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In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$…

We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous works of…

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on…

In this paper we discuss the obstacle problem for the $p$-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising…

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 any dimension $N \geq 1$, for given mass $m > 0$ and for the $C^1$ energy functional \begin{equation*} I(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}F(u)dx, \end{equation*} we revisit the classical problem of…

In the previous work [Interfaces Free Bound., 19, 351-369, 2017], de Queiroz and Shahgholian investigated the regularity of the solution to the obstacle problem with singular logarithmic forcing term \begin{equation*} -\Delta u = \log u \,…

We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $\Omega$}, & U|_{\partial\Omega}&=\phi. \end{eqnarray*} Here $\Omega$…

Let $\Omega $ be a bounded domain in $\mathbb{R}^{d}$ $\left( d\geq 2\right) $ pretty regular. We solve the variational Dirichlet problem for a class of quasi-linear elliptic systems.

We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function nondecreasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This includes, in…

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + \Omega(1/\sqrt{D})$ fraction of constraints, where…

We study the question whether Lipschitz minimizers of $\int F(\nabla u)\,dx$ in $\mathbb{R}^n$ are $C^1$ when $F$ is strictly convex. Building on work of De Silva-Savin, we confirm the $C^1$ regularity when $D^2F$ is positive and bounded…

We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective…

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…

We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is…

As a starting point of our research, we show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to…

We consider the minimization of an average distance functional defined on a two-dimensional domain $\Omega$ with an Euler elastica penalization associated with $\pd \Omega$, the boundary of $\Omega$. The average distance is given by…

We prove that, for any measurable phase space subset $\Omega\subset\mathbb{R}^{2d}$ with $0<|\Omega|<\infty$ and any $1\le p < \infty$, the nonlinear concentration problem $$ \sup_{f \in…