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Current contact: Irina Mitrea
The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.
Seick Kim, Yonsei University
Abstract: We consider an elliptic, double divergence form operator L*, which is the formal adjoint of the elliptic operator in non-divergence form L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L*u=0 and show that Schauder type estimates are available when the coefficients are of Dini mean oscillation and belong to certain function spaces. We will also discuss some applications and parabolic counterparts.
Brandon Sweeting, University of Alabama
Abstract: We discuss a kind of weak-type inequality for the Hardy-Littlewood maximal operator and Calderon-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior; in particular, as shown by Muckenhoupt and Wheeden, the class of weights characterizing such inequalities is strictly larger than $A_p$. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Rios for $p = 1$, both in the scalar and matrix weighted setting. I will also discuss recent work on the characterization of those weights for which these inequalities hold for the maximal operator.
Silvia Jimenez Bolanos, Colgate University
Abstract: First, we will discuss the periodic homogenization for a weakly coupled electroelastic system of a nonlinear electrostatic equation with an elastic equation enriched with electrostriction. Such coupling is employed to describe dielectric elastomers or deformable (elastic) dielectrics. We will show that the effective response of the system consists of a homogeneous dielectric elastomer described by a nonlinear weakly coupled system of PDEs whose coefficients depend on the coefficients of the original heterogeneous material, the geometry of the composite and the periodicity of the original microstructure. The approach developed here for this nonlinear problem allows obtaining an explicit corrector result for the homogenization of monotone operators with minimal regularity assumptions. Next, we will discuss the homogenization of high-contrast dielectric elastomer composites, The considered heterogeneous material consisting of an ambient material with inserted particles is described by a weakly coupled system of an electrostatic equation with an elastic equation enriched with electrostriction. It is assumed that particles gradually become rigid as the fine-scale parameter approaches zero. We will see that the effective response of this system entails a homogeneous dielectric elastomer, described by a weakly coupled system of PDEs. The coefficients of the homogenized equations are dependent on various factors, including the composite’s geometry, the original microstructure’s periodicity, and the coefficients characterizing the initial heterogeneous material. Particularly, these coefficients are significantly influenced by the high-contrast nature of the fine-scale problem’s coefficients. Consequently, as anticipated, the high-contrast coefficients of the original yield non-local effects in the homogenized response.
Katrina Morgan, Temple University
Abstract: A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.
Atilla Yilmaz, Temple University
Abstract: After giving a self-contained introduction to the homogenization of Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension, I will focus on the case where the Hamiltonian is nonconvex, and highlight some interesting differences between: (i) periodic vs. truly random media; (ii) dimension one vs. higher; and (iii) inviscid vs. viscous HJ equations. In particular, I will present a recent result (from joint work with E. Kosygina) on the loss of quasiconvexity which can happen only in the viscous case.
Cristian Rios, University of Calgary
Abstract: We will present an implementation of the Moser iteration method in a non-doubling geometry with applications to boundedness and continuity of solutions to elliptic equations in which the ellipticity degenerates to infinite order. This is the first implementation of the Moser iteration in such a degenerate setting, allowing us to obtain continuity of solutions when the right hand side is non-vanishing and admissible. This work is continuation of a project in which continuity was previously established via a De Giorgi iteration but only for vanishing right hand sides. The work is done in collaboration with Luda Korobenko, Eric Sawyer, and Ruipeng Shen.
Anna Zemlyanova, Kansas State University
Abstract: In this talk, we study a problem concerning a nano-sized material surface attached to the boundary of an elastic isotropic semi-plane. The material surface is modeled using the Steigmann-Ogden form of surface energy. The study of stationary points of the total elastic energy functional produces a boundary-value problem with non-classical boundary conditions. This problem is solved by using integral representations of stresses and displacements through certain unknown functions. With the help of these functions, the problem can be reduced to either a system of two singular integral equations or a single singular integral equation. The numerical solution of the system of singular integral equations is obtained by expanding each unknown function into a series based on Chebyshev polynomials. The accuracy of the numerical procedure is studied, and various numerical examples for different values of the surface energy parameters are considered.
Erich Stachura, Kennesaw State University
Abstract: I will discuss a basic model of passive intermodulation (PIM). PIM occurs when multiple signals are active in a passive device that exhibits a nonlinear response. It is known that certain nonlinearities (e.g. the electro-thermal effect) which are fundamental to electromagnetic wave interaction with matter should be accounted for. In this talk, I will discuss existence, uniqueness, and regularity of solutions to a simple model for PIM. This in particular includes a temperature dependent conductivity in Maxwell's equations, which themselves are coupled to a nonlinear heat equation. I will also discuss challenges related to a similar problem when the permittivity $\varepsilon$ also depends on temperature. This is joint work with Niklas Wellander and Elena Cherkaev.
Jacob Shapiro, University of Dayton
Abstract: We discuss recent results and work in progress on local energy decay for the acoustic wave equation in low regularity. The main challenge is to establish suitable control over the resolvent of the associated Helmholtz operator at both large and small frequencies. For large frequencies, we employ (after rescaling) a semiclassical Carleman estimate. Near zero frequency we obtain a resolvent expansion by perturbative methods. Both tools are sensitive to the decay of the coefficients near infinity.
Tiago Picon, University of Sao Paulo Ribeirao Prato
2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024