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We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels $K$ which are convex and kernels for which $K(\tau^{1/2})$ is a completely monotonic function of $\tau$. This last new class arose in our previous work on nonlocal Delaunay surfaces in $\mathbb{R}^n$. Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle $\mathbb{S}^1$ established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in $C^\beta$ and when $2s+\beta <1$ ($2s$ being the order of the operator), the solution is not always $C^{2s+\beta-\epsilon}$ for all $\epsilon>0$., Comment: 47 pages

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here ``minimality'' is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions., Comment: 34 pages, 1 figure

We consider analytic maps and vector fields defined in $\mathbb{R}^2 \times \mathbb{T}^d$, having a $d$-dimensional invariant torus $\mathcal{T}$. The map (resp. vector field) restricted to $\mathcal{T}$ defines a rotation of frequency $\omega$, and its derivative restricted to transversal directions to $\mathcal{T}$ does not diagonalize. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which $\mathcal{T}$ possesses stable and unstable invariant manifolds, and we show that such invariant manifolds are analyitic away from the invariant torus. We also provide effective algorithms to compute approximations of parameterizations of the invariant manifolds, and present some applications of the results., Comment: 42 pages

In this paper we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo-Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory., Comment: Minor changes in the acknowledgments and spelling in first author's name

We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is much more delicate. It requires the function to be a stable solution of a semilinear elliptic equation with a nonnegative, nondecreasing, and convex nonlinearity. As an application, our estimates provide quantitative proofs of two results established by contradiction-compactness arguments in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)]. We recall that this work proved the H\"older regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n \leq 9$., Comment: To appear in Amer. J. Math. arXiv admin note: substantial text overlap with arXiv:2205.11352

Auditory verbal hallucinations (AVH) in schizophrenia (SZ) have been associated with abnormalities of the left arcuate fasciculus and transcallosal white matter projections linking homologous language areas of both hemispheres. While most studies have used a whole-tract approach, here we focused on analyzing local alterations of the above-mentioned pathways in SZ patients suffering medication-resistant AVH.Fractional anisotropy (FA) was estimated along the left arcuate fasciculus and interhemispheric projections of the rostral and caudal corpus callosum. Then, potential associations between white matter tracts and SZ symptoms were explored by correlating local site-by-site FA values and AVH severity estimated via the Auditory Hallucinations Rating Scale (AHRS). Compared to a sample of healthy controls, SZ patients displayed lower FA values in the rostral portion of the left arcuate fasciculus, near the frontal operculum, and in the left and right lateral regions of the rostral portion of the transcallosal pathways. In contrast, SZ patients showed higher FA values than healthy controls in the medial portion of the latter transcallosal pathway and in the midsagittal section of the interhemispheric auditory pathway. Finally, significant correlations were found between local FA values in the left arcuate fasciculus and the severity of the AVH's attentional salience. Contributing to the study of associations between local white matter alterations of language networks and SZ symptoms, our findings highlight local alterations of white matter integrity in these pathways linking language areas in SZ patients with AVH. We also hypothesize a link between the left arcuate fasciculus and the attentional capture of AVH., Comment: Neuroscience, Elsevier - International Brain Research Organization, 2022

This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in $\mathbb{R}^n$ which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is even as well as nonincreasing on half its period. This monotonicity property solves an open problem and an obstruction which arose in an earlier attempt, by other authors, to show the existence of minimizers.

It is well known that linear vector fields defined in $\mathbb{R}^n$ can not have limit cycles, but this is not the case for linear vector fields defined in other manifolds. We study the existence of limit cycles bifurcating from a continuum of periodic orbits of linear vector fields on manifolds of the form $(\mathbb{S}^2)^m \times \mathbb{R}^n$ when such vector fields are perturbed inside the class of all linear vector fields. The study is done using the averaging theory. We also present an open problem concerning the maximum number of limit cycles of linear vector fields on $(\mathbb{S}^2)^m \times \mathbb{R}^n$., Comment: 15 pages

This article concerns the results obtained in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)], which established the H\"older regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n \leq 9$. For expository purposes, we provide self-contained proofs of all results. They only involve basic Analysis tools and can be understood by mathematicians who are not experts in PDEs. Two of the results in the 2020 article relied on compactness arguments. Here we present, instead, quantitative proofs from the more recent paper [Cabre, to appear in Amer. J. Math, arXiv:2211.13033]. They allow to quantify the H\"older regularity exponent and simplify significantly the treatment of boundary regularity. We also comment on similar progress and open problems for related equations.

