2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024
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
Atilla Yilmaz, Temple University
Cristian Rios, University of Calgary
Anna Zemlyanova, Kansas State University
Erich Stachura, Kennesaw State University
Jacob Shapiro, University of Dayton
Tiago Picon, University of Sao Paulo Ribeirao Prato
2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024