Elliptic measure with a lower dimensional boundary.
Abstract: We'll try to discuss work in progress with Max Engelstein, Joseph Feneuil, and Svitlana Mayboroda. We consider a domain in $R^n$ bounded by an Ahlfors regular set $E$, typically of dimension smaller than $n$, and want to study the elliptic measure associated to some degenerate elliptic operators $L$. Under rather weak size conditions on the coefficients of $L$ (how fast do they tend to infinity near $E$), there is a reasonable elliptic measure. Then we study the regularity of this measure (typically, its absolute continuity with respect to Hausdorff measure), in terms of the geometry of $E$ and the regularity of the coefficients. So far we have some extensions of the celebrated Dahlberg theorem, and hope to study the converse.
The strong cosmic censorship conjecture in general relativity
Abstract: The strong cosmic censorship conjecture is a fundamental conjecture in general relativity. It posits global uniqueness for solutions to the Einstein equations. In particular, certain failure of determinism exhibited in some explicit black hole solutions are expected to be non-generic. In the first lecture, I will explain the motivation and the formulation of the conjecture, as well as some recent progress. In the second lecture, I will discuss some techniques to understand singular solutions to the Einstein equations, and explain their relevance to the strong cosmic censorship conjecture.
Two questions of Landis and their applications
Abstract: We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on propagation of smallness from sets of positive measure, we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint works with A. Logunov, N. Nadirashvili, and F. Nazarov. --
Talk I: On Gluing Method I: Type II Blow-up for Fujita Equation in Matano-Merle Regime
Talk II: On Gluing Method II: Second Order Estimates of Allen-Cahn Equation
Abstract: These talks are concerned with applications of recently developed gluing methods. Gluing methods have been used widely in nonlinear elliptic equation, e.g. counterexamples of De Giorgi's Conjecture. In these series of talks, I will discuss new developments in the gluing methods. In the first talk, I will use the parabolic gluing method to construct various Type II blow-up solutions for nonlinear Fujita heat equation with exponent belonging to the Matano-Merle regime. Matano-Merle have shown that in the radially symmetric case all blow-ups are Type I. In the first example we prove a geometry-driven Type II blow-up in a non-convex domain. The second example will be Type II blow-up with thin cylinderical tubes with self-similar size when the exponent is 3 and dimension is greater than 5.
In the second talk, I will show how to use the reverse process of infinite dimensional gluing method to analyze the collapsing interfaces for Allen-Cahn equation. This corresponds to minimal surfaces with higher multiplicity. We will prove that curvature decaying and second order estimates and multiplicity one for stable solutions of Allen-Cahn in dimensions less than 11. To this end, we use the reverse gluing method to show that the obstruction of these estimates is the existence of Toda system. As a consequence we prove that finite Morse index solutions of two-dimensional Allen-Cahn implies finite ends.