A Brezis-Nirenberg result for non-local critical equations in low dimension
Raffaella Servadei Enrico Valdinoci
The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities \begin{eqnarray} (-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\ u=0, in R^n\setminus \Omega, \end{eqnarray} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators.
keywords: Critical nonlinearities integrodifferential operators variational techniques fractional Laplacian. Mountain Pass Theorem Linking Theorem best critical Sobolev constant
On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional
Isabeau Birindelli Enrico Valdinoci
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.
keywords: Allen-Cahn equation Grushin plane. Symmetry problmes
A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature
Alberto Farina Enrico Valdinoci
keywords: a-priori estimates. Geometric analysis of PDEs
Closed curves of prescribed curvature and a pinning effect
Matteo Novaga Enrico Valdinoci
We prove that for any $H: R^2 \to R$ which is $Z^2$-periodic, there exists $H_\varepsilon$, which is smooth, $\varepsilon$-close to $H$ in $L^1$, with $L^\infty$-norm controlled by the one of $H$, and with the same average of $H$, for which there exists a smooth closed curve $\gamma_\varepsilon$ whose curvature is $H_\varepsilon$. A pinning phenomenon for curvature driven flow with a periodic forcing term then follows. Namely, curves in fine periodic media may be moved only by small amounts, of the order of the period.
keywords: heterogeneous media. Prescribed curvature pinning phenomena
Asymptotics of the $s$-perimeter as $s\searrow 0$
Serena Dipierro Alessio Figalli Giampiero Palatucci Enrico Valdinoci
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ 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 $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.
keywords: Nonlinear problems minimal surfaces. fractional Laplacian fractional Sobolev spaces nonlocal perimeter
Variational methods for non-local operators of elliptic type
Raffaella Servadei Enrico Valdinoci
In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem $$ \left\{ \begin{array}{ll} \mathcal L_K u+\lambda u+f(x,u)=0        in   Ω \\ u=0                                 in   \mathbb{R}^n \backslash Ω , \end{array} \right. $$ where $\lambda$ is a real parameter and the nonlinear term $f$ satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional $\mathcal J_\lambda$ associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq \lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the operator $-\mathcal L_K$. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u)        in   Ω \\ u=0                                in   \mathbb{R}^n \backslash Ω. \end{array} \right. $$ Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.
keywords: Linking Theorem fractional Laplacian. Mountain Pass Theorem variational techniques integrodifferential operators
Preface: Special issue on rate-independent evolutions and hysteresis modelling
Stefano Bosia Michela Eleuteri Elisabetta Rocca Enrico Valdinoci
The interest in hysteresis and rate-independent phenomena is shared by scientists with a great variety of different backgrounds. We can encounter these processes in several situations of common life: for instance in elasto-plasticity, ferromagnetism, shape-memory alloys, phase transitions. Beyond physics, hysteresis and rate-independent phenomena appear also in engineering, biology, economics as well as in many other settings, playing an important role in many applications. The complexity arising in these fields necessarily requires a joint contribution of experts with different backgrounds and skills. Therefore, only synergy and cooperation among these several people can lead to concrete advances in the technological capabilities of our society.
    This special issue of Discrete and Continuous Dynamical Systems is devoted to the latest advances and trends in the modelling and in the analysis of this family of complex phenomena. In particular, we gathered contributions from different fields of science (mathematical analysis, mathematical physics, engineering) with the intent of presenting an updated picture of current research directions, offering a new and interdisciplinary perspective in the study of these processes.
    Motivated by the Spring School on Rate-independent Evolutions and Hysteresis Modelling, held at the Politecnico di Milano and University of Milano on May 27-31, 2013, this special issue contains different kinds of original contributions: some of them originate from the courses held in that occasion and from the discussions they stimulated, but are here presented in a new perspective; some others instead are original contributions in related topics. All the papers are written in the clearest possible language, accessible also to students and non-experts of the field, with the intent to attract and introduce them to this topic.
    Final acceptance of all the papers in this volume was made by the normal referee procedure and standard practices of AIMS journals.
    We wish to thanks all the referees, who kindly agreed to devote their time and effort to read and check all the papers carefully, providing useful comments and recommendations. We are also grateful to all the authors for their great job and the high quality of their contributions. We finally wish to express our gratitude to AIMS and in particular to Prof. Alain Miranville for the opportunity to publish this special issue and for the technical support.
On a fractional harmonic replacement
Serena Dipierro Enrico Valdinoci
Given $s\in(0,1)$, we consider the problem of minimizing the fractional Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$.
    We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$).
    Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.
keywords: Harmonic replacement fractional Sobolev spaces energy estimates.
The geometry of mesoscopic phase transition interfaces
Matteo Novaga Enrico Valdinoci
We consider a mesoscopic model of phase transitions and investigate the geometric properties of the interfaces of the associated minimal solutions. We provide density estimates for level sets and, in the periodic setting, we construct minimal interfaces at a universal distance from any given hyperplane.
keywords: plane-like solutions. Ginzburg-Landau-Allen-Cahn equation density estimates
(Non)local and (non)linear free boundary problems
Serena Dipierro Enrico Valdinoci

We discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin.

The main feature of these new free boundary problems is that they deeply take into account nonlinear energy superpositions and possibly nonlocal functionals.

The nonlocal parameter interpolates between volume and perimeter functionals, and so it can be seen as a fractional counterpart of classical free boundary problems, in which the bulk energy presents nonlocal aspects.

The nonlinear term in the energy superposition takes into account the possibility of modeling different regimes in terms of different energy levels and provides a lack of scale invariance, which in turn may cause a structural instability of minimizers that may vary from one scale to another.

keywords: Free boundary problems Gagliardo norm fractional perimeter nonlocal minimal surfaces Bernoulli's Law

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