arXiv subjects

Serena Dipierro

Publications and source records attributed to Serena Dipierro.

At least 181 records · Page 10Linked to original sources

arXiv2016

A logistic equation with nonlocal interactions

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, and transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

math.AP↗

arXiv2015

Stratification of free boundary points for a two-phase variational problem

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$ J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le 0\}}, \quad 1 0$. We prove the following dichotomy: if $x_0$ is a free boundary point then either the free boundary is smooth near $x_0$ or $u$ has linear growth at $x_0$. Furthermore, we show that for $p>1$ the free boundary has locally finite perimeter and the set of non-smooth points of free boundary is of zero $(N-1)$-dimensional Hausdorff measure. Our approach is new even for the classical case $p=2$.

math.AP↗

arXiv2015

Some monotonicity results for general systems of nonlinear elliptic PDEs

In this paper we show that minima and stable solutions of a general energy functional of the form $$ \int_Ω F(\nabla u,\nabla v,u,v,x)dx $$ enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy. Our results are quite general, and comprise some rigidity results which are known in the literature.

math.AP↗

arXiv2015

Nonlocal Delaunay surfaces

We construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

math.AP↗

arXiv2015

A nonlocal free boundary problem

Given~$s,σ\in(0,1)$ and a bounded domain~$Ω\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $σ$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(Ω^c)^2) } + \Per_σ\left(\{u>0\},Ω\right), $$ where~$[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and $\Per_σ$ is the fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper, as $s\nearrow1$, $σ\nearrow1$ or~$σ\searrow0$.

math.AP↗

arXiv2015

A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional $$ϕ_p(r,u,x_0)=\frac1{r^4}\int_{B_r(x_0)}\frac{|\nabla u^+(x)|^p}{|x-x_0|^{N-2}}dx\int_{B_r(x_0)}\frac{|\nabla u^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\in\partial\{u>0\}$ and $u$ is a minimizer of the functional $$J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le 0\}}, $$ for some bounded smooth domain $Ω\subset {\mathbb R}^N$ and positive constants $λ_\pm$ with $Λ:=λ_+^p-λ^p_->0$. Here we show the discrete monotonicity of $ϕ_p(r,u,x_0)$ in two spatial dimensions at non-flat points, when $p$ is sufficiently close to 2, and then establish the linear growth. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

math.AP↗

arXiv2015

Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential

We prove the existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$ (-Δ)^s u=\vartheta\frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb{R}^N).$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using of the moving plane method, in a nonlocal setting, on the whole $\mathbb{R}^N$ and by some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem into a different nonlocal problem in a weighted fractional space.

math.AP↗

arXiv2015

All functions are locally $s$-harmonic up to a small error

We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set. That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.

math.AP↗

arXiv2015

A density property for fractional weighted Sobolev spaces

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

math.AP↗

arXiv2014

Nonlocal problems with Neumann boundary conditions

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside $Ω$, decreasing energy, and convergence to a constant as $t\to \infty$. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition~$\partial_νu=0$ on~$\partialΩ$ consists in the nonlocal prescription $$ \int_Ω\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \ {\mbox{ for }} x\in\R^n\setminus\overlineΩ.$$

math.AP↗

arXiv2014

Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

For a smooth, bounded domain $Ω$, $s\in(0,1)$, $p\in \left(1,\frac{n+2s}{n-2s}\right)$ we consider the nonlocal equation $$ ε^{2s} (-Δ)^s u+u=u^p \quad {\mbox{in}}Ω$$ with zero Dirichlet datum and a small parameter $ε>0$. We construct a family of solutions that concentrate as $ε\to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function of $ ε^{2s} (-Δ)^s +1$ in the expanding domain $ε^{-1}Ω$ with zero exterior datum.

math.AP↗

arXiv2013

Strongly nonlocal dislocation dynamics in crystals

We consider the equation $$v_t=L_s v-W'(v)+σ_ε(t,x) \quad {\mbox{ in }} (0,+\infty)\times\R,$$ where $L_s$ is an integro-differential operator of order $2s$, with $s\in(0,1)$, $W$ is a periodic potential, and $σ_ε$ is a small external stress. The solution $v$ represents the atomic dislocation in the Peierls--Nabarro model for crystals, and we specifically consider the case $s\in(0,1/2)$, which takes into account a strongly nonlocal elastic term. We study the evolution of such dislocation function for macroscopic space and time scales, namely we introduce the function $$ v_ε(t,x):=v\left(\frac{t}{ε^{1+2s}},\frac{x}ε\right). $$ We show that, for small $ε$, the function $v_ε$ approaches the sum of step functions. From the physical point of view, this shows that the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. We also show that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.

math.AP↗

arXiv2013

Hardy inequalities on Riemannian manifolds and applications

We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator $Δ_{p}u := \Div(\abs{\nabla u}^{p-2}\nabla u)$. Namely, if $ρ$ is a nonnegative weight such that $-Δ_{p}ρ\geq0$, then the Hardy inequality $$c\int_{M}\frac{\abs{u}^{p}}{ρ^{p}}\abs{\nabla ρ}^{p} dv_{g} \leq \int_{M}\abs{\nabla u}^{p} dv_{g}, \quad u\in\Cinfinito_{0}(M)$$ holds. We show concrete examples specializing the function $ρ$.

math.AP↗

arXiv2013

A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian

We study the symmetry properties for solutions of elliptic systems of the type (-Δ)^{s_1} u = F_1(u, v), (-Δ)^{s_2} v= F_2(u, v), where $F\in C^{1,1}_{loc}(\R^2)$, $s_1,s_2\in (0,1)$ and the operator $(-Δ)^s$ is the so-called fractional Laplacian. We obtain some Poincaré-type formulas for the $α$-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions.

math.AP↗

arXiv2013

Geometric inequalities and symmetry results for elliptic systems

We obtain some Poincaré type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form {eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where $F\in C^{1,1}_{loc}(\R^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\R^2$.

math.AP↗

arXiv2012

Symmetry results for stable and monotone solutions to fibered systems of PDEs

We study the symmetry properties for solutions of elliptic systems of the type {ll}-\dive(a_1(x,|\nabla u^1|(X))\nabla u^1(X))=F_{1}(x, u^1(X),..., u^n(X)), ... -\dive(a_n(x,|\nabla u^n|(X))\nabla u^n(X))=F_{n}(x, u^1(X),..., u^n(X)), where $x\in \R^m$ with $1\leq m< N$, $X=(x,y)\in \R^m\times \R^{N-m}$, and $F_{1},..., F_{n}$ are the derivatives with respect to $ξ^1,..., ξ^n$ of some $F=F(x,ξ^1,..., ξ^n)$ such that for any $i=1,..., n$ and any fixed $(x,ξ^1,..., ξ^{i-1},ξ^{i+1},..., ξ^n)\in \R^m\times \R^{n-1}$ the map $ξ^i\to F(x,ξ^1,...,ξ^i,..., ξ^n)$ belongs to $C^2(\R)$. We obtain a Poincaré-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.

math.AP↗

arXiv2012

Asymptotics of the $s$-perimeter as $s\searrow 0$

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $Ω$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $Ω$. Moreover, we construct examples of sets for which the limit does not exist.

math.AP↗

arXiv2012

Concentration of solutions for a singularly perturbed mixed problem in non smooth domains

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $Ω\subset\R^{n}$ whose boundary has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

math.AP↗