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

In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n-1$. - The solution admits a $C^\infty$-expansion at all singular points, up to…

We study a random matching problem on closed compact $2$-dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume…

We study a class of Riemannian manifolds which are equipped with a singular metric. In particular we study a domain perturbation problem for the Dirichlet eigenvalues which depends on the best constant in the Hardy Inequality. However, we…

Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $\varphi$ (suitably smooth and decaying fast at…

In this paper, we prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraints. The focus is on the heat conduction problem studied by Aguilera, Caffarelli,…

On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein…

We show that in a typical sub-self-affine set, the Hausdorff and the Minkowski dimensions coincide and equal the zero of an appropriate topological pressure. This gives a partial positive answer to the question of Falconer. We also study…

We show that, on a $2$-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $L^2$-norm by identity plus the gradient of the solution to the Poisson problem $-\Delta…

In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous…

We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of…

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…

On any compact manifold of dimension $n\geq3$ with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface…

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers…

In this paper we study upper and lower bounds of the index and the nullity for sequences of harmonic maps with uniformly bounded Dirichlet energy from a two-dimensional Riemann surface into a compact target manifold. The main difficulty…

In this work we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null…

This article proves that the zero locus of a $\mathbb{Z}/2$ harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.

In this article we extend to generic $p$-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively…

Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and…

In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the…

In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the…