A priori estimates for second order elliptic equations and their applications

We study the equation (λ+H)u=f whereH is a self-adjoint operator associated with the Dirichlet form inL ²(IR ^d ,pdx). A priori estimates of the first and the second order derivatives of solutions are obtained under minimal restrictions on the coefficients of the operator and measure. As a consequence we give a criterion of the essential self-adjointness of the operatorH ↾C ₀ ^∞ (IR ^d ) with non-smooth coefficients. Recipient of a Dov Biegun Postdoctoral Fellowship.