Titles and Abstracts
Artur H. O. Andrade
Title: Overdetermined elliptic boundary value problems in uniformly rectifiable domains
Abstract: This talk focuses on the analysis of overdetermined boundary value problems (OBVP). These problems model various physical phenomena and are characterized by the imposition of both Dirichlet and Neumann type boundary conditions.
Specifically, we examine the OBVP for second-order, homogeneous, constant complex coefficient, weakly elliptic systems in non-smooth domains, with boundary data in Whitney–Lebesgue spaces with integrability exponent in the interval $(1, \infty)$. Our analysis contains integral representation formulas, jump formulas, characterization of admissible boundary data, and the existence and uniqueness of solutions for OBVP in uniformly rectifiable domains.
This is a joint work with Irina Mitrea (Temple University), Dorina Mitrea, and Marius Mitrea (Baylor University).
Matthew Badger
Title: Nodal Domains of Homogeneous Caloric Polynomials
Abstract: With a view towards confirming the existence of singular strata in Mourgoglou and Puliatti's two-phase free boundary regularity theorem for caloric measure, we identify the minimum number of nodal domains of homogeneous caloric polynomials (hcps) in R^{n+1} of degree d. We also provide estimates on the maximum number of nodal domains for all n and d. I'll survey the techniques that go into the proofs of the theorems, particularly the construction of hcps that realize the minimum number of nodal domains. This is joint work with Cole Jeznach.
Nicole Buczkowski
Title: Aspects of Two Nonlocal Biharmonic Operators
Abstract: Nonlocal operators are advantageous in modeling due to their flexibility in handling discontinuities, incorporating nonlocal effects, and modeling a range of interactions through different choices for kernels. Using these operators in models has several applications, notably peridynamics (fracture mechanics). The biharmonic operator appears in many models including deformations of beams and plates. The nonlocal biharmonic operator can be formulated in at least two ways: using a fourth difference operator or iterating the nonlocal Laplacian. In this talk, we discuss various aspects of the two operators, including convergence of the operators to their classical counterparts and the well-posedness of the nonlocal clamped boundary value problem.
Antonio De Rosa
Title: Surfaces with prescribed Gaussian images and applications to anisotropic minimal surfaces.
Abstract: We construct $d$-dimensional polyhedral chains in $\mathbb R^n$ such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the prescribed measure is centered) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure is a simple $d$-vector). If the measure on the Grassmannian is supported on the set of positively oriented $d$--planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove that, for anisotropic integrands, polyconvexity is equivalent to quasiconvexity of the associated $Q$-integrands (that is, ellipticity for Lipschitz multigraphs) and to show that strict polyconvexity is necessary for the atomic condition to hold. Joint work with Y. Lei and R. Young.
Nickolas Edelen
Title: Improved regularity for minimizing capillary hypersurfaces
Abstract: We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least 4 in the surface, and this estimate improves if the capillary angle is close to 0, \pi/2, or \pi. For capillary angles that are close to 0 or \pi, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem. This is joint work with Otis Chodosh and Chao Li.
Max Engelstein
Title: The Robin problem on rough domains.
Abstract: Robin boundary conditions for elliptic operators model a diffusion contained by a semipermeable membrane (think oxygen being absorbed into the lung). Despite huge advances in understanding both the Neumann and Dirichlet problems in rough domains, the Robin problem is still mostly not understood.
We construct a ``Robin harmonic measure" for any elliptic operator in a broad class of domains and prove the surprising fact that this measure is mutually absolutely continuous with respect to surface measure, even when the boundary of the domain is fractal. Along the way we will also address some older conjectures about partially reflecting Brownian motion. If time allows, I will also discuss work in progress in which we obtain extremely fine estimates on the associated Greens function.
This is joint work with Guy David (Paris Saclay), Stefano Decio (IAS), Svitlana Mayboroda (ETH/UMN) and Marco Michetti (Paris Saclay).
Dennis Kriventsov
Title: Stationary solutions to the Bernoulli free boundary problem
Abstract: Free boundary problems of Bernoulli type arise naturally in fluid dynamics, thermal models, shape optimization, and other contexts. We will focus on the simplest possible archetype problem, and consider many examples of solutions in one and two dimensions. Then we will look at the question of which kinds of solutions are closed under taking limits, and how those limits look like–this is a topic of practical importance for constructing solutions by any argument save for direct minimization. Thanks to a recent breakthrough in joint work with Georg Weiss, we can now give a very precise and descriptive answer to such questions.
Christos Mantoulidis
Title: Improved generic regularity for minimizing hypersurfaces
Abstract: I will discuss recent and ongoing work with O. Chodosh,F. Schulze, and Z. Wang showing that minimizing hypersurfaceshave, after a suitable perturbation, a smaller than codimension-7singular set.
