Related papers: On the singular set of free interface in an optima…

We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian…

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…

In the optimal partial transport problem, one is asked to transport a fraction $0<m \leq \min\{||f||_{L^1}, ||g||_{L^1}\}$ of the mass of $f=f \chi_\Omega$ onto $g=g\chi_\Lambda$ while minimizing a transportation cost. If $f$ and $g$ are…

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Assume that $M$ has…

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds…

In this paper, we study the spectral problem on a compact Finsler manifold with or without boundary. More precisely, given a certain collection of sets in Sobolev space $H^{1,2}(M)$ and a dimension-like function, we can define a…

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice…

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, in…

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.

Aronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong unique continuation result for differential forms, satisfying a certain first order differential inequality, on Riemannian manifolds with empty boundary. The present paper…

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and…

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_\lambda$ be a Dirichlet Laplace eigenfunction in $\Omega$ with eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set…

Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S of Y has a stratification S^0\subset S^1\subset\...\subset S, where y\in S^k if no tangent…

On a closed weighted Riemannian manifold with nonnegative Bakry-\'{E}mery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger…

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full…

The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to…

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $\varphi \colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\lim\varphi$. In…

A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has…

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary…

We study the Dirichlet problem for functions whose graphs are spacelike hypersurfaces with prescribed curvature in the Minkowski space and we obtain some new interior second order estimates for admissible solutions to the corresponding…