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Joaquim Serra

Publications and source records attributed to Joaquim Serra.

At least 37 records · Page 2Linked to original sources
arXiv2019

Regularity of minimal surfaces with lower dimensional obstacles

We study the Plateau problem with a lower dimensional obstacle in $\mathbb{R}^n$. Intuitively, in $\mathbb{R}^3$ this corresponds to a soap film (spanning a given contour) that is pushed from below by a "vertical" 2D half-space (or some smooth deformation of it). We establish almost optimal $C^{1,1/2-}$ estimates for the solutions near points on the free boundary of the contact set, in any dimension $n\ge 2$. The $C^{1,1/2-}$ estimates follow from an $\varepsilon$-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi's improvement of flatness. To prove it, we follow Savin's small perturbations method. A nontrivial difficulty in using Savin's approach for minimal surfaces with thin obstacles is that near a typical contact point the solution consists of two smooth surfaces that intersect transversally, and hence it is not very flat at small scales. Via a new "dichotomy approach" based on barrier arguments we are able to overcome this difficulty and prove the desired result.

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arXiv2017

On the fine structure of the free boundary for the classical obstacle problem

In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of $C^1$ manifolds of varying dimension. In two dimensions, this $C^1$ result has been improved to $C^{1,α}$ by Weiss \cite{W99}. In this paper we prove that, for $n=2$ singular points are locally contained in a $C^2$ curve. In higher dimension $n\ge 3$, we show that the same result holds with $C^{1,1}$ manifolds (or with countably many $C^2$ manifolds), up to the presence of some "anomalous" points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a $C^{1,α}$ manifold, thus extending to every dimension the result in \cite{W99}. We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for $n\ge3$ we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.

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arXiv2017

Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$

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.

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arXiv2017

Quantitative stability of the free boundary in the obstacle problem

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in $\mathbb{R}^n$ ($n \ge 2$) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field. To do so, working in the setting of the whole space, we examine the evolution of the free boundary $Γ^t$ corresponding to the boundary of the contact set for a family of obstacle functions $h^t$. Assuming that $h=h^t (x) = h(t,x)$ is $C^{k+1,α}$ in $[-1,1]\times \mathbb{R}^n$ and that the initial free boundary $Γ^0$ is regular, we prove that $Γ^t$ is twice differentiable in $t$ in a small neighborhood of $t=0$. Moreover, we show that the "normal velocity" and the "normal acceleration" of $Γ^t$ are respectively $C^{k-1,α}$ and $C^{k-2,α}$ scalar fields on $Γ^t$. This is accomplished by deriving equations for these velocity and acceleration and studying the regularity of their solutions via single and double layers estimates from potential theory.

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arXiv2017

Improvement of flatness for nonlocal phase transitions

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(-Δ)^{s/2} u = u-u^3$, with~$s\in(0,1)$, we obtain a result in the same spirit of a celebrated theorem of Savin for the equation $-Δu = u-u^3$. As a consequence, we deduce that entire solutions to~$(-Δ)^{s/2} u = u-u^3$ with asymptotically flat level sets are $1$D when~$s\in(0,1)$. The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.

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arXiv2017

On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1

We prove that every bounded stable solution of \[ (-Δ)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u(x)= ϕ(e\cdot x)$ for some $e\in \mathbb S^2$, where $ϕ:\mathbb R\to \mathbb R$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\mathbb R^{d+1}_+=\mathbb R^{d+1}\cap \{x_{d+1}\geq 0\}$ when $d = 4$, by proving that all critical points of $$ \int_{\{x_{d+1\geq 0}\}} \frac12 |\nabla U|^2 \,dx\, dx_{d+1} + \int_{\{x_{d+1}=0\}} \frac 1 4 (1-U^2)^2 \,dx $$ that are monotone in $\mathbb R^d$ (that is, up to a rotation, $\partial_{x_d} U>0$) are one dimensional. Our result is analogue to the fact that stable embedded minimal surfaces in $\mathbb R^3$ are planes. Note that the corresponding result about stable solutions to the classical Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still open.

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arXiv2017

Nonlocal phase transitions: rigidity results and anisotropic geometry

We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form $$ (-Δ)^{s/2} u=u-u^3,$$ with~$s\in(0,1)$. More generally, we can take into account equations like $$ L u = f(u),$$ where $f$ is a bistable nonlinearity and $L$ is an integro-differential operator, possibly of anisotropic type. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in the companion paper.

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arXiv2016

Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces

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 $n\ge 2$ 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.