Thialita M Nascimento
Title: Regularity in nonlinear elliptic equations via geometric paths
Abstract: We investigate the smoothness properties of solutions to fully nonlinear elliptic equations, focusing on features embedded within specific components of the equations. These components conceal a regularizing effect on the solutions, which we uncover by treating them as “non-physical” free boundaries of the equation. This novel geometric approach enables us to achieve higher-order regularity of solutions in regions where the behavior of the solution is initially unknown.
Stefania Patrizi
Title: Derivation of discrete dislocation dynamics of multiple dislocation loops from the Peierls-Nabarro model
Abstract: We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls-Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations. After suitably rescaling the equation with a small phase parameter, the rescaled solution solves a fractional Allen--Cahn equation. We show that, as the parameter goes to, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.
Patrick Phelps
Title: Asymptotic properties and separation rates for local energy solutions to the Navier-Stokes equations
Abstract: We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show asymptotics for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes. Motivated by recent work on non-uniqueness, we investigate how non-uniqueness of the velocity field would evolve in time in the local energy class. Specifically, by extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. We conclude by extending this separation rate to solutions with no scaling assumption. Joint work with Zachary Bradshaw.
Aleksandr Reznikov
Title: Equilibrium measures for Gaussian kernels
Abstract: a classical problem of potential theory is to find the (unique) equilibrium measure for a given decent kernel. Such a measure minimizes the energy defined by this kernel or, equivalently, produces a potential that is bigger than a constant quasi-everywhere, and smaller than the same constant on the support of the measure. For the Riesz kernels, this potential solves a PDE with a free boundary.
In the talk we will discuss Gaussian kernels which are known to be important building blocks for Riesz and other important kernels; we will, in particular, derive some properties of supports of equilibrium measures depending on the domain where the problem is defined.
Claudia R. Silveira
Title: Optimization of heat insulation
Abstract: Effective thermal management is essential across various engineering domains, from enhancing building insulation to optimizing thermal regulation in electronic devices. This study presents a novel methodology by examining the mathematical foundations of an optimization challenge aimed at minimizing heat flow with a moment constraint. This constraint accounts for both the scale and intensity of thermal insulation, thus offering a deeper and more nuanced understanding of insulation effectiveness.
Pedro Takemura
Title: The Neumann Problem on Beurling-Hardy Spaces
Abstract: In this talk we introduce the class Beurling-Hardy spaces in the general geometric setting of Ahlfors regular sets, and develop a Calderón-Zygmund theory for singular integral operators on this scale of spaces. This opens the door for the treatment of the Neumann Problem for the weakly elliptic systems, with boundary data arbitrarily prescribed in Beurling-Hardy spaces, via the method of layer potentials. This is joint work with Marius Mitrea.
Eduardo Teixeira
Title: Free boundary problems with varying geometries
Abstract: Free boundary problems refer to mathematical models in which a PDE drives the system within an a priori unknown subregion, leading to sharp alterations in the parameters that describe the problem. These abrupt changes leave a geometric signature, encoded within the sharp asymptotic behavior of the solution near the free boundary.In this talk, I will describe a recent program designed to investigate free boundary models describing a continuum of varying geometries. This includes Bernoulli-type problems with unbounded jumps as well as variational models with oscillatory singularities. The ultimate goal is to understand how the point-by-point change in free boundary geometries impacts the local regularity of the free boundary. We show that, at least in a homogeneous medium, if the singularities vary in a 𝑊1,𝑛+ fashion, then the free boundary is locally the graph of function with Holder continuous gradient.
Yuming Paul Zhang
Title: Convergence of Free Boundaries in the Incompressible Limit of Tumor Growth Models
Abstract: In this work, we investigate the general Porous Medium Equations with drift and source terms that model tumor growth. While much literature focuses on the incompressible limit of solutions, we prove the convergence of free boundaries in Hausdorff distance in this limit. For this purpose, we establish uniform-in-$m$ strict propagation and stability properties of the free boundaries. As a further consequence, we provide an upper bound for the Hausdorff dimension of the free boundary and show that the limiting free boundary has finite $(d-1)$-dimensional Hausdorff measure. This is joint work with Jiajun Tong.
Aida Khajavirad
Title: On the power of linear programming for data clustering
Abstract: We propose a linear programming (LP) relaxation for K-means clustering. We derive sufficient conditions under which this LP relaxation is tight and subsequently obtain recovery guarantees under a popular stochastic model for the input data. We present extensive computational experiments on both synthetic and real data sets. This is joint work with Antonio De Rosa.