arXiv subjects

Simone Carano

Publications and source records attributed to Simone Carano.

7 recordsLinked to original sources

arXiv2026

Energy maximum principle for vectorial higher order absolute minimisers in $L^\infty$ and $L^p$

We show that vectorial absolute minimisers of general $k$-th order supremal functionals in $W^{k,\infty}(\Omega,\mathbb R^N)$ satisfy a maximum principle of the form $$ \max_{\overline U} \mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big)=\max_{\partial U}\mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big), \qquad\forall\ U\subseteq\Omega \mbox{ open}, $$ suitably interpreted. This is only necessary for absolute minimisers, whilst it characterises a relevant weaker notion of absolute minimality involving compactly supported variations. Further, we obtain an existence result to the Dirichlet problem for such weaker absolute minimisers, as an application of the Baire Category method. Finally, via different methods, we supplement our results by establish a gradient maximum principle for $p$-harmonic maps for $p<\infty$.

math.AP↗

arXiv2025

An $L^\infty$-variational problem involving the Fractional Laplacian

For $s\in(0,1)$ and an open bounded set $\Omega\subset\mathbb R^n$, we prove existence and uniqueness of absolute minimisers of the supremal functional $$E_\infty(u)=\|(-\Delta)^s u\|_{L^\infty(\mathbb R^n)},$$ where $(-\Delta)^s$ is the Fractional Laplacian of order $s$ and $u$ has prescribed Dirichlet data in the complement of $\Omega$. We further show that the minimiser $u_\infty$ satisfies the (fractional) PDE $$ (-\Delta)^s u_\infty=E_\infty(u_\infty)\,\mathrm{sgn}f_\infty \qquad\mbox{in }\Omega, $$ for some analytic function $f_\infty\in L^1(\Omega)$ obtained as the restriction of an $s$-harmonic measure $\mu$ in $\Omega$.

math.AP↗

arXiv2025

Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order $L^\infty$-variational problem

We study a vectorial $L^\infty$-variational problem of second order, where the supremal functional depends on the vector function $u$ through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser $u_\infty$ under prescribed Dirichlet boundary conditions, together with a characterisation of $u_\infty$ as solution of a specific system of PDEs. Our result can be seen as a twofold extension of the one in Katzourakis-Moser (ARMA 2019): we generalise it to the vectorial setting and, at the same time, we consider more general elliptic operators in place of the Laplacian.

math.AP↗

arXiv2023

Strict BV relaxed area of Sobolev maps into the circle: the high dimension case

We deal with the relaxed area functional in the strict $BV$-convergence of non-smooth maps defined in domains of generic dimension and taking values into the unit circle. In case of Sobolev maps, a complete explicit formula is obtained. Our proof is based on tools from Geometric Measure Theory and Cartesian currents. We then discuss the possible extension to the wider class of maps with bounded variation. Finally, we show a counterexample to the locality property in case of both dimension and codimension larger than two.

math.AP↗

arXiv2023

Relaxed area of $0$-homogeneous maps in the strict $BV$-convergence

We compute the relaxed Cartesian area for general $0$-homogeneous map of bounded variation, with respect to the strict $BV$-convergence. In particular, we show that the relaxed area is finite for this class of maps and we provide an integral representation formula.

math.CA↗

arXiv2023

Relaxed area of graphs of piecewise Lipschitz maps in the strict $BV$-convergence

We compute the relaxed Cartesian area in the strict $BV$-convergence on a class of piecewise Lipschitz maps from the plane to the plane, having jump made of several curves allowed to meet at a finite number of junction points. We show that the domain of this relaxed area is strictly contained in the domain of the classical $L^1$-relaxed area.

math.CA↗

arXiv2023

The relaxed area of $\mathcal{S}^1$-valued singular maps in the strict $BV$-convergence

Given a bounded open set $\Omega \subset \mathbb{R}^2$, we study the relaxation of the nonparametric area functional in the strict topology in $BV(\Omega;\mathbb{R}^2)$, and compute it for vortex-type maps, and more generally for maps in $W^{1,1}(\Omega;\mathcal{S}^1)$ having a finite number of topological singularities. We also extend the analysis to some specific piecewise constant maps in $BV(\Omega;\mathcal{S}^1)$, including the symmetric triple junction map.

math.CA↗