arXiv subjects

Pablo Hidalgo-Palencia

Publications and source records attributed to Pablo Hidalgo-Palencia.

8 recordsLinked to original sources

arXiv2026

Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions

We show that the parabolic uniform rectifiability of a set can be characterized by an interior PDE property, namely Carleson measure estimates for bounded solutions to the heat equation. In particular, under very mild background hypotheses on an open set $Ω$ and its boundary $\partial Ω$, we show that the following are equivalent: (i) The quantity $|\nabla u(\cdot)|^2 \text{dist}(\cdot, \partialΩ)$ is the density of a Carleson measure on $Ω$ for all bounded solutions to the heat equation in $Ω$. (ii) The caloric measure for $Ω$ admits a corona decomposition. (iii) $\partial Ω$ is parabolic uniformly rectifiable. The implication (iii) implies (i) is the main result of the cited paper [Bortz, Hoffman, Hofmann, Luna-García, Nyström, Anal. PDE 2023]. The remaining implications are new, and are direct parabolic analogues of results of Garnett, Mourgoglou and Tolsa [Duke Math. J. 2018] in the elliptic setting for the Laplacian; however, our proofs require several new ideas. In particular, we must adapt arguments to account both for time-lag, and for the fact that some characterizations of (elliptic) uniform rectifiability have either not been developed or have been shown to be untrue in the parabolic setting. Our background assumptions, as noted above, are very mild: we assume that the domain satisfies the time-symmetric capacity density condition and has interior corkscrews, and that the boundary is (non time-directed) Ahlfors-David regular. These are weaker than the background assumptions of Bortz, Hofmann, Martell and Nyström in arXiv:2510.22047, and are essentially optimal for the boundary continuity properties of caloric functions on which our arguments rely.

math.AP↗

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.

math.AP↗

arXiv2025

The parabolic Dirichlet problem with continuous and Hölder boundary data, and rough coefficients

We provide very mild sufficient conditions for space-time domains (non-necessarily cylindrical) which ensure that the continuous Dirichlet problem and the Hölder Dirichlet problem are well-posed, for any parabolic operator in divergence form with merely bounded coefficients. Concretely, we show that the parabolic measure exists, even for unbounded domains, hence solving an open problem posed by Genschaw and Hofmann (2020). This problem has inherent difficulties because of its parabolic nature, as the behavior of solutions near the boundary may depend strongly on the values of the coefficients of the operator. One of our sufficient conditions, the time-backwards capacity density condition, is a quantitative version of the parabolic Wiener's criterion, and hence is adapted to the operator under consideration. The other condition, the time-backwards Hausdorff content condition, is (albeit slightly stronger) purely geometrical and independent of the operator, hence much easier to check in practice.

math.AP↗

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.

math.AP↗

arXiv2025

A Variable Coefficient Free Boundary Problem for $L^p$-solvability of Parabolic Dirichlet Problems in Graph Domains

We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nyström [BHMN25]. In particular, we show that if $Ω$ is the region above the graph of a Lip(1,1/2) (parabolic Lipschitz) function and $L$ is a parabolic operator in divergence form \[L = \partial_t - \text{div} A \nabla\] with $A$ satisfying an $L^1$ Carleson condition on its spatial and time derivatives, then the $L^p$-solvability of the Dirichlet problem for $L$ and $L^*$ implies that the graph function has a half-order time derivative in BMO. Equivalently, the graph is parabolic uniformly rectifiable. In the case of $A$ symmetric, we only require that the Dirichlet problem for $L$ is solvable, which requires us to adapt a clever integration by parts argument by Lewis and Nyström. A feature of the present work is that we must overcome the lack of translation invariance in our equation, which is a fundamental tool in similar works, including [BHMN25].

math.AP↗

arXiv2023

Elliptic operators in rough sets, and the Dirichlet problem with boundary data in Hölder spaces

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets $Ω\subset \mathbb{R}^{n+1}$, $n \ge 2$, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if $Ω$ is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when $Ω$ is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.

math.AP↗

arXiv2023

On the Kato problem for elliptic operators in non-divergence form

We consider the Kato square root problem for non-divergence second order elliptic operators $L =- a_{ij} D_iD_j$, and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real coefficients having sufficiently small BMO norm. We assume that the coefficients of the operator are smooth, but our estimates do not depend on the assumption of smoothness.

math.AP↗

arXiv2022

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Let $Ω$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $Ω$ is connected in some quantitative form one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, there is no Fefferman-Kenig-Pipher perturbation result available. In this paper, we work with a corona decomposition associated with the elliptic measure and show that it is equivalent to the fact that bounded null solutions satisfy partial/weak Carleson measure estimates, or to the fact that the Green function is comparable to the distance to the boundary in the corona sense. This characterization has profound consequences. We extend Fefferman-Kenig-Pipher's perturbation to non-connected settings. For the Laplacian, these corona decompositions or, equivalently, the partial/weak Carleson measure estimates are meaningful enough to characterize the uniform rectifiability of the boundary. As a consequence, we obtain that the boundary of the set is uniformly rectifiable if bounded null solutions for any Fefferman-Kenig-Pipher perturbation of the Laplacian satisfy Carleson measure estimates. For Kenig-Pipher operators any of the properties of the characterization is stable under transposition or symmetrization of the matrices of coefficients. As a result, we obtain that Carleson measure estimates for bounded null-solutions of non-symmetric variable operators satisfying an $L^1$-Kenig-Pipher condition occur if and only if the boundary of the open set is uniformly rectifiable. Our results generalize previous work in settings where quantitative connectivity.

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