Schedule

abstract: I am going to talk about some recent gradient estimates for solutions to variational problems involving different classes of non-uniformly elliptic operators.

Fri, 2:40-3:20pm: Fernando Charro - The Monge-Ampère equation: Classical local applications and recent nonlocal developments

abstract: In this talk we will present the classical local Monge-Ampère equation and some of its applications to optimal transport and differential

geometry. We will discuss the degeneracy of the equation and the challenges it poses for regularity of solutions. Finally, we will consider a

nonlocal analogue of the Monge-Ampere operator, recently introduced in a joint work with Luis Caffarelli.

Fri, 3:20-4:40pm: Coffee Break - Departmental Colloquium: J. Douglas Wright (Drexel)

Department of Mathematics, Colloquium 3:30-4:30 in Thack 704

Fri, 4:40-5:20pm: Bianca Stroffolini - Minimizers of a Landau-de Gennes Energy with a Subquadratic Elastic Energy

abstract: I will present a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in

the gradient. In fact, there is little experimental evidence to conjecture that the elastic energy density may be subquadratic near defects matched by a

quadratic growth away from defects.

The analysis of the behaviour of global minimizers in two and three dimensional domains, subject to uniaxial boundary conditions, in the asymptotic

regime, is performed using tools of the regularity theory for functionals with general growth.

The results presented in this talk have been obtained in collaboration with Giacomo Canevari (Verona) and Apala Majumdar (Bath).

Fri, 5:20-6:00pm: Alexis Vasseur - Holder regularity up to the boundary for SQG on bounded domains

abstract: We consider the dissipative SQG equation in bounded domains first introduced by Constantin and Ignatova. We show global Holder regularity up to the boundary of the solution. The method is based on the De Giorgi techniques. The boundary introduces several difficulties, as the lack of translation invariance of the Laplacian operator, or the fact that the gradient does not commute with the Dirichlet Laplacian. This is a joint work with Logan Stokols.

Dinner

Saturday, Oct 12 (Thack 704)

Sat, 8:30-9:10am: Ovidiu Savin - The singular set in the fully nonlinear obstacle problem

abstract: For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\eps}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

Sat, 9:10-9:50am: Rodrigo Bañuelos - Littlewood-Paley for Levy processes

abstract: $L^p$ inequalities for Littlewood-Paley functionals arising from L\’evy processes will be discussed. These are motivated by applications to the $L^p$ boundedness of Fourier multiples similar to those for the classical Riesz transforms. The relevant Fourier multiples have been studied using the sharp martingale inequalities of Burkholder. The proofs here, although not sharp, are elementary and use nothing more than It\^o’s formula for processes with jumps. Based on work with Daesung Kim.

Sat, 9:50-10:20pm: Coffee Break

Sat, 10:20-11:00am: Robin Neumayer - Anisotropic liquid drop models

abstract: Anisotropic surface energies are a natural generalization of the perimeter functional that arise, for instance, in scaling limits for certain probabilistic models on lattices. Smoothness and ellipticity assumptions are sometimes imposed on the energy. These assumptions improve certain analytic aspects of associated isoperimetric problems, but are not always reasonable for applications or checkable when the problem comes from a scaling limit. We consider an anisotropic variant of a model for atomic nuclei and show that minimizers behave in a fundamentally different way dependent on whether or not the energy is smooth and elliptic. This is joint work with Choksi and Topaloglu.

Sat, 11:00-11:40am: Jessica Lin - Stochastic Homogenization for Nondivergence Form Equations: Error Estimates and Regularity Theory

abstract: I will present an overview of quantitative stochastic homogenization and its connection to regularity theory for nondivergence form elliptic equations. The key to quantitative stochastic homogenization is to establish a large-scale, coarsened regularity theory which holds with high probability. Coupled with classical probabilistic tools, this allows one to obtain optimal error estimates on the approximate correctors, which can then be used to analyze the full Dirichlet problem. This talk is based on joint work with Scott Armstrong.

Sat, 11:40-1:50pm: lunch

Sat, 1:50pm-2:30pm: Christine Breiner - Harmonic maps to metric spaces

abstract: We consider harmonic maps from a compact Riemann surface to a metric space with upper curvature bounds in the sense of Alexandrov. We will discuss existence and compactness theory. We will then prove an analogue of the Measurable Riemann Mapping Theorem for the singular setting.

