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We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. We discuss the spectral properties associated to a weighted eigenvalue problem and present a global bound for subsolutions. The Neumann condition that we take into account comprises, as a particular case, the one that has been recently introduced in [S. Dipierro, X. Ros-Oton, E. Valdinoci, Rev. Mat. Iberoam. (2017)]. Also, the results that we present here find a natural application to a logistic equation motivated by biological problems that has been recently considered in [S. Dipierro, E. Proietti Lippi, E. Valdinoci, preprint (2020)].

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 introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition which can be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth Hölder norm of the solution is controlled in terms of the seminorm of the nonlinearity. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.

In this note, we showcase some recent results obtained in [DSV19] concerning the stickiness properties of nonlocal minimal graphs in the plane. To start with, the nonlocal minimal graphs in the planeenjoy an enhanced boundary regularity, since boundary continuity with respect to the external datum is sufficient to ensure differentiability across the boundary of the domain. As a matter of fact, the Hoelder exponent of the derivative is in this situation sufficiently high to provide the validity of the Euler-Lagrange equation at boundary points as well. From this, using a sliding method, one also deduces that the stickiness phenomenon is generic for nonlocal minimal graphs in the plane, since an arbitrarily small perturbation of continuous nonlocal minimal graphs can produce boundary discontinuities (making the continuous case somehow ``exceptional'' in this framework., Bruno Pini Mathematical Analysis Seminar, Vol 11, No 1 (2020): Something about nonlinear problems

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ)s, with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

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.

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators. Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.

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

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 define a family of functionals, called \begin{document}$ p $\end{document} -oscillation functionals, that can be interpreted as discrete versions of the classical total variation functional for \begin{document}$ p = 1 $\end{document} and of the \begin{document}$ p $\end{document} -Dirichlet functionals for \begin{document}$ p>1 $\end{document} . We introduce the notion of minimizers and prove existence of solutions to the Dirichlet problem. Finally we provide a description of Class A minimizers (i.e. minimizers under compact perturbations) in dimension \begin{document}$ 1 $\end{document} .

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation. In addition, our result applies also to the $\psi$-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions.

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.

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviors in their homecoming, which are interesting to observe and to describe analytically. In this paper, we present a simple mathematical formulation to describe the little penguins parade in Phillip Island. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters" such as the eye-sight of the penguins, their cruising speed and the possible "fear" of animals. On the one hand, this favors the formation of conglomerates of penguins that gather together, but, on the other hand, this may lead to the "panic" of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behavior of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a "stop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins freeze due to panic). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguins parade., 13 figures

We present a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement 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.

The aim of this paper is to study the following problem \begin{document}$(P_{\lambda}) \equiv\left\{\begin{array}{rcll}(-\Delta)^s u& = &\lambda u^{q}+u^{p}&{\text{ in }}\Omega,\\ u&>&0 &{\text{ in }} \Omega, \\ \mathcal{B}_{s}u& = &0 &{\text{ in }} \mathbb{R}^{N}\backslash \Omega,\end{array}\right.$ \end{document} with \begin{document}$0 , \begin{document}$N>2s$\end{document} , \begin{document}$λ> 0$\end{document} , \begin{document}$Ω \subset \mathbb{R}^{N}$\end{document} is a smooth bounded domain, \begin{document}$(-Δ)^su(x) = a_{N,s}\;P.V.∈t_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,$ \end{document} \begin{document}$a_{N,s}$\end{document} is a normalizing constant, and \begin{document}$\mathcal{B}_{s}u = uχ_{Σ_{1}}+\mathcal{N}_{s}uχ_{Σ_{2}}.$\end{document} Here, \begin{document}$Σ_{1}$\end{document} and \begin{document}$Σ_{2}$\end{document} are open sets in \begin{document}$\mathbb{R}^{N}\backslash Ω$\end{document} such that \begin{document}$Σ_{1} \cap Σ_{2} = \emptyset$\end{document} and \begin{document}$\overline{Σ}_{1}\cup \overline{Σ}_{2} = \mathbb{R}^{N}\backslash Ω.$\end{document} In this setting, \begin{document}$\mathcal{N}_{s}u$\end{document} can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [ 20 ], and \begin{document}$\mathcal{B}_{s}u$\end{document} is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem ( \begin{document}$P_{λ}$\end{document} ) for suitable ranges of \begin{document}$λ$\end{document} and \begin{document}$p$\end{document} and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

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 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 transition layers which, as time flows, approaches either a constant function or a single heteroclinic (depending on the algebraic properties of the orientations of the initial transition layers). The results are endowed with explicit and quantitative estimates and, as a byproduct, we show that the ODE systems of particles that govern the evolution of the transition layers does not admit stationary solutions (i.e. roughly speaking, transition layers always move).

