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Given a bounded open set $ \Omega\subseteq{\mathbb{R}}^n $, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $ \Omega $. We prove that the second eigenvalue $ \lambda_2(\Omega) $ is always strictly larger than the first eigenvalue $ \lambda_1(B) $ of a ball $ B $ with volume half of that of $ \Omega $. This bound is proven to be sharp, by comparing to the limit case in which $ \Omega $ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.

In this paper we use viscosity solution methods to show that almost integrable Hamiltonian systems are not smoothly integrable at rotation numbers which are exponentially close to resonances.

We prove that $C^{1,\alpha}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.

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 consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum., Comment: Journal d'Analyse Math\'ematique

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.

We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

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.

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

These are the handouts of an undergraduate minicourse at the Universita di Bari (see Fig. 1), in the context of the 2017 INdAM Intensive Period “Contemporary Research in elliptic PDEs and related topics”. Without any intention to serve as a throughout epitome to the subject, we hope that these notes can be of some help for a very initial introduction to a fascinating field of classical and modern research.

We construct entire solutions of an elliptic equation whose nonlinearity depends on the gradient of u. Such solutions are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.

This survey presents some applications of KAM theory and variational methods to some concrete problems arising in celestial mechanics and PDEs. While the details of the proofs, sometimes lengthy or technical, are published elsewhere, we provide here a general overlook of results and techniques, with the aim of accenting the analogies between the finite and infinite dimensional cases and the interplay of the KAM-Nash-Moser iteration techniques with the Lyapunov-Schmidt variational methods. In this review, we deal with four topics: Spatial planetary three-body problem. We consider one “star” and two “planets”, modelled by three massive points, interacting through gravity in a three-dimensional space. Near the limiting solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system is proved to have two-dimensional, elliptic, quasiperiodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semiaxes belong to a two-dimensional set of density close to one. Planar planetary many-body problem. As above, but one “star” and N “planets”, the interior two ones bigger than the others (as in the exterior solar system). Near the limiting solutions given by the N planets revolving around the star on Keplerian ellipses with small eccentricity and zero mutual inclination, the system is proved to have N -dimensional, elliptic, quasiperiodic solutions. Periodic orbits approaching lower dimensional elliptic KAM tori. By a general Birkhoff-Lewis-Conley-Zehnder-type result, we prove the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori of Hamiltonian systems. As an application, periodic orbits close to the quasiperiodic ones of the above planetary problems are constructed.

We prove for some singular kernels $K(x,y)$ that viscosity solutions of the integro-differential equation $\int_{\mathbb{R}^n} \left[u(x+y)+u(x-y)-2u(x)\right]\,K(x,y)dy=f(x)$ locally belong to some Gevrey class if so does $f$. The fractional Laplacian equation is included in this framework as a special case., 15 pages

In this paper, we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation, and it is obtained using variational and topological methods. We treat both the non-resonant case and the resonant one. Copyright © 2015 John Wiley & Sons, Ltd.

Aim of this paper is to give the details of the proof of some density properties of smooth and compactly supported functions in the fractional Sobolev spaces and suitable modifica- tions of them, which have recently found application in variational problems. The arguments are rather technical, but, roughly speaking, they rely on a basic technique of convolution (which makes functions C 1 ), joined with a cut-off (which makes their support compact), with some care needed in order not to exceed the original support.

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the \(t\)-perimeter, up to multiplicative constants, controls from above that of the \(s\)-perimeter, with \(s\) smaller than \(t\). To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the \(t\)-perimeter and the \(s\)-perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on \(t-s\), while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all \(s,\,t\). When \(s=0\) this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in Ω , u = 0 in R n ∖ Ω where L K is an integrodifferential operator with kernel K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.

We consider a nonlocal (or fractional) curvature and we investigate similarities and differences with respect to the classical local case. In particular, we show that the nonlocal mean curvature can be seen as an average of suitable nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case. Nevertheless, different from the classical cases, minimal and maximal nonlocal directional curvatures are not in general attained at perpendicular directions and, in fact, one can arbitrarily prescribe the set of extremal directions for nonlocal directional curvatures. Also the classical directional curvatures naturally enjoy some linear properties that are lost in the nonlocal framework. In this sense, nonlocal directional curvatures are somewhat intrinsically nonlinear. © 2014 Copyright Taylor & Francis Group, LLC.

The main result of the paper [BCV] Elliptic two-dimensional invariant tori for the planetary three-body problem by the authors, published in Arch. Ration. Mech. Anal. 170, 91–135 (2003), namely, Theorem 1.2, exploits a classical result (Theorem 1.1) essentially due to Delaunay and Poincare. In Appendix C of [BCV] a detailed proof of Theorem 1.1 is discussed. However, this proof contains a flaw. In fact, the statement following equation (C.44) in Proposition C.3, Section C.2.2, Appendix C is not correct. The Z∗ j ’s (called in [BCV] “Poincare integrals”) are not global integrals. Therefore, Section C.2.2 requires amendment. What now follows, replaces Section C.2.2 in [BCV], correcting the above mentioned flaw. Notations are as in [BCV]. Recall that in [BCV] we consider the 10-dimensional invariant symplectic manifold, Mver, defined by taking the non-vanishing total angular momentum to be vertical, i.e., C:=C(1) + C(2) = ck3 where C denotes the angular momentum of the ith planet with respect to the origin of an inertial heliocentric frame {k1, k2, k3} and c ∈ R\{0}; compare (C.30) and (C.31). Recall, also, that in [BCV] we use the “osculating Poincare variables” ( ∗, η∗, p∗, λ∗, ξ∗, q∗) as described in Section C.2.1 of [BCV], where ∗ = ( ∗1, ∗2), η∗ = (η∗ 1, η∗ 2), etc., and the index i = 1, 2 refers to the two-body system ith planet-star. Finally, recall that −ζ ∗ i = θi is the longitude of the ith node on the plane {k1, k2}, while −Z∗ i = i = C · k3. Let us now describe, analytically, Mver. From the definition of the Delaunay variables (Li, i,Gi, gi, i, θi) (compare to Section C.1.1) it follows that the relations defining Mver, i.e., C · k1 = 0 = C · k2, are equivalent to ζ ∗ 2 − ζ ∗ 1 = π and G1 − 1 = G2 − 2. Therefore, since Gi = ∗i − H ∗ i , with H ∗ i :=(η∗ i 2 + ξ∗ i )/2 (compare to (C.5), (C.15) and (C.17)), we find

In this paper we consider a resonance problem driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation {(-δ)su=λa(x)u + f(x; u) in u = 0 in Rn n ; when λ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s 2 (0; 1) is fixed, is an open bounded set of Rn, n > 2s, with Lipschitz boundary, a is a Lipschitz continuous function, while f is a suffciently smooth function. This existence theorem extends to the non-local setting some results, already known in the literature in the case of the Laplace operator delta;.

For any $Ngeq 2$ we prove the existence of quasi--periodic orbits lying on $N$--dimensional invariant elliptic tori for the planetary planar $(N+1)$--body problem. For small planetary masses, such orbits are close to the limiting solutions given by the $N$ planets revolving around the sun on planar circles. The eigenvalues of the linearized secular dynamics are also computed asymptotically. The proof is based on an appropriate averaging/KAM theory which overcomes the difficulties caused by the intrinsic degeneracies of the model. For concreteness, we focus on a caricature of the outer solar system.