Global continuity of variational solutions weakening the one-sided bounded slope condition
Thomas Stanin (University Salzburg)
Abstract: In this talk, we have a look at regularity properties of variational solutions to a class of Cauchy-Dirichlet problems of the form
$$ \begin{cases} \partial_t u - \mathrm{div}_x (D_\xi f(Du)) = 0 & \text{in}\ \Omega_T, \\ u = u_0 & \text{on}\ \partial_\mathcal{P}\Omega_T. \end{cases} $$
We do not impose any growth conditions from above on $f \colon \R^n \to \R$ but require it to be convex and coercive. The domain $\Omega \subset \R^n$ is supposed to be bounded and convex and for the time-independent boundary datum $u_0 \colon \overline\Omega \to \R$, we require continuity. These assumptions on $u_0$ are weaker than a one-sided version of the bounded slope condition. We present a result showing variational solutions $u \colon \Omega_T \to \R$ to these problem class to be globally continuous.
mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control
Audience: researchers in the topic
Online Seminar "Geometric Analysis"
Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.
Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php
Organizers: | Simon Blatt*, Philipp Reiter*, Armin Schikorra*, Guofang Wang |
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