Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

TBA

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.

We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to  nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.