The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue question in differential geometry: the regularity of stable minimal surfaces., Comment: To appear in Proceedings of the 8ECM. Note of 6 pages for non-experts

We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-\Delta)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and the convergence of blow-downs to stable nonlocal $s$-minimal cones. As a consequence, we obtain a new classification result: if for some pair $(n,s)$, with $n\ge 3$, hyperplanes are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^n\setminus\{0\}$, then every stable solution to the fractional Allen-Cahn equation in $\mathbb{R}^n$ is 1D, namely, its level sets are parallel hyperplanes. Combining this result with the classification of stable $s$-minimal cones in $\mathbb{R}^3\setminus\{0\}$ for $s\sim 1$ obtained by the authors in a recent paper, we give positive answers to the "stability conjecture" in $\mathbb{R}^3$ and to the "De Giorgi conjecture" in $\mathbb{R}^4$ for the fractional Allen-Cahn equation when the order $s\in (0,1)$ of the operator is sufficiently close to $1$.

We study stable solutions to the equation $(-\Delta)^{1/2} u = f(u)$, posed in a bounded domain of $\mathbb{R}^n$. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions $n\leq 4$. This result, which was known only for $n=1$, follows from a new interior H\"older estimate that is completely independent of the nonlinearity $f$. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabr\'e, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the H\"older bound for $n\leq 4$, a universal $H^{1/2}$ estimate in all dimensions. Our $L^\infty$ bound is expected to hold for $n\leq 8$, but this has been settled only in the radial case or when $f(u) = \lambda e^u$. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when $f(u) = \lambda e^u$, even in the radial case., Comment: 38 pages, 1 figure. Version 2 contains a new Liouville result (Corollary 1.5) which is a corollary of our main estimate. Version 3 contains a short comment (Remark 3.1) on the difficulties of extending the result to other powers of the Laplacian. Version 4 is the final version after referee report and proofs, we have added a comment to facilitate the interpretation of the colors of Figure 1

The Atacama Desert has long been established as an excellent site for submillimeter observations. Yet identifying potentially optimal locations for a new facility within this region can require long field campaigns that rely on the construction of weather stations and radiometer facilities to take data over sufficiently long timescales. Meanwhile, high-level remote sensing data products from satellites have generally only been available at >25 km resolution, limiting their utility for astronomical site selection. We aim to improve and expedite the process of site selection through the use of km-resolution satellite data. We analyze the daytime precipitable water vapor (PWV) values inferred using near-IR measurements from the MODIS Aqua and Terra satellites, comparing the level-2 satellite products to those from existing ground-based measurements from the radiometer at the Atacama Pathfinder Experiment (APEX) site. Since the APEX radiometer data has been extensively tested and compared to atmospheric transmission models, particularly in low-PWV conditions of interest for astronomy, we use these data to provide a re-calibration of the MODIS data for the entire region. After re-calibration, the satellite data allow mapping of the PWV across the region, and we identify several promising sites. We demonstrate a potentially powerful method for siting new facilities such as AtLAST and extensions to global very long baseline interferometry networks like the EHT. Further, our findings identify a trend in both the APEX and MODIS data, and indicate that the PWV has increased moderately over the past two decades. We verify this trend exists in the Coupled Model Intercomparison Project Phase 6 (CMIP6) climate models, and show that it is worse in the case of greater increase in the average global temperature., Comment: significantly revised and resubmitted version. includes climate model predictions for PWV worldwide

We consider a map $F$ of class $C^r$ with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $F$, there exist invariant curves of class $C^r$ away from the fixed point, and that they are analytic when $F$ is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of $F$ on them., Comment: 43 pages

We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two -- for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the "pure" linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels., Comment: To appear in Arch. Rat. Mech. Anal

We consider the equation $-\Delta_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $\Delta_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when $n

The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an additional $L^p$ term weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.

