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We investigate the convergence as $p\searrow 1$ of the minimizers of the $W^{s,p}$-energy for $s\in (0,1)$ and $p\in (1,\infty )$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $\Gamma $-convergence. We also address the convergence of the corresponding Euler–Lagrange equations and the equivalence between minimizers and weak solutions. As ancillary results, we study some regularity issues regarding minimizers of the $W^{s,1}$-energy.

We consider an elliptic equation in a cone, endowed with (possibly inhomogeneous) Neumann conditions. The operator and the forcing terms can also allow non-Lipschitz singularities at the vertex of the cone. In this setting, we provide unique continuation results, both in terms of interior and boundary points. The proof relies on a suitable Almgren-type frequency formula with remainders. As a byproduct, we obtain classification results for blow-up limits.

We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily small perturbations of the far-away data produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation of the prescribed conditions produces boundary discontinuities. The proof relies on a sliding method combined with a fine boundary regularity analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is Holder continuous up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal graphs in the plane “jumps” from discontinuous to $$C^{1,\gamma }$$, with no intermediate possibilities allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up to the boundary. As an interesting byproduct of our analysis, one obtains a detailed understanding of the “switch” between the regime of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and hence with differentiable inverse) ones.

We provide an existence result for stationary fractional mean field game systems, with fractional exponent greater than 1/2. In the case in which the coupling is a nonlocal regularizing potential, we obtain existence of solutions under general assumptions on the Hamiltonian. In the case of local coupling, we restrict to the subcritical regime, that is the case in which the diffusion part of the operator dominates the Hamiltonian term. We consider both the case of local bounded coupling and of local unbounded coupling with power-type growth. In this second regime, we impose some conditions on the growth of the coupling and on the growth of the Hamiltonian with respect to the gradient term., Comment: 19 pages

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., Comment: minor corrections

We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.

We consider a fractional version of Gauss capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional minimizing cone, the one determined by the fractional version of Young's law.

We present a quantitative estimate for the radially symmetric configuration concerning a Serrin-type overdetermined problem for the torsional rigidity in a bounded domain Ω ⊂ R N , when the equation is known on Ω ∖ ω ¯ only, for some open subset ω ⋐ Ω . The problem has concrete motivations in optimal heating with malfunctioning, laminar flows and beams with small inhomogeneities.

We consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions.

The authors are members of INdAM/GNAMPA. The second author is supported by the Australian Research Council Discovery Project 170104880 NEW "Nonlocal Equations at Work". We obtain symmetry results for solutions of an elliptic system of equationpossessing a cooperative structure. The domain in which the problem is set maypossess "holes" or "small vacancies" (measured in terms of capacity) along which thesolution may diverge.The method of proof relies on the moving plane technique, which needs to besuitably adapted here to take care of the complications arising from the vacancies inthe domain and the analytic structure of the elliptic system.

This paper recalls some classical motivations in fluid dynamics leading to a partial differential equation which is prescribed on a domain whose boundary possesses two connected components, one endowed with a Dirichlet datum, and the other endowed with a Neumann datum. The problem can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum. A series of recent symmetry results are presented and compared with the existing literature.

We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions. The method is based on a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. From this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant. As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano. In addition, we will obtain as a consequence a new proof of a result recently established by Bandle, Mastrolia, Monticelli, and Punzo.

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that, for any dimension n ≥ 2, there exist embedded hypersurfaces in Rn which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n ≥ 3. Interestingly, when n=2, our result provides instead a counterexample in the nonlocal framework to the well known Grayson's Theorem [17], which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.

We consider a general energy functional for phase coexistence models, which comprises the case of Banach norms in the gradient term plus a double-well potential. We establish density estimates for $Q$-minima. Namely, the state parameters close to both phases are proved to occupy a considerable portion of the ambient space. From this, we obtain the uniform convergence of the level sets to the limit interface in the sense of Hausdorff distance. The main novelty of these results lies in the fact that we do not assume the double-well potential to be non-degenerate in the vicinity of the minima. As far as we know, these types of density results for degenerate potentials are new even for minimizers and even in the case of semilinear equations, but our approach can comprise at the same time quasilinear equations, $Q$-minima and general energy functionals.

We study a Bernoulli type free boundary problem with two phases J [ u ] = ∫ Ω | ∇ u ( x ) | 2 d x + Φ ( M − ( u ) , M + ( u ) ) , u − u ¯ ∈ W 0 1 , 2 ( Ω ) , where u ¯ ∈ W 1 , 2 ( Ω ) is a given boundary datum. Here, M 1 and M 2 are weighted volumes of { u ⩽ 0 } ∩ Ω and { u > 0 } ∩ Ω , respectively, and Φ is a nonnegative function of two real variables. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem). Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem. Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit. As a special case, we can deal with the energy levels generated by the volume term Φ ( 0 , r 2 ) ≃ r 2 n − 1 n , which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality. In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, with s¿(0,1). These identities involve local boundary terms, in which the quantity (Formula presented.) plays the role that ¿u/¿¿ plays in the second-order case. Here, u is any solution to Lu = f(x,u) in O, with u = 0 in RnO, and d is the distance to ¿O.

