arXiv subjects

Tommaso Bertin

Publications and source records attributed to Tommaso Bertin.

3 recordsLinked to original sources

arXiv2025

Relaxation for highly discontinuous, possibly unbounded, integral functionals

We consider the functional \[ F(u)=\int_Ω f(\nabla u)\,dx\qquad u\inφ+W^{1,1}_0(Ω) \] where $Ω$ is a Lipschitz bounded open set of $\R^N$, $f:\R^N\to\R\cup \{+\infty\}$ is a superlinear Borel function, $φ\in W^{1,\infty}(Ω)$. We prove that, if $f$ is superlinear and satisfies very weak assumptions, then the Lavrentiev phenomenon does not occur. We underline that our assumptions include the case of non continuous, non convex, and unbounded Lagrangians.

math.AP↗

arXiv2025

Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_Ωf(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev phenomenon. We determine a formulation of the lower semicontinuous envelope of $E[f]$ with respect to various topologies and with fixed Lipschitz Dirichlet boundary conditions.

math.AP↗

arXiv2025

Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians

We consider the functional $$F_\infty(u)=\int_Ωf(x,u(x),\nabla u(x)) dx \quad\quad u\in φ+ W_0^{1,\infty}(Ω,\mathbb{R})$$ where $Ω$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $φ\in W^{1,\infty}(Ω)$. We do not assume neither convexity or continuity of the Lagrangian w.r.t. the last variable. We prove that, under suitable assumptions, the lower semicontinuous envelope of $F_\infty$ both in $φ+W^{1,\infty}(Ω)$ and in the larger space $φ+W^{1,p}(Ω)$ can be represented by means of the bipolar $f^{**}$ of $f$. In particular we can also exclude Lavrentiev Phenomenon between $W^{1,\infty}(Ω)$ and $W^{1,1}(Ω)$ for autonomous Lagrangians.

math.AP↗