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73

Sep 22, 2013
09/13

by
Enrico Valdinoci

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We review some recent results on minimisers of a non-local perimeter functional, in connection with some phase coexistence models whose diffusion term is given by the fractional Laplacian.

Source: http://arxiv.org/abs/1210.5612v1

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116

Sep 22, 2013
09/13

by
Enrico Valdinoci

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This note illustrates how a simple random walk with possibly long jumps is related to fractional powers of the Laplace operator. The exposition is elementary and self-contained.

Source: http://arxiv.org/abs/0901.3261v1

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2.0

Jun 30, 2018
06/18

by
Alberto Farina; Enrico Valdinoci

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We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1406.7223

56
56

Sep 18, 2013
09/13

by
Isabeau Birindelli; Enrico Valdinoci

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We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate. However, they are not necessarily one-dimensional, as a counter-example shows.

Source: http://arxiv.org/abs/0806.4139v1

57
57

Sep 23, 2013
09/13

by
Luis Caffarelli; Enrico Valdinoci

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We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.

Source: http://arxiv.org/abs/1105.1158v2

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45

Jul 20, 2013
07/13

by
Ovidiu Savin; Enrico Valdinoci

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We discuss the $\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. When $s\in [1/2,\,1)$, we show that the energy $\Gamma$-converges to the classical minimal surface functional -- while, when $s\in(0,\,1/2)$, it is easy to see that the functional $\Gamma$-converges to...

Source: http://arxiv.org/abs/1007.1725v3

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8.0

Jun 27, 2018
06/18

by
Claudia Bucur; Enrico Valdinoci

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We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena. Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schr\"{o}dinger equations. Furthermore, an example of an $s$-harmonic function, the harmonic extension and some insight on a fractional version of a classical conjecture formulated by De Giorgi are...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1504.08292

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11

Jun 27, 2018
06/18

by
François Hamel; Enrico Valdinoci

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We consider an integral equation in the plane, in which the leading operator is of convolution type, and we prove that monotone (or stable) solutions are necessarily one-dimensional.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1506.00109

2
2.0

Jun 29, 2018
06/18

by
Jürgen Sprekels; Enrico Valdinoci

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In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the $s$th power of a positive definite operator having a discrete spectrum in $\R^+$. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter $s$. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1601.00568

9
9.0

Jun 27, 2018
06/18

by
Stefania Patrizi; Enrico Valdinoci

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We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls-Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1504.00044

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67

Sep 23, 2013
09/13

by
Andrea Pinamonti; Enrico Valdinoci

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We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if $\epsilon\ge0$ and $\bar u_\epsilon$ minimizes the functional $$ \int_\Omega(\epsilon+|\nabla_{\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $\psi$, then 0 \le \div_{\H^n}\, \Big((\epsilon+|\nabla_{\H^n}\bar u_\epsilon|^2)^{(p/2)-1}...

Source: http://arxiv.org/abs/1105.5075v1

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42

Sep 23, 2013
09/13

by
Ovidiu Savin; Enrico Valdinoci

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We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension.

Source: http://arxiv.org/abs/1202.0973v1

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4.0

Jun 30, 2018
06/18

by
Serena Dipierro; Enrico Valdinoci

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Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors. Also, a classical mathematical feature is that (fractional) parabolic equations possess smoothing effects, in contrast with the case of hyperbolic equations, which typically exhibit shocks and discontinuities. In this paper, we show how a...

Topics: Quantitative Biology, Neurons and Cognition, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1702.05553

9
9.0

Jun 27, 2018
06/18

by
Annalisa Massaccesi; Enrico Valdinoci

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We study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under which a nonlocal dispersal behavior is favored. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource....

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1503.01629

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53

Sep 18, 2013
09/13

by
Yannick Sire; Enrico Valdinoci

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We consider a quasilinear equation given in the half-space, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincar\'e inequality and, as a byproduct of this inequality, a result on the symmetry of low-dimensional bounded stable solutions, under some suitable assumptions on the nonlinearities. More precisely, we analyze the following boundary problem $$ \left\{\begin{matrix} -{\rm div} (a(x,|\nabla u|)\nabla u)+g(x,u)=0 \qquad {on $\R^n\times(0,+\infty)$} -a(x,|\nabla...

Source: http://arxiv.org/abs/0803.1382v1

50
50

Sep 22, 2013
09/13

by
Ovidiu Savin; Enrico Valdinoci

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We consider the minimizers of the energy $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer $u$ in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.