We consider a possibly anisotropic integrodifferential semilinear equation, driven by a non-decreasing nonlinearity. We prove that if the solution grows less than the order of the operator at infinity, then it must be affine (possibly constant).

We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ \Gamma $-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $\Gamma$-converges to the classical perimeter, while for $a\in(0,1)$ the $\Gamma$-limit is a new nonlocal operator, that in dimension $n=1$ interpolates the classical and the nonlocal perimeter.

The formation of singularities in finite time in non-local Burgers' equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise estimates on the blow-up time are provided. The employment of the present equation to model a problem arising in job market is also analyzed., 18 pages, 1 figure

We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.

We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally $s$-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in ℝ 3 {\mathbb{R}^{3}} , i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum. This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error. We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

In this paper, we consider the asymptotic behavior of the fractional mean curvature when $s\to 0^+$. Moreover, we deal with the behavior of $s$-minimal surfaces when the fractional parameter $s\in(0,1)$ is small, in a bounded and connected open set with $C^2$ boundary $\Omega\subset \mathbb{R}^n$. We classify the behavior of $s$-minimal surfaces with respect to the fixed exterior data (i.e. the $s$-minimal set fixed outside of $\Omega$). So, for $s$ small and depending on the data at infinity, the $s$-minimal set can be either empty in $\Omega$, fill all $\Omega$, or possibly develop a wildly oscillating boundary. Also, we prove the continuity of the fractional mean curvature in all variables, for $s\in (0,1]$. Using this, we see that as the parameter $s$ varies, the fractional mean curvature may change sign., Comment: 43 pages, 4 figures

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. ( arXiv:1808.07696 , 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.

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).

Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain H\"older estimates for the solution.

We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function. These two nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincar\'e-Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation., Comment: To appear in Calc. Var. Partial Differential Equations

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.

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurringin 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 simple toy-model of a highly ramified structure, somehow inspired by that of the Purkinje cells, may produce a fractional diffusion via the superposition of travelling waves that solve a hyperbolic equation. This could suggest that the high ramification of the Purkinje cells might have provided an evolutionary advantage of "smoothing" the transmission of signals and avoiding shock propagations (at the price of slowing a bit such transmission). Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media. The case of a travelling concave parabola with sufficiently small curvature is explicitly computed. The new link that we propose between time fractional diffusion and hyperbolic equation also provides a novelty with respect to the usual paradigm relating time fractional diffusion with parabolic equations in the limit. This paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, both complete explanations of the objects considered and detailed lists of references are provided., Comment: Bull. Math. Biol

We provide an integral estimate for a non-divergence (non-varia-tional) form second order elliptic equation \begin{document}$a_{ij}u_{ij} = u^p$\end{document} , \begin{document}$u≥ 0$, $p∈[0, 1)$\end{document} , with bounded discontinuous coefficients \begin{document}$a_{ij}$\end{document} having small BMO norm. We consider the simplest discontinuity of the form \begin{document}$x\otimes x|x|^{-2}$\end{document} at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when \begin{document}$p = 0$\end{document} ) cannot be smooth at the points of discontinuity of \begin{document}$a_{ij}(x)$\end{document} . To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

We consider an anisotropic Levy operator I s of any order s ∈ ( 0 , 1 ) and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain are different according to the cases s 1 / 2 and s > 1 / 2 . In the isotropic one dimensional 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.

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $$\mathcal {L}_K$$ $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal {L}_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &{} \hbox {in } \Omega \\ u=0 &{} \hbox {in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here $$s\in (0,1),\, \Omega $$ is an open bounded set of $${\mathbb {R}}^n,\, n>2s$$ , with continuous boundary, $$\lambda $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is a fractional critical Sobolev exponent and $$f$$ is a lower order perturbation of the critical power $$|u|^{2^*-2}u$$ , while $$\mathcal {L}_K$$ is the integrodifferential operator defined as $$\begin{aligned} \mathcal {L}_Ku(x)= \int _{{\mathbb {R}}^n}\left( u(x+y)+u(x-y)-2u(x)\right) K(y)\,dy, \quad x\in {\mathbb {R}}^n. \end{aligned}$$ Under suitable growth condition on $$f$$ , we show that this problem admits non-trivial solutions for any positive parameter $$\lambda $$ . This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when $$K(x)=|x|^{-(n+2s)}$$ (this gives rise to the fractional Laplace operator $$-(-\Delta )^s$$ ), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow. In particular, we give examples of neckpinch singularity formation, and we discuss convexity properties of the evolution. We also take into account traveling waves for this geometric flow, showing that a new family of $C^{1,1}$ and convex traveling sets arises in this setting.

