Related papers: Some minimization problems in the class of convex …

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise…

We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$ where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$…

The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions…

The $\Gamma $-limit of a family of functionals $u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ is obtained for $s=1,2$ and when the integrand $f=f\left( y,z,v\right) $ is a continous…

In this paper, we investigate the minimization problem : $$ \inf_{ \displaystyle{\begin{array}{lll} u \in H_0^1(\Omega), v \in H_0^1(\Omega),\\ \quad \| u \|_{L^{q}} =1, \quad \| v \|_{L^{q}} = 1 \end{array}}} \left[ \frac{1}{2}…

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and…

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the…

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal…

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in \Omega} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(\Omega)}$ when the supremand $f$ is not…

We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$…

Let $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(\Omega;\mathbb R^N)$ that satisfies \[ \big\|…

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $\Phi(0,0)=0$,…

In this paper, we consider the following minimizing problem with two constraints: \[ \inf \left\{ E(u) | u=(u_1,u_2), \ \| u_1 \|_{L^2}^2 = \alpha_1, \ \| u_2 \|_{L^2}^2 = \alpha_2 \right\}, \] where $\alpha_1,\alpha_2 > 0$ and $E(u)$ is…

In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{x \in \Omega}{\text{ess sup}} \, f (x, u(x), Du(x)) \] is proved by using Perron's method. The function is assumed to be quasiconvex and…

We describe a `discretize-then-relax' strategy to globally minimize integral functionals over functions $u$ in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends…

Given $\Omega$ bounded open set of $\mathbb R^{n}$ and $\alpha\in \mathbb R$, let us consider \[ \mu(\Omega,\alpha)=\min_{\substack{v\in W_{0}^{1,2}(\Omega)\\v\not\equiv 0}} \frac{\displaystyle\int_{\Omega} |\nabla v|^{2}dx+\alpha…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u)\, dx$$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the…

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $\Omega\subset\mathbb{R}^N$ with given boundary datum $\varphi$. We prove Lipschitz regularity up to the boundary for solutions of this…

The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…