Source: http://arxiv.org/abs/1103.6205v1

48
48

Sep 18, 2013
09/13

by
Ovidiu Savin; Enrico Valdinoci

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We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$ \int F(\nabla u, u, x) dx $$ under the assumption that a certain integral grows at most quadratically at infinity. As a consequence we obtain several rigidity results of global solutions in low dimensions.

Source: http://arxiv.org/abs/1209.1517v1

4
4.0

Jun 30, 2018
06/18

by
Stefania Patrizi; Enrico Valdinoci

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We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well).

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1407.0620

9
9.0

Jun 27, 2018
06/18

by
Serena Dipierro; Enrico Valdinoci

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We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are H\"older continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1504.05569

2
2.0

Jun 28, 2018
06/18

by
Alberto Farina; Enrico Valdinoci

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We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order~$2$ in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1512.06509

2
2.0

Jun 30, 2018
06/18

by
Nicola Soave; Enrico Valdinoci

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We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1412.5074

4
4.0

Jun 29, 2018
06/18

by
Stefania Patrizi; Enrico Valdinoci

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We describe the asymptotic states for the solutions of a nonlocal equation of evolutionary type, which have the physical meaning of the atom dislocation function in a periodic crystal. More precisely, we can describe accurately the "smoothing effect" on the dislocation function occurring slightly after a "particle collision" (roughly speaking, two opposite transitions layers average out) and, in this way, we can trap the atom dislocation function between a superposition of...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1609.04441

3
3.0

Jun 29, 2018
06/18

by
Francesco Maggi; Enrico Valdinoci

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We explore the possibility of modifying the classical Gauss free energy functional used in capillarity theory by considering surface tension energies of nonlocal type. The corresponding variational principles lead to new equilibrium conditions which are compared to the mean curvature equation and Young's law found in classical capillarity theory. As a special case of this family of problems we recover a nonlocal relative isoperimetric problem of geometric interest.

Topics: Optimization and Control, Analysis of PDEs, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1606.08610

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13

Jun 27, 2018
06/18

by
Fabio Punzo; Enrico Valdinoci

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We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1504.06265

4
4.0

Jun 28, 2018
06/18

by
Mariel Sáez; Enrico Valdinoci

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In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain quantities (such as positive fractional curvature) and smoothness of graphical evolutions.

Topics: Differential Geometry, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1511.06944

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6.0

Jun 29, 2018
06/18

by
Serena Dipierro; Enrico Valdinoci

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We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1607.06872

2
2.0

Jun 30, 2018
06/18

by
Alberto Farina; Enrico Valdinoci

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We study bounded, monotone solutions of$\Delta u=W'(u)$ in the whole of$\R^n$, where$W$ is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, $u$ is $1$D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of $\Gamma$-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1410.3206

47
47

Sep 23, 2013
09/13

by
Yannick Sire; Enrico Valdinoci

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We deal with symmetry properties for solutions of nonlocal equations of the type $(-\Delta)^s v= f(v)\qquad {in $\R^n$,}$ where $s \in (0,1)$ and the operator $(-\Delta)^s$ is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation ${-div (x^\a \nabla u)=0 \qquad {on $\R^n\times(0,+\infty)$} -x^\a u_x = f(u) \qquad {on $\R^n\times\{0\}$} $ where $\a \in (-1,1)$. This equation is related to the...

Source: http://arxiv.org/abs/0801.2355v1

49
49

Sep 21, 2013
09/13

by
Yannick Sire; Enrico Valdinoci

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We consider a functional obtained by adding a trace term to the Allen-Cahn phase segregation model and we prove some density estimates for the level sets of the interfaces. We treat in a unified way also the cases of possible degeneracy and singularity of the ellipticity of the model and the quasiminimal case.

Source: http://arxiv.org/abs/1011.6617v1

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15

Jun 26, 2018
06/18

by
Serena Dipierro; Enrico Valdinoci

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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.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1501.04918

100
100

Jul 20, 2013
07/13

by
Ovidiu Savin; Enrico Valdinoci

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We prove density estimates for level sets of minimizers of the energy $$\eps^{2s}\|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. As a consequence we obtain, as $\eps \to 0$, the uniform convergence of the level sets of $u$ to either a $H^s$-nonlocal minimal surface if $s\in(0,\frac 1 2)$, or to a classical minimal surface if $s \in[\frac 1...