We show that nonlocal minimal cones which are non-singular subgraphs outside the origin are necessarily halfspaces. The proof is based on classical ideas of~\cite{DG1} and on the computation of the linearized nonlocal mean curvature operator, which is proved to satisfy a suitable maximum principle. With this, we obtain new, and somehow simpler, proofs of the Bernstein-type results for nonlocal minimal surfaces which have been recently established in~\cite{FV}. In addition, we establish a new nonlocal Bernstein-Moser-type result which classifies Lipschitz nonlocal minimal subgraphs outside a ball.

In this chapter, we complete the proof of Theorem 2.2.4. The computations needed for this are delicate and somehow technical. Many calculations are based on general ideas of Taylor expansions and can be adapted to other types of nonlinearities (though other estimates do take into account the precise growth conditions of the main term of the nonlinearity and of its perturbation). Rather than trying to list abstract conditions on the nonlinearity which would allow similar techniques to work (possibly at the price of more careful Taylor expansions), we remark that the case treated here is somehow classical and motivated from geometry.

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., Comment: To appear in Annales de l'Institut Henri Poincar\'e Analyse Non Lin\'eaire

We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution. As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problems. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possibly nonlocality of the problem, but it is due to the nonlinear character of the energy functional., Analysis & PDE 2017

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 example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.

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). The complete proofs and the full details of the results presented here are given in [17, 31, 35, 26, 28, 39].

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels $$|z|^{-n-s}$$ , with $$s\in (0,1)$$ and n the dimension of the ambient space. The fractional Young’s law (contact angle condition) predicted by these models coincides, in the limit as $$s\rightarrow 1^-$$ , with the classical Young’s law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient $$\sigma $$ is negative, and larger if $$\sigma $$ is positive. In addition, we address the asymptotics of the fractional Young’s law in the limit case $$s\rightarrow 0^+$$ of interaction kernels with heavy tails. Interestingly, near $$s=0$$ , the dependence of the contact angle from the relative adhesion coefficient becomes linear.

We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions., Comment: The paper supercedes the papers arXiv:1505.06919 and arXiv:1506.00109. Final version, accepted for publication in Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire

We consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.

We prove pointwise gradient bounds for entire solutions of pde’s of the form ℒu(x) = ψ(x, u(x), ∇u(x)) ,where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.

We consider some elliptic PDEs with Dirichlet and Neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classication of the solution and of the domain. In particular, we nd mild conditions under which a partially overdetermined problem is, in fact, globally overdetermined: this enables to use several classical results in order to classify all the domains that admit a solution of suitable, general, partially overdetermined problems. These results may be seen as solutions of suitable inverse problems { that is to say, given that an overdetermined system possesses a solution, we nd the shape of the admissible domains. Models of these type arise in several areas of mathematical physics and shape optimization.

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, referring to the papers [CRS10, SV13, CV13, BFV14, FV, DSV15, CSV16] for full details.

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.

The type of problems introduced in this chapter are connected to solitary solutions of nonlinear dispersive wave equations (such as the Benjamin-Ono equation, the Benjamin-Bona-Mahony equation and the fractional Schrodinger equation). In this chapter, only stationary equations are studied. This type of equation arises in the study of the fractional Schrodinger equation when looking for standing waves. The first section deals with the existence of a solution that concentrates at interior points of the domain, points that depend on the global geometry of the domain. In the last section, we point out a simple consequence of the Uncertainty Principle, which can be seen as a fractional Sobolev inequality in weighted spaces.

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 Levy 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: $ \bullet $bounded domains, $ \bullet $periodic environments, $ \bullet $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.

We prove that, if E is the Engel group and u is a stable solution of ∆Eu = f(u), then ˆ {∇Eu 6=0}  |∇Eu|2 {( p + 〈 (Hu) ν, v 〉 |∇Eu| )2 + h } − J   η ≤ ˆ E |∇Eη||∇Eu| for any test function η ∈ C∞ 0 (E). Here above, h is the horizontal mean curvature, p is the imaginary curvature and J := 2(X3X2uX1u−X3X1uX2u) + (X4u)(X1u−X2u) This can be interpreted as a geometric Poincare inequality, extending the work of [21, 22, 13] to stratified groups of step 3. As an application, we provide a non-existence result.

We consider a singular or degenerate elliptic problem in a proper domain and we prove a gradient bound and some symmetry results.