Related papers: A uniqueness criterion and a counterexample to reg…

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact…

In this paper, we consider minimizers of integral functionals of the type \begin{equation*} \mathcal{F}(u):= \int_\Omega \dfrac{1}{p} \bigl( |Du(x)|_{\gamma(x)}-1\bigr)_+^p \ \mathrm{d}x, \end{equation*} for $p >1$, where $u : \Omega…

We consider $L^2$ minimizing geodesics along the group of volume preserving maps $SDiff(D)$ of a given 3-dimensional domain $D$. The corresponding curves describe the motion of an ideal incompressible fluid inside $D$ and are (formally)…

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals $\Phi(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\in C^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler than the…

Let $B_L$ the open ball in ${\bf R}^n$ centered at $0$, of radius $L$, and let $\phi$ be a homeomorphism from $B_L$ onto ${\bf R}^n$ such that $\phi(0)=0$ and $\phi=\nabla\Phi$, where the function $\Phi:\bar {B_L}\to ]-\infty,0]$ is…

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let $Y$ be a…

Let $J \in C(\mathbb{R})$, $J\ge 0$, $\int_{\tiny$\mathbb{R}$} J = 1$ and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where $f$…

We prove existence, uniqueness and initial time regularity for variational solutions to nonlocal total variation flows associated with image denoising and deblurring. In particular, we prove existence of parabolic minimisers $u$, that is,…

In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity.…

We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable…

We consider a nonlocal functional $J_K$ that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $u\colon \mathbb{R}^d \to \mathbb{R}$, we define $J_K(u)$ as the integral of weighted…

This paper proposes a universal algorithm for convex minimization problems of the composite form $g_0(x)+h(g_1(x),\dots, g_m(x)) + u(x)$. We allow each $g_j$ to independently range from being nonsmooth Lipschitz to smooth, from convex to…

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla \u|^2+2|\u|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta…

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing"…

Using a calibration method we prove that, if $\Gamma\subset \Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma$ and regular outside, then the function $u_{\beta}$ which solves $$ \begin{cases}…

We consider the following problem: minimize the functional $\int_\Omega f(\nabla u(x))\, dx$ in the class of concave functions $u: \Omega \to [0,M]$, where $\Omega \subset \mathbb{R}^2$ is a convex body and $M > 0$. If $f(x) = 1/(1 +…

We consider the functional $$J(v) = \int_\Omega [f(|\nabla v|) - v] dx,$$ where $\Omega$ is a bounded domain and $f:[0,+\infty)\to \mathbb{R}$ is a convex function vanishing for $s\in [0,\sigma]$, with $\sigma>0$. We prove that a minimizer…

We study the problem of minimizing the functional $$ I(\varphi)=\int\limits_{\Omega} W(x,D\varphi)\,dx $$ on a new class of mappings. We relax summability conditions for admissible deformations to $\varphi\in W^1_n(\Omega)$ and growth…