Related papers: Geometry and interior nodal sets of Steklov eigenf…

We study boundary properties of plurisubharmonic functions near real submanifolds of almost complex manifolds.

We investigate the interaction between systolic geometry and positive scalar curvature through spinorial methods. Our main theorem establishes an upper bound for the two-dimensional stable systole on certain high-dimensional manifolds with…

We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of…

We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for…

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic…

Let $\Omega$ be a bounded domain with $C^\infty$ boundary in an $n$-dimensional $C^\infty$ Riemannian manifold, and let $\varrho$ be a non-negative bounded function defined on $\partial \Omega$. It is well-known that for the biharmonic…

Given two planar, conformal, smooth open sets $\Omega$ and $\omega$, we prove the existence of a sequence of smooth sets $\Omega_n$ which geometrically converges to $\Omega$ and such that the (perimeter normalized) Steklov eigenvalues of…

We investigate the approximation of weighted integrals over $\mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$…

We obtain geometric estimates for the first eigenvalue and the fundamental tone of the p-laplacian on manifolds in terms of admissible vector fields. Also, we defined a new spectral invariant and we show its relation with the geometry of…

We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to…

The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on…

This monograph develops the theory of Besov spaces for abelian group actions on semifinite von Neumann algebras and then proves Peller criteria for traceclass properties of associated Hankel operators. This allows to extend known index…

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply…

We study the first nontrivial Steklov eigenvalue of perimeter-normalized regular \(N\)-gons and show that it is strictly increasing in \(N\). The proof mainly relies on an analytic framework that establishes a refined asymptotic expansion…

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schr\"odinger equations) on a compact, smooth Riemannian manifold,…

We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex…

We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand…

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal…

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic…

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…