arXiv subjects

María Soria-Carro

Publications and source records attributed to María Soria-Carro.

8 recordsLinked to original sources

arXiv2026

An Elliptic-Parabolic Free Boundary Problem with Discontinuous Data

We consider an elliptic-parabolic free boundary problem that models the fluid flow through a partially saturated porous medium. The free boundary arises as the interface separating the saturated and unsaturated regions. Our main goal is to investigate, for the 1+1 dimensional model, how jump discontinuities on the boundary and initial data influence the regularity of both the solution and the free boundary. We show that if the data is merely bounded, then weak solutions are Lipschitz in space and $C^{1/2}$ in time in the unsaturated region. Moreover, the free boundary is locally the graph of a $C^{1/2}$ function, and this regularity is optimal. We view this analysis as a stepping stone towards the study of local regularization for higher-dimensional elliptic-parabolic free boundaries.

math.AP↗

arXiv2026

A geometric approach to nonlocal 2-Hessian equations

We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}Δ^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.

math.AP↗

arXiv2025

Fully nonlinear parabolic fixed transmission problems

We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is $C^{1,α}$, in the parabolic sense, then viscosity solutions are piecewise $C^{1,α}$ up to the interface. As byproducts, we obtain a new ABP-Krylov-Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.

math.AP↗

arXiv2024

Lipschitz regularity for Poisson equations involving measures supported on $C^{1,\operatorname{Dini}}$ interfaces

We prove optimal Lipschitz regularity of solutions to Poisson's equation with measure data supported on a $C^{1,\operatorname{Dini}}$ interface and with $C^{0,\operatorname{Dini}}$ density. We achieve this by deriving pointwise gradient estimates on the interface, further showing the piecewise differentiability of solutions up to this surface. Our approach relies on perturbation arguments and estimates for the Green's function of the Laplacian. Additionally, we provide sharp counterexamples highlighting the minimality of our assumptions.

math.AP↗

arXiv2024

Transmission problems for simply connected domains in the complex plane

We study existence and uniqueness of a transmission problem in simply connected domains in the plane with data in weighted Lebesgue spaces by first investigating solvability results of a related novel problem associated to a homeomorphism in the real line and domains given by the upper and lower half planes. Our techniques are based on the use of conformal maps and Rellich identities for the Hilbert transform, and are motivated by previous works concerning the Dirichlet, Neumann and Zaremba problems.

math.AP↗

arXiv2024

Rellich identities for the Hilbert transform

We prove Hilbert transform identities involving conformal maps via the use of Rellich identity and the solution of the Neumann problem in a graph Lipschitz domain in the plane. We obtain as consequences new $L^2$-weighted estimates for the Hilbert transform, including a sharp bound for its norm as a bounded operator in weighted $L^2$ in terms of a weight constant associated to the Helson-Szegö theorem.

math.FA↗

arXiv2024

A parabolic free transmission problem: flat free boundaries are smooth

We study a two-phase parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Each phase is governed by a heat equation with distinct thermal conductivity, and a transmission-type condition is imposed on the free interface. We establish strong regularity properties of the free boundary: first, we prove that flat free boundaries are $C^{1,α}$ by means of a linearization technique and compactness arguments. Then we use the Hodograph transform to achieve higher regularity. To this end, we prove a new Harnack-type inequality and develop the Schauder theory for parabolic linear transmission problems.

math.AP↗

arXiv2022

On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Ampère

We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Ampère that are given by infimums of integro-differential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem, prescribing data at infinity, and prove existence, uniqueness, and $C^{1,1}$-regularity of solutions in the full space.

math.AP↗