arXiv subjects

Xavier Ros-Oton

Publications and source records attributed to Xavier Ros-Oton.

At least 19 recordsLinked to original sources

arXiv2026

The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and $L^{2}$-solvability

Given $s\in (0,1)$ and a bounded Lipschitz domain $Ω\subset \mathbb{R}^{n}$, we establish a quantitative Dahlberg theory for the $s$-harmonic measure of $Ω$, $ω_s^x$. In the nonlocal setting, the natural reference measure is an integral weight $σ_{s}$ in $Ω^{c}$ that behaves like $(1-s)\text{dist}(\cdot, \partial Ω)^{-s}$ close to the boundary. Our main result is a scale-invariant reverse-Hölder estimate for the density $dω_{s}^{x}/dσ_{s}$ on boundary-centered balls. As a consequence, we obtain $L^2(Ω^c,σ_s)$-solvability of the exterior Dirichlet problem, with estimates for a nonlocal non-tangential maximal function and uniqueness in the natural distributional class. A weighted Gehring argument improves the reverse-Hölder exponent beyond $2$ and consequently yields $L^{q}$-solvability for a range of exponents extending strictly below $2$. Our results apply to general symmetric stable operators comparable to the fractional Laplacian. Moreover, the proofs are compatible with the limit $s\to 1^-$ and thus yield the corresponding results for the Laplacian in the nonlocal-to-local limit. The main new step is to convert a fractional Pohozaev identity for the Green function into uniform square estimates on distance level sets of a Lipschitz boundary. As applications, we derive optimal Sobolev regularity estimates for the homogeneous weighted Dirichlet problem and for the inhomogeneous Poisson problem with zero exterior data.

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arXiv2026

Boundary regularity for general elliptic operators of order $2s$

We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider Lévy operators that are symmetric and its Fourier symbol satisfies $\mathcal{A}(ξ)\asymp |ξ|^{2s}$ in $\mathbb{R}^d$. This was only known when the kernel of the operator (or Lévy measure) is either homogeneous or comparable to that of the fractional Laplacian, with different proofs in each case. Our new proofs extend both at the same time, and work in a very general class of domains, under a $C^1$-Dini-type condition.

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arXiv2026

Dimension of the singular set in the parabolic obstacle problem

In this paper we study the singular set in the parabolic obstacle problem for general obstacles $φ\in C^{2,1}$. We prove that the singular set has parabolic Hausdorff dimension at most $n-1$. Prior to our result, this was only known when $Δφ\equiv -1$. Our approach combines a truncated parabolic frequency formula and monotonicity estimates with an iterative argument showing that the frequency is saturated for all values of the truncation parameter between $2$ and $3$.

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arXiv2026

Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems

Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-Δu = f$ in $Ω$, $u = 0$ on $\partialΩ$, $\partial_νu = Q$ on $\partialΩ$. Our main result establishes that if $Ω$ is Lipschitz, then it is actually $C^{\infty}$ (provided that $f$ and $Q$ are smooth). This was known before only for viscosity solutions. As a corollary, we obtain an alternative solution of Serrin's problem in the case of Lipschitz domains. We also discuss the characterisation of the regularity of Lipschitz domains in terms of their Poisson kernel.

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arXiv2025

The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform

We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-Δ)^s u=h$ in $Ω$, with the external condition $\mathcal N^s u=0$ in $Ω^c$. For this, a key point is to establish a 1D Liouville theorem for functions with growth, which we prove by using complex analysis and the Mellin transform. More precisely, we prove a ``meta-theorem'' relating the classification of 1D solutions to general linear homogeneous equations of the type $Lu=0$ in $(0,\infty)$ to the (complex) roots of an explicit meromorphic function $f(z)$ that depends on $L$. In case of the fractional Laplacian with Neumann conditions, we show that all solutions are $C^{2s+α}$ when $s\leq 1/2$, and $C^{s+\frac12+α}$ when $s\geq1/2$. Moreover, quite surprisingly, we prove that even in 1D there exist highly oscillating solutions of the type $u(x)=x^{a} \cos(b \log x)$ for $x>0$, with $a>0$ and $b>0$ that depend on $s$, and $a<2s$ for $s\sim1$.

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arXiv2025

Higher regularity in nonlocal free boundary problems

We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,α}$ free boundaries are $C^\infty$. This is new even for the fractional Laplacian, as it was only known in case $s=\frac12$. We also establish a general result for overdetermined problems, showing that if the boundary condition is smooth, then so is $\partialΩ$. Our approach is very robust and works as well for the nonlocal obstacle problem, where it yields a new proof of the higher regularity of free boundaries, completely different from the one in [AbRo20]. In order to prove our results, we need to develop, among other tools, new integration by parts formulas and delicate boundary Hölder estimates for nonlocal equations with (local) Neumann boundary conditions that had not been studied before and are of independent interest.

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arXiv2025

Positive solutions to general semilinear overdetermined boundary problems

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $Ω\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical hypotheses on the coefficients and the nonlinearity, we show that there exist open sets $Ω\subset\mathbb{R}^n$ with smooth boundary and of any prescribed volume where the overdetermined problem admits a positive solution. The proof builds on ideas of Alt and Caffarelli on variational problems for functions defined on a bounded region. In our case, we need to consider functions defined on the whole $\mathbb{R}^n$, so the key challenge is to obtain uniform bounds for the minimizer and for the diameter of its support. Our methods extend to higher dimensions, although in this case the free boundary $\partialΩ$ could have a singular set of codimension 5. The results are new even in the case of the Poisson equation $-Δv =g(x)$ with constant Neumann data.

