Johns Hopkins

Tittle: On uniqueness of excited states and related questions

Abstract: This talk will present the long-standing problem of excited states uniqueness for the nonlinear Schroedinger equation. We will describe the history of the problem, it's relevance to long-term dynamics of nonlinear wave equations, related spectral problems, and progress on the uniqueness question via rigorous numerics. The recent breakthrough by Moxun Tang, who found an analytical proof, will be discussed. 

Title:

Fundamental solution to the heat equation with a dynamical boundary condition

Abstract:

We give an explicit representation of the fundamental solution to the heat equation on a half-space with the homogeneous dynamical boundary condition, and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space with the homogeneous dynamical boundary condition. This talk is based on a joint work with Kazuhiro Ishige (University of Tokyo) and Sho Katayama (University of Tokyo).

Title: A Counterexample to the Mizohata-Takeuchi Conjecture

Abstract: We derive a family of Lp estimates of the X-Ray transform of positive measures in Rd, which we use to construct a logR-loss counterexample to the Mizohata-Takeuchi conjecture for every C2 hypersurface in Rd that does not lie in a hyperplane. In particular, multilinear restriction estimates at the endpoint cannot be sharpened directly by the Mizohata-Takeuchi conjecture

Title: Regularity theory for fractional elliptic equations in nondivergence form 

Abstract: We present recent advances in the regularity theory for solutions to fractional nonlocal equations driven by fractional powers of nondivergence form elliptic operators in bounded domains, where sharp regularity on the coefficients is assumed. The results include Harnack inequality, H\”older regularity of solutions and Schauder estimates. These are proved using the characterization of solutions by a degenerate/singular extension problem. The sharp form of the extension problem has been recently obtained in joint work with Animesh Biswas (Missouri State University). This is joint work with Mary Vaughan (Texas State University). 

Title: Critical sets of harmonic functions and Almgren's frequency function

Abstract: Given a harmonic function on a nice domain in n-dimensional Euclidean space, its critical set (the set of points where its gradient vanishes) is known to have dimension at most n-2, with locally finite (n-2)-dimensional Hausdorff measure. For harmonic polynomials, the Hausdorff measure can be bounded in terms of the degree of the polynomial. For more general harmonic functions, Almgren's frequency function serves as an analog of the degree of a polynomial, measuring local growth properties of the harmonic function.

Lin conjectured that for a harmonic function with frequency at most N, the (n-2)-dimensional Hausdorff measure of the critical set is (locally) at most CN^2. The first (and only) quantitative bound in this direction was due to Naber and Valtorta, who showed a bound of the form exp(CN^2). In this talk, I'll discuss an improvement of this bound to the nearly polynomial threshold N^(C log N) via a multiscale analysis, some linear algebra, and geometric combinatorics. This is joint work with Josep Gallegos and Eugenia Malinnikova.

Title: A degenerate one-phase free boundary problem arising from the Alt-Phillips equation for negative exponents 

Abstract: We study viscosity solutions for a degenerate one-phase free boundary problem of the form $\Delta w = \frac{h(\nabla w)}{w}$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This type of degenerate one-phase free boundary problem arises when a canonical transformation is performed to a semilinear equation $\Delta u = f(u)$, and $f$ morally behaves like $u^{-(\gamma + 1)}$ for some $\gamma > 0$. In this case, known as the Alt-Phillips equation for negative exponents, $h(\nabla u) = c(|\nabla u|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ degenerates to $2$, the free boundary becomes a minimal surface.  

Title: Rigidity of critical points of hydrophobic capillary functionals

Abstract: We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between 90◦ and 120◦. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the “tangential” part of the capillary boundary is H^n-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between 90◦ and 180◦. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds. This is joint work with R. Neumayer and R. Resende.

Title: Ricci Solitons and Hamiltonian Dynamics

Abstract: In this talk, we will discuss a specific connection between a Kähler Gradient Ricci soliton (GRS) and Hamiltonian dynamics. The former is a singularity model of the Kahler version of Ricci flows and a generalization of Einstein metrics while the latter comes from classical mechanics. In particular, we show that a Kähler GRS with a non-trivial potential function in complex dimension two is an integrable Hamiltonian system. This novel perspective directly exploits the non-triviality of the potential function, a distinctive feature of non-trivial GRS. Thus, it leads to strong rigidity which might not hold for the trivial case of Einstein metrics. For instance, we discuss a conjecture stating that a Kahler GRS in complex dimension two has toric symmetry and a positive confirmation under a generic condition. 

Title: Bounds for spectral projectors on the three-dimensional torus

Abstract: Periodic functions can be expanded in Fourier series, a linear combination of exponentials indexed by the integers. Despite the explicitness of the involved expressions, it is a difficult problem to take information about the frequency support of a function and deduce information on the physical side. We study the L^p norms of spectral projection operators on the torus, with special attention given to the three dimensional case, and prove sharp estimates for various values of p and sizes of the window. Our methods include harmonic analysis via decoupling, the geometry of numbers, exponential sum estimates, and bilinear techniques. This work is joint with Pierre Germain and Simon Myerson.

Title: On singular points in the supercooled Stefan problem

Abstract: The supercooled Stefan problem models how water, cooled below freezing, forms into ice. Experiments have shown that this process can exhibit nucleation (pieces of ice forming spontaneously) and that fractal-like patterns can emerge in the ice. This stands in sharp contrast with the melting problem, in which the ice is expected to be smooth at most points and times. This was recently demonstrated rigorously by Figalli, Ros Oton and Serra. 

In joint work with Inwon Kim (UCLA) and Sebastian Munoz (UCLA), we analyze a subclass of solutions to the supercooled problem which are connected to a parabolic obstacle-type problem. For this subclass we show that some of the experimentally observed behavior can occur (such as nucleation) but that the singular set cannot be completely pathological (i.e. we give bounds on the size of the singular points).