Bounding the area of submanifolds with prescribed boundary in terms of its curvature energy
Dienstag, 3.6.25, 16:15-17:45, Seminarraum 125
Given an (m-1)-dimensional, embedded, compact submanifold \(\Gamma\) in \(\mathbb{R}^n\), consider any compact, immersed m-dimensional submanifold whose boundary is exactly given by \(\Gamma\). In this talk, we show how the area of such an m-submanifold is controlled in terms of its curvature energy. The talk is based on joint work with Prof. Ernst Kuwert.
Sharp quantitative estimates of Struwe’s decomposition
Donnerstag, 5.6.25, 14:00-15:00, Seminarraum 125
Suppose \(u\in \dot{H}^1(\mathbb{R}^n)\). In a fundamental paper, Struwe proved that if \(u\geq 0\) and \(\|\Delta u+u^{\frac{n+2}{n-2}}\|_{H^{-1}}:=\Gamma(u)\to 0\) then \(dist(u,\mathcal{T})\to 0\), where \(dist(u,\mathcal{T})\) denotes the \(\dot{H}^1(\mathbb{R}^n)\)-distance of \(u\) from the manifold of sums of Talenti bubbles. In this talk, I will talk about a quantitative version of this Struwe’s decomposition. Precisely, we proved nonlinear quantitative estimates for dimension while Figalli-Glaudo proved a linear estimate for dimension . Furthermore, we showed that these estimates are sharp in the sense of the exponents are optimal. It is joint work with Liming Sun and Juncheng Wei.
Liouville type theorem for a class quasi-linear \(p\)-Laplace equation on the half space
Donnerstag, 5.6.25, 15:00-16:00, Seminarraum 125
In this talk, we will study the positive solutions of p-Laplacian with subcrtical exponent in the half space with the Neumann boundary condition. We deduce a Liouville Theorem via the method of vector field and integral by part motivated by Obata. This is a joint work with Xinan Ma and Yang Zhou.
TBA
Dienstag, 17.6.25, 16:15-17:45, Seminarraum 125