I will present a joint work with Leonid Kovalev and Jani Onninen. The proofs are based on topological arguments (degree theory) and the properties of planar quasiconformal mappings. These new ideas apply well to inner variational equations of conformally invariant energy integrals; in particular, to the Hopf-Laplace equation for the Dirichlet integral.
In this talk, we use Carleman type techniques to address uniqueness of self-shrinkers of mean curvature flow with given asymptotic behaviors.
A theory of Sobolev inequalities in arbitrary open sets in $R^n$ is offered. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. This is a joint work with V. Maz'ya.
We will focus on vortex gradient fields of unit-length. The associated stream function solves the eikonal equation, more precisely it is the distance function to a point. We will prove a kinetic formulation characterizing such vector fields in any dimension.
We consider Euler equations on a fixed Lorentzian manifold. The fluid is initially supported on a compact domain and the boundary between the fluid and the vacuum is allowed to move. Imposing the so-called physical vacuum boundary condition, we will explain how to obtain a priori estimates for this problem. In particular, our functional framework allows us to track the regularity of the free boundary. This is joint work with S. Shkoller and J. Speck.
We analyze wave propagation in random media in the so-called paraxial regime, which is a special high-frequency regime in which the wave propagates along a privileged axis. We show by multiscale analysis how to reduce the problem to the Ito-Schrodinger stochastic partial differential equation. We also show how to close and solve the moment equations for the random wave field. Based on these results we propose to use correlation-based methods for imaging in complex media and consider two examples: virtual source imaging in seismology and ghost imaging in optics.
The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model, which generalizes the heat flow for harmonic maps into the $2$-sphere. In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in 2D.
In this talk, we consider two disparate questions involving wave equations: (1) how singularities of solutions of subconformal focusing nonlinear wave equations form, and (2) when solutions of (linear and nonlinear) wave equations are determined by their data at infinity. In particular, we will show how tools from solving the second problem - a new family of global nonlinear Carleman estimates - can be used to establish some new results regarding the first question. Previous theorems by Merle and Zaag have established both upper and lower bounds on the local H¹-norm near noncharacteristic blow-up points for subconformal focusing NLW. In our main result, we show that this H¹-norm cannot concentrate along past timelike cones emanating from the blow-up point, i.e., that a significant amount of the action must occur near the corresponding past null cones.
These are joint works with Spyros Alexakis.
We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are $L^1$-close. As an application, we address the global and local minimality of certain lamellar configurations.
