Related papers: Extremal metrics for Laplace eigenvalues in pertur…

We consider sensitivity of a semidefinite program under perturbations in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. When the coefficient matrices are perturbed, the optimal values can…

This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight…

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into…

The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their…

We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal…

The existence of extremal functions for the Sobolev trace inequalities is studied using the concentration compactness theorem. The conjectured extremal, the function of conformal factor, is considered and is proved to be an actual extremal…

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional…

We consider extremal eigenvalues of sparse random matrices, a class of random matrices including the adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs $\mathcal{G}(N,p)$. Recently, it was shown that the leading order fluctuations of extremal…

Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing with time dependent problems, but…

We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed…

Given two Riemann surfaces with boundary and a homotopy class of topological embeddings between them, there is a conformal embedding in the homotopy class if and only if the extremal length of every simple multi-curve is decreased under the…

We introduce an explicit description of the $\Phi^4_3$ measure on a bounded domain. Our starting point is the interpretation of its Laplace transform as the value function of a stochastic optimal control problem along the flow of a scale…

The normalized eigenvalues $\Lambda_i(M,g)$ of the Laplace-Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics…

We consider epimorphisms and several variant notions -- split, effective, regular, strong, and extremal -- and determine which of these coincide in the metric coarse and coarsely Lipschitz categories. In particular, we characterise extremal…

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to…

We prove a regularity result for unit volume conformal metrics with integral scalar curvature bounds for $p>n/2$ and first eigenvalue of $\Delta$ bounded from below by a constant $B > \Lambda_1(S^n,[g_{st.}]).$

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order…

A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Tur\'an extremal problems. A slight variation of the extremal problem is introduced and the…

We discuss conformal deformation and warped products on some open manifolds. We discuss how these can be applied to construct Riemannian metrics with specific scalar curvature functions.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$. If $s$ is a positive integer satisfying $2s<n$, we let $P_g^s$ be the GJMS operator of order $2s$ in $M$. We investigate in this paper the extremal values taken by fixed…