We study the set of periods of the Morse--Smale diffeomorphisms on the $n$-dimensional sphere $\mathbb{S}^n$, on products of two spheres of arbitrary dimension $\mathbb{S}^m \times \mathbb{S}^n$ with $m \neq n$, on the $n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and on the $n$-dimensional quaternion projective space $\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz periods for such Morse--Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function., Comment: 12 pages

In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael-Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a "ground state" substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy-Poincar\'e inequality., Comment: In version 2, an equality within the proof of the Michael-Simon inequality has been corrected and it is now an inequality. This is explained in a new footnote. The correction does not alter at all the rest of the proof

We study the following natural question that, apparently, has not been well addressed in the literature: Given functions $u$ with support in the unit ball $B_1\subset\mathbb{R}^n$ and with gradient in the Morrey space $M^{p,\lambda}(B_1)$, where $1\lambda p/(\lambda-p)$. The function is basically a negative power of the distance to a set of Hausdorff dimension $n-\lambda$. When $\lambda\notin\mathbb{Z}$, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent $q$ can go up to $np/(\lambda-p)$., Comment: 30 pages, 4 figures. Version 2 contains new references and some comments to them. A few minor misprints have been corrected in version 3

In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. This result, that was only known to be true for $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n \leq 9$. This answers to two famous open problems posed by Brezis and Brezis-V\'azquez., Comment: To appear in Acta Mathematica

We consider the class of stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain of $\mathbb{R}^n$. Since 2010 an interior a priori $L^\infty$ bound for stable solutions is known to hold in dimensions $n \leq 4$ for all $C^1$ nonlinearities $f$. In the radial case, the same is true for $n \leq 9$. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions ---for instance $L^p$ bounds for every finite~$p$ in dimension 5. Since the mid nineties, the existence of an $L^\infty$ bound holding for all $C^1$ nonlinearities when $5 \leq n \leq 9$ was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper., Comment: To appear in a special volume of DCDS dedicated to Luis Caffarelli. While version 1 was written approximately one year ago, version 2 contains updated bibliography and comments to an important recent result. Some few additional comments added in version 3

For nonnegative even kernels $K$, we consider the $K$-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated $K$-nonlocal mean curvature equation in an open set $\Omega\subset\mathbb{R}^n$, we built a calibration for the nonlocal perimeter inside $\Omega\subset\mathbb{R}^n$. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in $\Omega$ of each leaf of the foliation. As an application, we prove the minimality of $K$-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions., Comment: To appear in Annali di Matematica Pura ed Applicata

These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces., Comment: To appear in Lecture Notes in Mathematics, CIME Foundation Subseries, Springer. Formulas (16) and (17) have been corrected

We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the $C^\infty$ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for $n = 1, 2$ (but without a quantitative bound), in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi & Miranda for solutions of the classical mean curvature equation., Comment: To appear in Duke Math. J

We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its proof can not rely on a compactness argument perturbing from $s=1$. In fact, our proof gives a quantifiable value for the required closeness of $s$ to $1$. We use the geometric formula for the second variation of the fractional $s$-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.

We report on the cryogenic characterization of Red Green Blue - High Density (RGB-HD) SiPMs developed at Fondazione Bruno Kessler (FBK) as part of the DarkSide program of dark matter searches with liquid argon time projection chambers. A dedicated setup was used to measure the primary dark noise, the correlated noise, and the gain of the SiPMs at varying temperatures. A custom-made data acquisition system and analysis software were used to precisely characterize these parameters. We demonstrate that FBK RGB-HD SiPMs with low quenching resistance (RGB-HD-LR$_q$) can be operated from 40 K to 300 K with gains in the range $10^5$ to $10^6$ and noise rates on the order of a few Hz/mm$^2$.

This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case.

We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension $M\leq N$. Our CNMC sets form a $C^2$ branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point. Thus, the new translated near-balls (or near-spheres) appear from infinity. We find their exact asymptotic shape as the parameter tends to infinity., Comment: 44 pages. Only the funding acknowledgements changed

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of them with the same constant nonlocal mean curvature, and bifurcating from a straight cylinder. These are Delaunay type cylinders in the nonlocal setting. The proof uses the Crandall-Rabinowitz theorem applied to a quasilinear type fractional elliptic equation., Comment: Minor changes have been made. To appear in "J. Math. Pures Appl."