January 28, 2020
2:00PM - 2:50PM
Cockins Hall 240
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2020-01-28 15:00:00
2020-01-28 15:50:00
Analysis and Operator Theory Seminar- Hana Turcinova
Title: Characterization of Sobolev functions with zero traces via the distance function from the boundary
Speaker: Hana Turcinova - Charles University
Abstract: Consider a domain $\Omega \subset R^N$ having the outer cone property and let $d(x)=\operatorname{dist}(x,\partial\Omega)$. A classical result of late 1980's states that for $p\in (1,\infty)$ and $m \in N$, $u$ belongs to the Sobolev space $W^{m,p}_0(\Omega)$ if and only if $u/d^m\in L^p(\Omega)$ and $\left|\nabla^m u\right|\in L^p(\Omega)$. During the consequent decades, several authors have spent considerable effort in order to relax the characterizing condition. Recently, it was proved that $u\in W^{m,p}_0(\Omega)$ if and only if $u/d^m\in L^1(\Omega)$ and $\left|\nabla^m u\right|\in L^p(\Omega)$. We will present a new, yet more relaxed condition on the function $u/d$, expressed in terms of Lorentz spaces, which together with $\left|\nabla u\right|\in L^p(\Omega)$ still guarantees that $u\in W^{1,p}_0(\Omega)$. Moreover, we will investigate optimality of our condition.
Cockins Hall 240
OSU ASC Drupal 8
ascwebservices@osu.edu
America/New_York
public
Date Range
Add to Calendar
2020-01-28 14:00:00
2020-01-28 14:50:00
Analysis and Operator Theory Seminar- Hana Turcinova
Title: Characterization of Sobolev functions with zero traces via the distance function from the boundary
Speaker: Hana Turcinova - Charles University
Abstract: Consider a domain $\Omega \subset R^N$ having the outer cone property and let $d(x)=\operatorname{dist}(x,\partial\Omega)$. A classical result of late 1980's states that for $p\in (1,\infty)$ and $m \in N$, $u$ belongs to the Sobolev space $W^{m,p}_0(\Omega)$ if and only if $u/d^m\in L^p(\Omega)$ and $\left|\nabla^m u\right|\in L^p(\Omega)$. During the consequent decades, several authors have spent considerable effort in order to relax the characterizing condition. Recently, it was proved that $u\in W^{m,p}_0(\Omega)$ if and only if $u/d^m\in L^1(\Omega)$ and $\left|\nabla^m u\right|\in L^p(\Omega)$. We will present a new, yet more relaxed condition on the function $u/d$, expressed in terms of Lorentz spaces, which together with $\left|\nabla u\right|\in L^p(\Omega)$ still guarantees that $u\in W^{1,p}_0(\Omega)$. Moreover, we will investigate optimality of our condition.
Cockins Hall 240
Department of Mathematics
math@osu.edu
America/New_York
public
Title: Characterization of Sobolev functions with zero traces via the distance function from the boundary
Speaker: Hana Turcinova - Charles University
Abstract: Consider a domain $\Omega \subset R^N$ having the outer cone property and let $d(x)=\operatorname{dist}(x,\partial\Omega)$. A classical result of late 1980's states that for $p\in (1,\infty)$ and $m \in N$, $u$ belongs to the Sobolev space $W^{m,p}_0(\Omega)$ if and only if $u/d^m\in L^p(\Omega)$ and $\left|\nabla^m u\right|\in L^p(\Omega)$. During the consequent decades, several authors have spent considerable effort in order to relax the characterizing condition. Recently, it was proved that $u\in W^{m,p}_0(\Omega)$ if and only if $u/d^m\in L^1(\Omega)$ and $\left|\nabla^m u\right|\in L^p(\Omega)$. We will present a new, yet more relaxed condition on the function $u/d$, expressed in terms of Lorentz spaces, which together with $\left|\nabla u\right|\in L^p(\Omega)$ still guarantees that $u\in W^{1,p}_0(\Omega)$. Moreover, we will investigate optimality of our condition.
