Related papers: A sharp uniqueness result for a class of variation…

In this paper we prove that the shape optimization problem $$\min\left\{\lambda_k(\Omega):\ \Omega\subset\R^d,\ \Omega\ \hbox{open},\ P(\Omega)=1,\ |\Omega|<+\infty\right\},$$ has a solution for any $k\in\N$ and dimension $d$. Moreover,…

The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions.…

In this paper, we give a uniqueness result to a transport equation fulfilled by probability measure on a infinite dimensional Hilbert space. Main arguments are based on projective aspects and a probabilistic representation of the solutions.…

We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the…

We consider the functional $\int_\Omega g(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(\Omega)$ of the associated Dirichlet problem. We prove that, under…

Motivated by Gamow's liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by $P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ a nonlocal…

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main…

We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(\Omega)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is…

We obtain sufficient conditions for solutions of the $m$th-order differential inequality $$ \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, u) \ge f (x) g (|u|) \quad \mbox{in } B_1 \setminus \{ 0 \} $$ to have a removable singularity at…

Consider a real line equipped with a (not necessarily intrinsic) distance. We deal with the minimum-weight perfect matching problem for a complete graph whose points are located on the line and whose edges have weights equal to distances…

We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the…

We develop a variational approach to the minimization problem of functionals of the type $\frac12\left\lVert \nabla \phi \right\rVert^2_2 + \beta \left\lVert \phi \right\rVert_1$ constrained by $\left\lVert \phi \right\rVert_2 = 1$ which is…

In this paper, we develop the theory of Perelman's $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated…

We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = \log \frac{1}{\langle x, y \rangle}$. We calculate the variation of the corresponding Kantorovich functional…

This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal…

We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order…

We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values…

We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. These problems consist of minimizing a linear objective function over a high-dimensional polytope, here…

We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the…

We study the minimizers of a functional on the set of partitions of a domain $\Omega \subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $\Omega$, whose main term is $\sum_{j=1^N} \int_{\Omega \cap \partial W_j} a(x)…