Johns Hopkins

Title: Boundaries of boundary value problems.

Abstract. We will describe a novel approximation result about how solutions to the Dirichlet problem for second-order real elliptic PDEs with boundary data in $L^p$ on rough domains may be globally approximated, up to the boundary in a precise sense, by a family of solutions to certain corresponding inhomogeneous Poisson problems with null boundary data. This result is new even for the Laplacian and on the unit ball, but is shown in high geometric generality, as well as with minimal assumptions on the coefficients.

This approximation result was inspired by a duality result: we fully characterize the dual space to the space of functions whose Kenig-Pipher modified non-tangential maximal function lies in $L^p$, answering a question of Hytonen and Rosen. We show that the dual space consists of exactly two components, and each of them correspond to the Banach spaces in which the Dirichlet and Poisson problems are solved with control of the $L^p$ norm of the non-tangential maximal function. Moreover, one component is the weak-$*$ boundary of the other component. This relationship at the level of the data in fact “lifts” to the level of solution spaces to the boundary value problems, at least partially, allowing us to make the following statement precise: the Dirichlet problem lies on the boundary of the Poisson problem, in a certain topology.

These results may be thought of as deeper instances of the robust relationship between (singular) boundary value problems and the inhomogeneous problems recently investigated in the literature. We also consider the validity of analogues of the approximation result for the Neumann problem and the Dirichlet Regularity problem. The results in this talk are from joint works with Mihalis Mourgoglou and Martin Ulmer.

Professor Poggi's website: Website.

Title: Vector balancing: a PDE perspective 

Abstract: A classical problem emerging in combinatorics and analysis is that of vector balancing: we are given a family of n vectors in a given set of m-dimensional Euclidean space, say the unit ball. Then we are asked to find n signs such that when we add the vectors multiplied by the respective signs the resulting vector is as small as possible, most commonly in the maximum-norm. I will discuss how a two player game version of this problem can be related to flow by mean curvature, building up on ideas of Spencer and of Kohn and Serfaty. This characterization allows us to characterize the long time asymptotics of the game in terms of the extinction time for  the cube under mean curvature flow, from where the large n asymptotics can be computed. Joint work with Vlad Kobzar. 

Professor Guillen's Website: Website.

Title: Nonconcentration of eigenfunctions in microlocal Kakeya-Nikodym norms: a phase space approach

Abstract: Previous works of the speaker and Sogge showed the significance of nonconcentration of microlocal Kakeya-Nikodym averages towards improving L^p bounds on eigenfunctions of the Laplacian. These averages are formed by taking the L^2 norm of an eigenfunction when localized in phase space to a small, frequency-dependent tube about a geodesic segment via a pseudodifferential operator.  We will review previous results on these norms, which concerned manifolds with nonpositive sectional curvatures.   Then we consider more recent developments which instead center on the dynamics of the geodesic flow.

Professor Blair's:  Website.

Title: Vorticity in shock reflection problems and transport equations.

Abstract: In this talk, we discuss self-similar solutions to two-dimensional Riemann problems with transonic shocks, focusing on regular shock reflection. We begin with a brief review of known results on global self-similar solutions in the potential flow framework. We then discuss the extension to the isentropic Euler system, where vorticity plays a central role. Existence of global regular reflection solutions in this setting remains an outstanding open problem. We show that such solutions necessarily have low regularity and establish the existence, uniqueness, and stability of renormalized solutions to the associated vorticity transport equation in this low-regularity setting. These results follow as a special case of a more general analysis of transport equations without a time variable in domains of arbitrary dimension, allowing for vector fields with stagnation points.

Professor Feldman's Website.

Title: Well-posedness for Dispersive PDE with Combined Nonlinearities.

Abstract: We discuss local and global well-posedness for the 1D NLS with combined nonlinearities in a weighted Sobolev space, scattering for initial data with a quadratic phase, and discuss a possible threshold for the global vs finite time existence. We then comment on how the same quadratic phase can lead to blow up solutions by showing numerical examples for exponential type nonlinearities which motivate analytical results. Finally, we emphasize the flexibility of the weighted space technique by presenting higher dimensional studies and extensions to other dispersive PDE such as the nonlinear Klein-Gordon equation. 

Alex's Website.

Title: Polynomial bounds for the size of the critical set of harmonic functions

Abstract: It is well-known that the complexity of a solution to an elliptic PDE is intricately related to the spatial growth rate of the solution. One instance of this phenomenon is the fact that the size of the zero set of a solution is bounded by a power of the solution's 'doubling index', a global quantity measuring growth. Obtaining analogous bounds for the size of the critical set turns out to be a much more delicate problem, and the best currently available bounds grow superexponentially in the doubling index. We will explore the reasons behind this difficulty and sketch a proof of a polynomial bound for the critical set of harmonic functions on a real-analytic manifold. This is joint work (in progress) with Benjamin Foster, Eugenia Malinnikova and Fedor Nazarov.

Josef's Website.

Title: Fine properties of free boundaries in the supercooled Stefan problem

Abstract: The supercooled Stefan problem describes the freezing of water below its freezing temperature. Unlike melting, this process can develop highly irregular interfaces, with spontaneous nucleation and fractal freezing patterns. I will discuss the construction of such evolutions, followed by free boundary regularity results for maximal solutions. The analysis combines stochastic optimization and PDE methods. This talk is based on joint works with Raymond Chu, Max Engelstein, and Inwon Kim.

Prof. Munoz's Website.

TBD: Website.

TBD: Website.