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Berardo Ruffini

Publications and source records attributed to Berardo Ruffini.

30 records · Page 2Linked to original sources
arXiv2017

On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving well-posedness of the model, we show C 1,1-regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.

math.AP↗
arXiv2017

Schrödinger operators with negative potentials and Lane-Emden densities

We consider the Schrödinger operator $-Δ+V$ for negative potentials $V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of $-Δ+V$ is positive, provided that $V$ is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation $-Δu=u^{q-1}$ (for some $1\le q< 2$). In this case, the ground state energy of $-Δ+V$ is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper.

math.AP↗
arXiv2016

Some results on anisotropic fractional mean curvature flows

We show the consistency of a threshold dynamics type algorithm for the anisotropic motion by fractional mean curvature, in the presence of a time dependent forcing term. Beside the consistency result, we show that convex sets remain convex during the evolution, and the evolution of a bounded convex set is uniquely defined.

math.NA↗
arXiv2015

Compact Sobolev embeddings and torsion functions

For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a {\it torsional Hardy inequality} that we investigate in details.

math.AP↗
arXiv2014

Existence and stability for a non-local isoperimetric model of charged liquid drops

We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits global minimizers with respect to $L^1$ perturbations preserving the volume. This leads us to study it in more regular classes of competitors, for which we show existence of minimizers. We then prove that the ball is the unique solution for sufficiently small charges.

math.AP↗
arXiv2014

Non-Local Isoperimetric Problems

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $t-s$, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $s,\,t$. When $s=0$ this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

math.AP↗
arXiv2014

A note on the hausdorff dimension of the singular set for minimizers of the mumford-shah energy

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two Authors in [4, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing sets.

math.AP↗
arXiv2014

Stability theorems for GNS inequalities: a reduction principle to the radial case

A symmetrization techique, introduced by Cianchi, Fusco, Maggi and Pratelli concerning the Sobolev inequality, is adapted to the Gagliardo-Nirenberg-Sobolev inequality (GNS) to obtain a reduction step of the problem of showing its quantitative version. More precisely we prove a stability result for the GNS inequality under the hypothesis that it holds, in turn, in the smaller class of radial symmetric decreasing functions.

math.AP↗
arXiv2013

Shape Optimization Problems for Metric Graphs

We consider the shape optimization problem $$\min\big\{{\mathcal E}(Γ)\ :\ Γ\in{\mathcal A},\ {\mathcal H}^1(Γ)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\mathcal D}=\{D_1,\dots,D_k\}\subset{\mathbb R}^d$. The cost functional ${\mathcal E}(Γ)$ is the Dirichlet energy of $Γ$ defined through the Sobolev functions on $Γ$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

math.OC↗
arXiv2013

Optimal Potentials For Schrodinger Operators

We consider the Schrodinger operator a given domain. Our goal is to study some optimization problems where an optimal (non-negative) potential V has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

math.AP↗