Georgia Institute of Technology College of Sciences

We discuss bilateral obstacle problems for fully nonlinear second order uniformly elliptic partial differential equations (PDE for short) with merely continuous obstacles. Obstacle problems arise not only in minimization of energy functionals under restriction by obstacles but also stopping time problems in stochastic optimal control theory. When the main PDE part is of divergence type, huge amount of works have been done. However, less is known when it is of non-divergence type. Recently, Duque showed that the Holder continuity of viscosity solutions of bilateral obstacle problems, whose PDE part is of non-divergence type, and obstacles are supposed to be Holder continuous. Our purpose is to extend his result to enable us to apply a much wider class of PDE. This is a joint work with Shota Tateyama (Tohoku University).

We give derivative estimates for solutions to divergence form elliptic equations with piecewise smooth coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.

I will discuss the limit shapes for local minimizers of the Alt-Caffarelli energy. Fine properties of the associated pinning intervals, continuity/discontinuity in the normal direction, determine the formation of facets in an associated quasi-static motion. The talk is partially based on joint work with Charles Smart.

The boundary value and mixed value problems for linear and nonlinear degenerate abstract elliptic and parabolic equations are studied. Linear problems involve some parameters. The uniform L_{p}-separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems. In this talk, the classes of boundary value problems (BVPs) and mixed value problems (MVPs) for regular and singular degenerate differential operator equations (DOEs) are considered. The main objective of the present talk is to discuss the maximal regularity properties of the BVP for the degenerate abstract elliptic and parabolic equation We prove that for f∈L_{p} the elliptic problem has a unique solution u∈ W_{p,α}² satisfying the uniform coercive estimate ∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)} where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be defined later. We also prove that the differential operator generated by this elliptic problem is R-positive and also is a generator of an analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for the corresponding degenerate parabolic problem. This fact is used to obtain the existence and uniqueness of maximal regular solution of the MVP for the nonlinear parabolic equation.