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 present some long-range interaction models for phase coexistence which have recently appeared in the literature, recalling also their relation to classical interface and capillarity problems. In this note, the main focus will be on the Γ-convergence methods, emphasizing similarities and differences between the classical theory and the new trends of investigation. In doing so, we also obtain some new, more precise Γ-convergence results in terms of ``interior'' and ``exterior'' contributions. We also discuss the structural differences between Γ-limits and ``pointwise'' limits, especially concerning the ``boundary terms''., Bruno Pini Mathematical Analysis Seminar, Vol 11, No 1 (2020): Something about nonlinear problems

We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimensions 2 and 3. More precisely, we prove that stable solutions in dimension 2 and minimizers and monotone solutions in dimension 3 depend on only one Euclidean variable. Monotone solutions in the 2-dimensional case without weights were studied in de la Llave and Valdinoci (Math Res Lett 16(5):909–918, 2009). In our paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.

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.

As well known, harmonic functions satisfy the mean value property, namely the average of the function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a characterizing feature of balls, namely whether a set for which all harmonic functions satisfy the mean value property is necessarily a ball. This question was investigated by several authors, and was finally elegantly, completely and positively settled by \"Ulk\"u Kuran, with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, Nicola Fusco, Ermanno Lanconelli and Xiao Zhong who proved a quantitative stability result for the mean value formula, showing that a suitable "mean value gap" (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the "gap" between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain "almost" satisfies the mean value property, then it must be necessarily close to a ball. Here we investigate the nonlocal counterparts of these results. In particular we will prove a classification result and a stability result, establishing that: if fractional harmonic functions enjoy a suitable exterior average property for a given domain, then the domain is necessarily a ball, a suitable "nonlocal mean value gap" is bounded from below by an appropriate measure of the difference between the set and the ball. Differently from the classical case, some of our arguments rely on purely nonlocal properties, with no classical counterpart, such as the fact that "all functions are locally fractional harmonic up to a small error".

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.

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.

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

where K : Rn × Rn → [0,+∞] is a measurable kernel comparable to that of the fractional Laplacian of order 2s, with s ∈ (0, 1), and W : Rn × R→ [0,+∞) is a smooth double-well potential, with zeroes at u = ±1. Both K and W are assumed to be Zn-periodic. For any vector ω ∈ Rn \{0}, we prove the existence of a minimizer uω of E that is directed along ω and whose interface {|uω| 0. Moreover, uω enjoys a suitable periodicity/almost-periodicity property, in dependence of whether ω is rational or not. As a result, we obtain the existence of plane-like entire solutions to the integro-differential Euler-Lagrange equation corresponding to E .

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.

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).

We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space. We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, \begin{document}$ \frac12 $\end{document} . In this situation, we prove that the speed of invasion of the fundamental solution is at least "almost of square root type", namely it is larger than \begin{document}$ ct^\beta $\end{document} for any given \begin{document}$ c>0 $\end{document} and \begin{document}$ \beta\in\left(0,\frac12\right) $\end{document} .

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 consider an evolution equation inspired by a model in peridynamics, with a singular pairwise interaction force term, and we give global in time existence, uniqueness and stability results for the Cauchy problem.

We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a divergent fractional Laplace equation which is also prescribed in a neighborhood of infinity. Secondly, we take into account the Dirichlet problem for the divergent fractional Laplace equation, proving the existence of a solution and characterizing its multiplicity. Finally, we take into account the case of nonlinear equations, obtaining a new approximation results. These results maintain their interest also in the case of functions for which the fractional Laplacian can be defined in the usual sense.

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.

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 $n \geqslant 2$ 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 surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb{R}^3$. ¶ On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n = 2, 3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ – with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

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

International audience; This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact ``obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.

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.

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 the interior regularity of solutions to the Dirichlet problem L u = g in Ω, u = 0 in R n ∖ Ω , for anisotropic operators of fractional type L u ( x ) = ∫ 0 + ∞ d ρ ∫ S n − 1 d a ( ω ) 2 u ( x ) − u ( x + ρ ω ) − u ( x − ρ ω ) ρ 1 + 2 s . 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 ∈ C ∞ ( S n − 1 ) and g is C ∞ ( Ω ) , solutions are known to be C ∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L ∞ ( S n − 1 ) , solutions are only known to be C 3 s inside Ω. We prove here that, for general measures a, solutions are C 1 + 3 s − ϵ inside Ω for all ϵ > 0 whenever Ω is convex. When a ∈ L ∞ ( S n − 1 ) , we show that the same holds in all C 1 , 1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C 3 s + ϵ for any ϵ > 0 – even if g and Ω are C ∞ .

We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi [6] stating that the validity of Bernstein’s theorem in dimension n + 1 {n+1} is a consequence of the nonexistence of n-dimensional singular minimal cones in ℝ n {\mathbb{R}^{n}} .

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 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.

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