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David Jesus

Publications and source records attributed to David Jesus.

13 recordsLinked to original sources
arXiv2026

A transmission problem arising from the two-phase Stefan problem

We study a parabolic transmission problem whose interface condition depends on the time derivative of the solution. This model arises naturally as the linearized limiting profile of a two-phase inhomogeneous Stefan problem after the hodograph transform. Our main result shows the existence and uniqueness of a classical solution. We also prove a Harnack inequality for a general class of transmission problems and establish $C^{1,\alpha}$ estimates up to the interface from both sides. These results can be used to obtain $C^{1,\alpha}$ regularity of flat free boundaries for the two-phase Stefan problem with distributed sources.

math.AP↗
arXiv2026

Fractional $p$-caloric functions are Lipschitz

We study the parabolic fractional $p-$Laplace equation $\partial_t u+(-\Delta_p)^su = 0$ in the degenerate range $2 < p < 2/(1-s)$. We show that weak solutions are Lipschitz continuous in space and, if $p > 1/(1-s)$, also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.

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 $C^{1,\alpha}$, in the parabolic sense, then viscosity solutions are piecewise $C^{1,\alpha}$ 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↗
arXiv2025

On the Geometry of Solutions of the Fully Nonlinear Inhomogeneous One-Phase Stefan Problem

In this paper, we characterize the geometry of solutions to one-phase inhomogeneous fully nonlinear Stefan problem with flat free boundaries under a new nondegeneracy assumption. This continues the study of regularity of flat free boundaries for the linear inhomogeneous Stefan problem started in [9], as well as justifies the definition of flatness assumed in [15].

math.AP↗
arXiv2024

Boundary regularity for a fully nonlinear free transmission problem

We examine boundary regularity for a fully nonlinear free transmission problem. We argue using approximation methods, comparing the operators driving the problem with a limiting profile. Working natural conditions on the data of the problem, we produce regularity estimates in Sobolev and $C^{1,{\rm Log-Lip}}$-spaces. Our findings extend recent developments in the literature to the free boundary setting.

math.AP↗
arXiv2024

A fully nonlinear transmission problem degenerating on the interface

In this paper we prove that solutions to a transmission problem degenerating on the interface are H\"older differentiable up to the interface with universal estimates. Furthermore, we obtain a sharper pointwise $C^{1,\alpha(\cdot)}$ with optimal variable exponent and uniform estimates.

math.AP↗
arXiv2024

Boundary Estimates for Doubly Nonlinear Parabolic Equations

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions vanish continuously at the Lipschitz boundary of a parabolic cylinder $\Omega_T$, they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain $\Omega_T$, we obtain a power-like decay at the boundary and a boundary Harnack inequality.

math.AP↗
arXiv2024

Free boundary regularity for the inhomogeneous one-phase Stefan problem

In this paper, we prove that flat free boundaries of solutions to inhomogeneous one-phase Stefan problem are $C^{1,\alpha}$. The method consists of employing a hodograph transform and deriving the regularity via a linearization technique, following the approach introduced by De Silva, Forcillo, and Savin in \cite{DFS23}.

math.AP↗
arXiv2022

Fully nonlinear Hamilton-Jacobi equations of degenerate type

We examine Hamilton-Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii-Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

math.AP↗
arXiv2021

A degenerate fully nonlinear free transmission problem with variable exponents

We study degenerate fully nonlinear free transmission problems, where the degeneracy rate varies in the domain. We prove optimal pointwise regularity depending on the degeneracy rate. Our arguments consist of perturbation methods, relating our problem to a homogeneous, fully nonlinear, uniformly elliptic equation.

math.AP↗