Related papers: Existence and regularity of minimizers for a varia…

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further…

In this paper, we prove the existence of minimizers of a class of multi-constrained variational problems. We consider systems involving a nonlinearity that does not satisfy compactness, monotonicity, neither symmetry properties. Our…

Nonconvex functionals with spherical symmetry are studied. Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation.

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a…

This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior…

In this paper we prove the uniqueness and radial symmetry of minimizers for variational problems that model several phenomena. The uniqueness is a consequence of the convexity of the functional. The main technique is Fourier transform of…

We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.

We provide a simple proof of the radial symmetry of any nonnegative minimizer for a general class of quasi-linear minimization problems.

The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.

A class of variational models describing ecological systems of k species competing for the same resources is investigated. The occurrence of coexistence in minimal energy solutions is discussed and positive results are proven for suitably…

We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by…

In this paper, we investigate an optimal design problem motivated by some issues arising in population dynamics. In a nutshell, we aim at determining the optimal shape of a region occupied by resources for maximizing the survival ability of…

Minimizers in the least gradient problem with discontinuous boundary data need not be unique. However, all of them have a similar structure of level sets. Here, we give a full characterization of the set of minimizers in terms of any one of…

We continue the analysis of some modifications of the total variation image inpainting method formulated on the space $BV(\Omega)^M$ in the sense that we generalize the main results of [32] to the case that a more general data fitting term…

The geometrical features of the (non-convex) loss landscape of neural network models are crucial in ensuring successful optimization and, most importantly, the capability to generalize well. While minimizers' flatness consistently…

A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or…

We prove local boundedness of generalized solutions to a large class of variational problems of linear growth including boundary value problems of minimal surface type and models from image analysis related to the procedure of…

We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $\Omega$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold…

We study the isoperimetric problem with a potential energy $g$ in $\mathbb{R}^n$ weighted by a radial density $f$ and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast…

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…