Related papers: Minimization of a fractional perimeter-Dirichlet i…
We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs…
In this paper we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider the displacement problem in which the…
We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the…
We prove existence and regularity of optimal shapes for the problem$$\min\Big\{P(\Omega)+\mathcal{G}(\Omega):\ \Omega\subset D,\ |\Omega|=m\Big\},$$where $P$ denotes the perimeter, $|\cdot|$ is the volume, and the functional $\mathcal{G}$…
The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity.…
We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…
In this manuscript we study the following optimization problem with volume constraint: \[ \min\left\{\frac{1}{p}\int_{\Omega} |\nabla v|^pdx- \int_{\partial \Omega} gv\,dS \colon v \in W^{1, p} \left(\Omega\right), \text{ and } |\{v>0\}|…
We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the…
We consider the optimization problem of minimizing $\int_{\mathbb{R}^n}|\nabla u|^2\,\mathrm{d}x$ with double obstacles $\phi\leq u\leq\psi$ a.e. in $D$ and a constraint on the volume of $\{u>0\}\setminus\overline{D}$, where…
We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ \Gamma $-limit. The minimizers of the limiting functional are closely related to minimizing…
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…
We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla u|)+\lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,G}(\Omega)$ with $u-\phi_0\in W_0^{1,G}(\Omega)$, for a given $\phi_0\geq 0$ and bounded.…
This paper deals with the eigenvalue problem for the operator $L=-\Delta -x\cdot \nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $\lambda_k$ of $L$ under a suitable…
Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers…
The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of…
This work presents a numerical study of the Dirichlet problem for the fractional Laplacian $(-\Delta)^s$ with $s\in(0,1)$ using Finite Element methods with non-standard bases. Classical approaches based on piece-wise linear basis yield…
When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$:…
We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$ u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0<s<1. $$ The problem is posed in $\{x\in\ren, t\in…
We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the…
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.…