to tackle the singular term.In the second part,we investigate the Dirichlet boundary value problems to linear second order elliptic equations whose coefficients are signed measures,which belong to some Kato class.Since there is no Dirichlet form associated with this elliptic operator,the previous method using probabilistic approach to solve the boundary problems are not valid here.Especially,the Girsanov transformation and Kato-type inequality can not be used anymore.Our idea is as follows.When the boundary and the boundary function are smooth,we will approximate the coefficients by smooth functions through a sequence of mollifiers.We then study the solutions of the boundary value problems corresponding to the smooth coefficients.We show that the solutions and the gradients of the solutions of the approximating problems converge uniformly on compact subsets to the corresponding solution and its gradient.To this end,we employed heavily proba-bilistic techniques combined with the estimates and convergence of heat kernels of the killed processes.To improve conditions on coefficients and to drop the restrictions on the boundary,we will use the global and interior C1,α0-estimates we obtained for the solutions.The results on the Holder continuity of the heat kernel of the killed process and the results on the existence and uniqueness of continuous additive functionals are also of independent interest.