Related papers: A relaxation result in the vectorial setting and $…

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation \[ f(x)-af(bx)=g(x), \] where $a\in\mathbb{R}$, $b\in\mathbb{R} \setminus…

We study higher regularity for weak solutions of the $p$-Laplace equation $-\Delta_p u = f$ in a domain $\Omega \subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and H\"older…

We give a necessary and sufficient condition for non-local functionals on vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous. Here a non-local functional shall have the form of a double integral of a density which…

We consider degenerate nonautonomous energies $$ \int_\Omega f(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(\Omega, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption…

We investigate time-dependent optimization problems in fractional Sobolev spaces with the sparsity promoting $L^p$-pseudo norm for $0<p<1$ in the objective functional. In order to avoid computing the fractional Laplacian on the time-space…

In this paper, we introduce methods from convex optimization to solve the multimarginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of…

In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply…

In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought…

A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few…

For a fixed constant $\lambda > 0$ and a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type…

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function $W\colon\operatorname{SL}(2)\to\mathbb{R}$ with $W(RF)=W(FR)=W(F)$ for all $F\in\operatorname{SL}(2)$ and all…

We study spaces $\mathcal{CV}^{k}(\Omega,E)$ of $k$-times continuously partially differentiable functions on an open set $\Omega\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space…

In this note, we study the approximation of singular plurifinely plurisubharmonic function $u$ defined on a plurifinely domain $\Omega$. Under some conditions, we prove that $u$ can be approximated by an increasing sequence of…

Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, $E$ a separable real Banach space and $p\geq 1$. Given a sequence of functions $f, f_1, f_2,...$ from $T\times E$ to ${\bf R}$, under general assumptions, we prove that, for each…

This article is the continuation of the work [DK] where we had proved maximal estimates $$\left\|\sup_{t > 0} |m(tA)f| \right\|_{L^p(\Omega,Y)} \leq C \|f\|_{L^p(\Omega,Y)}$$ for sectorial operators $A$ acting on $L^p(\Omega,Y)$ ($Y$ being…

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the…

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{ in $\Omega$}, \\ \quad u>0 &\text{ in $\Omega$}, \\ \quad u=0 & \text{ on $\partial \Omega$}, \end{cases} \] where $L(v)=-{\rm…

We establish new approximation results in the sense of Lusin for Sobolev functions $f$ with $|\nabla f| \in L\log L$ on infinite-dimensional spaces equipped with Gaussian measures. The proof relies on some new pointwise estimate for the…

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, $\Omega$ satisfies the interior Corkscrew and Harnack chain conditions, which are respectively…

We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$…