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arXiv2025

Regularity for the Boltzmann equation conditional to pressure and moment bounds

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in $L^\infty$ in the case of hard potentials. As a consequence, we derive $C^{\infty}$ estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our $L^\infty$ estimates are uniform in the limit $s \nearrow 1$ and hence we recover the same results also for the Landau equation.

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arXiv2025

$L^p$ estimates for the Laplacian via blow-up

In this note we provide a new proof of the $W^{2,p}$ Calderón-Zygmund regularity estimates for the Laplacian, i.e., $Δu=f$ and its parabolic counterpart $\partial_t u-Δu=f$. Our proof is an adaptation of a contradiction and compactness argument that so far had been only used to prove estimates in Hölder spaces. This new approach is simpler than previous ones, and avoids the use of any interpolation theorem.

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arXiv2025

Optimal regularity for kinetic Fokker-Planck equations in domains

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $Ω\subset \mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\partial_t f +v\cdot\nabla_x f-Δ_v f=h$. Our main results establish the following: - Solutions are always $C^\infty$ in $t,v,x$ away from the grazing set $\{x\in\partialΩ,\ v\cdot n_x=0\}$. - They are $C^{4,1}_{\text{kin}}$ up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not $C^5_{\text{kin}}$. These results show for the first time that solutions are classical up to boundary, i.e. $C^1_{t,x}$ and $C^2_v$.

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arXiv2025

Regularity for nonlocal equations with local Neumann boundary conditions

In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in $C^{k,γ}$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v \asymp d^{s-1}$ and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients $v/d^{s-1}$, depending on the regularity of the domain and on the data of the problem. The results of this article will be important in a forthcoming work on nonlocal free boundary problems.

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arXiv2025

Extinction rates for nonradial solutions to the Stefan problem

We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and (more surprisingly) lower estimates for its evolution. In 2D, these bounds almost match the best known ones for radial solutions, but hold for all solutions to the Stefan problem, with no extra assumption on the initial or boundary data. On the other hand, as a consequence of our results, we also characterize the global regularity of the free boundary, as follows: it can be written as a graph $t = Γ(x)$, where $Γ$ is $C^1$ (and not $C^2$) near any singular points in the lower strata $Σ_m$, $m \leq n - 2$. Moreover, $Γ$ is not $C^1$ at singular points in $Σ_{n-1}$.

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arXiv2025

Improvement of flatness for nonlocal free boundary problems

In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,α}$ near regular points, and that the set of regular free boundary points is open and dense. Moreover, in 2D we classify all blow-up limits and prove that the free boundary is $C^{1,α}$ everywhere. The main technical tool of our proof is an improvement of flatness scheme, which we establish in the general framework of viscosity solutions, and which is of independent interest. All of these results were only known for the fractional Laplacian, and are completely new for general nonlocal operators. In contrast to previous works on the fractional Laplacian, our method of proof is purely nonlocal in nature.

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arXiv2024

Integro-Differential Elliptic Equations

This book aims to provide a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. Such a class of equations often arises in analysis, probability theory, mathematical physics, and in several contexts in applied sciences. The authors give a detailed presentation of all the necessary techniques, primarily focusing on the main ideas rather than proving all results in their greatest generality. The book starts from the very basics, studying the square root of the Laplacian and weak solutions to linear equations. Then, the authors develop the theory of viscosity solutions to nonlinear equations and prove the main known results in this context. Finally, they study obstacle problems for integro-differential operators and establish the regularity of solutions and free boundaries. Almost all the covered material appears in book form for the first time, and several proofs are different (and shorter) than those in the original papers. Moreover, several open problems are listed throughout the book. This is a draft of the book ''Integro-Differential Elliptic Equations''. The final version has been published in Progress in Mathematics, vol. 350, Birkhäuser, 2024.

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arXiv2024

Semiconvexity estimates for nonlinear integro-differential equations

In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro-differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré-Dipierro-Valdinoci [CDV22]. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.

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arXiv2024

$C^\infty$ regularity in semilinear free boundary problems

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $Δu=u^{γ-1}$, with $γ\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. In addition $u/d^{\frac{2}{2-γ}}$ and $u^{\frac{2-γ}{2}}$ are $C^\infty$ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-Δv = κv/d^2$ in $Ω$, where $d$ is the distance to the boundary and $κ\leq\frac{1}{4}$. Interestingly, we need to include even the critical constant $κ=\frac{1}{4}$, which corresponds to $γ=\frac{2}{3}$.

math.AP↗

arXiv2024

Optimal regularity for nonlocal elliptic equations and free boundary problems

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.

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arXiv2023

Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels

We study integro-differential elliptic equations (of order $2s$) with variable coefficients, and prove the natural and most general Schauder-type estimates that can hold in this setting, both in divergence and non-divergence form. Furthermore, we also establish Hölder estimates for general elliptic equations with no regularity assumption on $x$, including for the first time operators like $\sum_{i=1}^n(-\partial^2_{\textbf{v}_i(x)})^s$, provided that the coefficients have ``small oscillation''.

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