Sat, 2:30pm-3:10pm: Pablo Stinga - On nonlocal Monge-Ampere equations

abstract: We report on recent progress on some nonlocal Monge--Amp\`ere equations. Our results include the interior Harnack inequality for the fractional linearized Monge--Amp\`ere equation (with Diego Maldonado from Kansas State University) and the regularity for the obstacle problem for the Caffarelli--Charro fractional Monge--Amp\`ere equation (with Yash Jhaveri from Institute for Advanced Study).

Sat, 3:10pm-3:40pm: Coffee Break

Sat, 3:40pm-4:20pm: Russell Schwab - Analyzing a special case of the Hele-Shaw flow using integro-differential operators

abstract: We will demonstrate that the solutions of an evolutionary free boundary problem, called Hele-Shaw, can be analyzed by an equivalent fractional parabolic equation in one fewer dimensions (a nonlinear version of the fractional (1/2)-heat equation). In order to obtain useful results from the parabolic interpretation of the problem, one must have precise information about the drift vectors and Levy measures that characterize the fractional operator that drives the equation. We will explain what happens in this context in the case of Hele-Shaw with an initial boundary that is the graph of a function whose derivative enjoys a Dini modulus of continuity. This is ongoing and joint work with Farhan Abedin.

Sat, 4:20pm-5:00pm: Ali Hyder - Blow-up phenomena to a higher order Liouville equation

abstract: Contrary to the two dimensional situation where blow-up occurs only on a finite set, in an open Euclidean domain of dimension four or higher it is possible to have blow-up on larger sets. It can be written as a union of a finite set and the zero set of a poly-harmonic function. I will talk about the role of the zero set in quantization of energy.

Sat, 5:00pm-6:00pm: Poster Session

Posters:

Animesh Biswas (Iowa State) - Regularity Theory for non-local space-time Master equation
Stefano Buccheri (University of Brasilia) - Large solutions to quasilinear problems involving the p–laplacian as p diverges
Luis Carlos Garcia Lirola (Kent State) - Volume product and Lipschitz-free Banach spaces
Antoine Remond-Tiedrez (CMU) - Viscous surface waves with generalized surface energies
Michael Roysdon (Kent State) - On Rogers-Shephard type inequalities for geneal measures
Mary Vaughan (Iowa State) - Fractional derivatives and Laplacians in one and two-sided weighted Sobolev spaces
Difan Yuan (Chinese Academy of Sciences) - Global Entropy Solutions to the Gas Flow in General Nozzle

Sunday, Oct 13 (Thack 704)

Sun, 8:30am-9:10am: Tadele Mengesha - A fractional Korn-type inequality

abstract: I will present a fractional version of the classical Korn’s inequality. The inequality allows us to characterize fractional Sobolev spaces of vector fields via a norm that involves only the measure of the magnitude of projected difference quotients.

The result is used to describe the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the characterization permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability. This is based on ongoing and joint work with James M. Scott.

Sun, 9:10am-9:50am: Henrik Schumacher - Gradient Flows for the Möbius Energy

abstract: Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to certain fractional-order Sobolev scalar products that are adapted to the Möbius energy. In contrast to $L^2$-gradient flows, the resulting flows are ordinary differential equations on an infinite-dimensional manifold of embedded curves. In the fully discrete setting, this allows us to completely decouple the time step size from the spatial discretization, resulting in a very robust optimization algorithm that is orders of magnitude faster than following the discrete $L^2$-gradient flow.

Based on joint work with Philipp Reiter.

Sun, 9:50am-10:20am: Coffee Break

Sun, 10:20am-11:00am: Julio D. Rossi - Convex functions

abstract: In this talk we deal with the notion of convexity in different ambient spaces (including $\mathbb{R}^N$ and trees). Our main goal is to show how this notion (convexity) depends on two things: what is a "segment" and what is a "mean value". We will also introduce a probabilistic interpretation of convexity.

Based on joint works with P. Blanc, L. Del Pezzo, N. Frevenza and C. Esteve.