Global continuity of variational solutions weakening the one-sided bounded slope condition

Thomas Stanin (University Salzburg)

25-Jan-2022, 17:00-18:00 (2 years ago)

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.

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Organizers: Simon Blatt*, Philipp Reiter*, Armin Schikorra*, Guofang Wang
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