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arXiv2016

Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries

We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,γ}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $x_0\in\partial\{u=φ\}$: (i) either $u(x)-φ(x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+α})$ for some $c>0$, (ii) or $u(x)-φ(x)=o(|x-x_0|^{1+s+α})$, where $d$ is the distance to the contact set $\{u=φ\}$. Moreover, we show that the set of free boundary points $x_0$ satisfying (i) is open, and that the free boundary is $C^{1,γ}$ and $u\in C^{1,s}$ near those points. These results were only known for the fractional Laplacian \cite{CSS}, and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators.

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arXiv2016

The boundary Harnack principle for nonlocal elliptic operators in non-divergence form

We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $Ω\cap B_1$, $u_1=u_2=0$ in $B_1\setminusΩ$, and $u_1,u_2\geq0$ in $\mathbb R^n$, then $u_1$ and $u_2$ are comparable in $B_{1/2}$. The result applies to arbitrary open sets $Ω$. When $Ω$ is Lipschitz, we show that the quotient $u_1/u_2$ is Hölder continuous up to the boundary in $B_{1/2}$.

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arXiv2016

Boundary regularity estimates for nonlocal elliptic equations in $C^1$ and $C^{1,α}$ domains

We establish sharp boundary regularity estimates in $C^1$ and $C^{1,α}$ domains for nonlocal problems of the form $Lu=f$ in $Ω$, $u=0$ in $Ω^c$. Here, $L$ is a nonlocal elliptic operator of order $2s$, with $s\in(0,1)$. First, in $C^{1,α}$ domains we show that all solutions $u$ are $C^s$ up to the boundary and that $u/d^s\in C^α(\barΩ)$, where $d$ is the distance to $\partialΩ$. In $C^1$ domains, solutions are in general not comparable to $d^s$, and we prove a boundary Harnack principle in such domains. Namely, we show that if $u_1$ and $u_2$ are positive solutions, then $u_1/u_2$ is bounded and Hölder continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in non-divergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators \cite{CRS-obstacle}.

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arXiv2016

Pohozaev identities for anisotropic integro-differential operators

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity $u/d^s|_{\partialΩ}$ plays the role that $\partial u/\partialν$ plays in the second order case. Here, $u$ is any solution to $Lu=f(x,u)$ in $Ω$, with $u=0$ in $\mathbb R^n\setminusΩ$, and $d$ is the distance to $\partialΩ$.

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arXiv2015

An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

We study nonlinear elliptic equations for operators corresponding to non-stable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) Lévy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen-Cahn type equations with a non-stable Lévy diffusion. These operators may still be realized as local operators using a system of PDEs ---in the spirit of the extension problem of Caffarelli and Silvestre.

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arXiv2015

$C^{σ+α}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

We establish $C^{σ+α}$ interior estimates for concave nonlocal fully nonlinear equations of order $σ\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^α(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(σ)$, then $u$ belongs to $C^{σ+α}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^α$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.

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arXiv2015

Boundary regularity for fully nonlinear integro-differential equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which consists of infinitesimal generators of stable Lévy processes belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that solutions to $I u=f$ in $Ω$, $u=0$ in $\mathbb R^n\setminusΩ$, satisfy $u/d^s\in C^{s+γ}(\overlineΩ)$, where $d$ is the distance to $\partialΩ$ and $f\in C^γ$. We expect the class $\mathcal L_*$ to be the largest scale invariant subclass of $\mathcal L_0$ for which this result is true. In this direction, we show that the class $\mathcal L_0$ is too large for all solutions to behave like $d^s$. The constants in all the estimates in this paper remain bounded as the order of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.

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arXiv2015

The Pohozaev identity for the fractional Laplacian

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $(-Δ)^s u = f(u)$ in $Ω$, $u \equiv 0$ in $\mathbb R^n\setminusΩ$. Here, $s\in(0,1)$, $(-Δ)^s$ is the fractional Laplacian in $\mathbb R^n$, and $Ω$ is a bounded $C^{1,1}$ domain. To establish the identity we use, among other things, that if $u$ is a bounded solution then $u/δ^s|_Ω$ is $C^α$ up to the boundary $\partialΩ$, where $δ(x)={\rm dist}(x,\partialΩ)$. In the fractional Pohozaev identity, the function $u/δ^s|_{\partialΩ}$ plays the role that $\partial u/\partialν$ plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over $\partialΩ$) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.

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arXiv2015

Local integration by parts and Pohozaev identities for higher order fractional Laplacians

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-Δ)^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$. As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-Δ)^sϕ=λϕ$ in $Ω$, $ϕ\equiv0$ in $\mathbb R^n\setminusΩ$.

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