arXiv subjects

Clara Torres-Latorre

Publications and source records attributed to Clara Torres-Latorre.

12 recordsLinked to original sources

arXiv2026

Boundary regularity of harmonic functions in $C^1$ slit domains

We establish precise upper and lower estimates for harmonic functions vanishing on the slit of a $C^1$ slit domain, with no assumption that the slit lies in a hyperplane. The classical $\sqrt{d}$ growth near the edge, $d$ being the distance to the slit, persists in this generality, up to an explicit factor \[\exp\Big( \pm C \int_ρ^r ω(s)\, \frac{ds}{s} \Big)\] determined by the $C^1$-modulus of continuity $ω$ of the slit, where $ρ< r$ are the two scales being compared. The upper estimates allow a right-hand side and non-zero boundary data. The correction factors remain bounded above and below by positive constants as $ρ\to0$ precisely when $ω$ satisfies the Dini condition. Moduli of continuity beyond the Dini regime, as is the case for the logarithmic moduli arising at singular sets in relevant free boundary problems, were not covered by the previous $C^{1,α}$ theory. Previously, the $\sqrt{d}$ growth was known for slits contained in a hyperplane, which additionally have a $C^{1,α}$ edge (De Silva, Savin). For Lipschitz slits, there are boundary Harnack principles, but no growth rate is identified precisely. The main technical ingredient is a change of coordinates flattening a Lipschitz slit domain onto the model half-hyperplane slit, with quantitative estimates up to second order. The construction is geometric and does not use the equation, so we expect it to be useful for other boundary regularity problems.

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arXiv2026

Minimizers for Coulomb gases constrained to a halfspace

We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.

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arXiv2026

Boundary estimates for parabolic non-divergence equations in $C^1$ domains

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for parabolic Lipschitz domains, unifying both regimes with a single proof.

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arXiv2026

Existence and nonexistence of infinitely many solutions to elliptic problems with oscillating nonlinearities

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-Δ_p u = λf(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function with a sequence of positive zeros converging to zero or diverging to infinity. Under a growth condition on the primitive $F(s) = \int_0^s f(t)dt$, we establish ranges of the parameter $λ$ for which infinitely many small or large solutions exist, as well as ranges where no bifurcation from zero or infinity can occur. The existence result is obtained via variational methods for a general class of divergence form operators, while the nonexistence result is established both for the $p$-Laplacian and for uniformly elliptic operators in non-divergence form via an ODE argument.

math.AP↗

arXiv2026

The Pohozaev identity for the Spectral Fractional Laplacian

In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-Δ|_Ω)^su = f(u)$ in star-shaped domains. The first such identity for non-local operators was established by Ros-Oton and Serra in 2014 for the Restricted Fractional Laplacian (RFL). However, the SFL differs fundamentally from the RFL, and the integration by parts strategy of Ros-Oton and Serra cannot be applied. Instead, we develop a novel spectral approach that exploits the underlying quadratic structure. Our main result expresses the identity as a Schur product of the classical Pohozaev quadratic form and a transition matrix that depends on the eigenvalues of the Laplacian and the fractional exponent.

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arXiv2025

Boundary estimates for non-divergence equations in $C^1$ domains

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence equations in $C^1$ domains, providing an explicit modulus of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for flat Lipschitz domains, unifying both theories with a single proof.

math.AP↗

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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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.

math.AP↗

arXiv2024

Parabolic boundary Harnack inequalities with right-hand side

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing for the first time a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side $f \in L^q$ for $q > n+2$. In the case of the heat equation, we also show the optimal $C^{1-\varepsilon}$ regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are $C^{1,α}$ in the parabolic obstacle problem and in the parabolic Signorini problem.

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arXiv2023

Generic regularity of free boundaries for the thin obstacle problem

The free boundary for the Signorini problem in $\mathbb{R}^{n+1}$ is smooth outside of a degenerate set, which can have the same dimension ($n-1$) as the free boundary itself. In [FR21] it was shown that generically, the set where the free boundary is not smooth is at most $(n-2)$-dimensional. Our main result establishes that, in fact, the degenerate set has zero $\mathcal{H}^{n-3-α_0}$ measure for a generic solution. As a by-product, we obtain that, for $n+1 \leq 4$, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in $\mathbb{R}^3$ and $\mathbb{R}^4$ for the thin obstacle problem.

math.AP↗

arXiv2021

Optimal regularity for supercritical parabolic obstacle problems

We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-Δ)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result in this context was due to Caffarelli and Figalli, who established the $C^{1,s}_x$ regularity of solutions for the case $L = (-Δ)^s$, the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually \textit{more} regular than in the elliptic case. More precisely, we show that they are $C^{1,1}$ in space and time, and that this is optimal. We also deduce the $C^{1,α}$ regularity of the free boundary. Moreover, at all free boundary points $(x_0,t_0)$, we establish the following expansion: $$(u - φ)(x_0+x,t_0+t) = c_0(t - a\cdot x)_+^2 + O(t^{2+α}+|x|^{2+α}),$$ with $c_0 > 0$, $α> 0$ and $a \in \mathbb R^n$.

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arXiv2021

New boundary Harnack inequalities with right hand side

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in $L^q$ with $q > n$. Our approach is based on the scaling and comparison arguments of \cite{DS20}, and we show that all our assumptions are sharp. As a consequence of our results, we deduce the $\mathcal{C}^{1,α}$ regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

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