Source: http://arxiv.org/abs/1007.2114v3

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52

Sep 21, 2013
09/13

by
Milena Chermisi; Enrico Valdinoci

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In $\R^m\times\R^{n-m}$, endowed with coordinates $X=(x,y)$, we consider the PDE $$ -{\rm div} \big(\alpha(\x) |\nabla u(\X)|^{p(x)-2}\nabla u(\X)\big)=f(x,u(\X)).$$ We prove a geometric inequality and a symmetry result.

Source: http://arxiv.org/abs/0808.1835v1

5
5.0

Jun 29, 2018
06/18

by
Stefania Patrizi; Enrico Valdinoci

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We consider an anisotropic L\'evy operator $\mathcal{I}_s$ of any order $s\in(0,1)$ and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the cases $s 1/2$. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1610.00203

11
11

Jun 27, 2018
06/18

by
Matteo Cozzi; Enrico Valdinoci

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We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1505.02304

3
3.0

Jun 30, 2018
06/18

by
Carina Geldhauser; Enrico Valdinoci

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We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter $s$ is the $s$-th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter $s$ are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.

Topics: Probability, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1703.09329

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48

Sep 22, 2013
09/13

by
Daniela De Silva; Enrico Valdinoci

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We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments.

Source: http://arxiv.org/abs/0811.1476v1

97
97

Sep 18, 2013
09/13

by
Hannes Junginger-Gestrich; Enrico Valdinoci

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Using theorems of Bangert, we prove a rigidity result which shows how a question raised by Bangert for elliptic integrands of Moser type is connected, in the case of minimal solutions without self-intersections, to a famous conjecture of De Giorgi for phase transitions.

Source: http://arxiv.org/abs/0707.3327v1

6
6.0

Jun 26, 2018
06/18

by
Xavier Ros-Oton; Enrico Valdinoci

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We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{ 2u(x)-u(x+\rho\omega)-u(x-\rho\omega)}{\rho^{1+2s}}.$$ Here, $a$ is any measure on~$S^{n-1}$ (a prototype example for~$L$ is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When $a\in C^\infty(S^{n-1})$ and $g$ is...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.00782

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36

Sep 21, 2013
09/13

by
Mouhamed Moustapha Fall; Enrico Valdinoci

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We consider the equation $\Ds u+u=u^p$, with $s\in(0,1)$ in the subcritical range of $p$. We prove that if $s$ is sufficiently close to 1 the equation possesses a unique minimizer, which is nondegenerate.

Source: http://arxiv.org/abs/1301.4868v2

9
9.0

Jun 30, 2018
06/18

by
Serena Dipierro; Aram Karakhanyan; Enrico Valdinoci

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We present a series of recent results on some new classes of free boundary problems. Differently from the classical literature, the problems considered have either a "nonlocal" feature (e.g., the interaction or/and the interfacial energy may depend on global quantities) or a "nonlinear" flavor (namely, the total energy is the nonlinear superposition of energy components, thus losing the standard additivity and scale invariances of the problem).

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1701.07897

5
5.0

Jun 29, 2018
06/18

by
Luis Caffarelli; Serena Dipierro; Enrico Valdinoci

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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 L\'evy 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...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1601.05552

8
8.0

Jun 28, 2018
06/18

by
Serena Dipierro; Ovidiu Savin; Enrico Valdinoci

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In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension~$3$, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1506.04281

2
2.0

Jun 30, 2018
06/18

by
Serena Dipierro; Ovidiu Savin; Enrico Valdinoci

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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.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1404.3652

13
13

Jun 28, 2018
06/18

by
Serena Dipierro; Nicola Soave; Enrico Valdinoci

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We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1509.04001

5
5.0

Jun 29, 2018
06/18

by
Serena Dipierro; Matteo Novaga; Enrico Valdinoci

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We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers. We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1604.07219

63
63

Sep 21, 2013
09/13

by
Giampiero Palatucci; Enrico Valdinoci; Ovidiu Savin

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We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$ where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$ and $W$ is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $\mathbb{R}^n$. The results collected here will also be useful for forthcoming papers, where the second...

Source: http://arxiv.org/abs/1104.1725v2

19
19

Jun 27, 2018
06/18

by
Luca Rossi; Andrea Tellini; Enrico Valdinoci

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In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder....

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1504.04698

3
3.0

Jun 28, 2018
06/18

by
Serena Dipierro; Stefania Patrizi; Enrico Valdinoci

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We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1511.06799

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Jun 29, 2018
06/18

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Serena Dipierro; Nicola Soave; Enrico Valdinoci

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We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1604.07755

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Jun 29, 2018
06/18

by
Eleonora Cinti; Joaquim Serra; Enrico Valdinoci

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We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $n\ge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.00540