The goal of my research is to understand regularity in nonlinear and nonlocal partial differential equations in regimes where the classical theory breaks down. These include degenerate, and free boundary problems, where new ideas are required to capture the intricate interaction between geometry, nonlocality, and nonlinearity.
This program has led to several distinct but interconnected research directions:
the resolution of a standing open problem on C1+alpha regularity for the fractional p-Laplacian,
the development of viscosity methods for the inhomogeneous Stefan problem,
the analysis of degenerate equations in both divergence and nondivergence form, and
the study of transmission problems.
Through these contributions, I aim to advance the understanding of challenging and fundamental problems in this area.
D. Giovagnoli, D. Jesus, and L. Silvestre. C1+α regularity for fractional p-harmonic functions. arXiv:2509.26565.
D. Jesus, M. Soria-Carro. Fully nonlinear parabolic fixed transmission problem. Nonlinear Anal., 264, (2026).
D. Giovagnoli, D. Jesus, and F. Ferrari. On the geometry of solutions of the fully nonlinear inhomogeneous one-phase Stefan problem. arXiv:2504.12912.
D. Jesus, E. Pimentel, and D. Stolnicki. Boundary regularity for a fully nonlinear free transmission problem. arXiv:2411.15335.
D. Giovagnoli, D. Jesus. A fully nonlinear transmission problem degenerating on the interface. arXiv:2410.16957.
F. Ferrari, N. Forcillo, D. Giovagnoli, D. Jesus. Free boundary regularity for the inhomogeneous one-phase Stefan problem. arXiv:2404.07535.
U. Gianazza, D. Jesus. Boundary estimates for doubly nonlinear parabolic equations. Nonlinear Differ. Equ. Appl. 32, 17 (2025).
D. Jesus, Y. Sire. Gradient regularity for fully nonlinear equations with degenerate coefficients. Ann. Mat. Pura Appl. (2024).
D. Jesus, E. Pimentel, and J. M. Urbano. Fully nonlinear Hamilton-Jacobi equations of degenerate type. Nonlinear Anal. 227 (2023).
D. Jesus. A degenerate fully nonlinear free transmission problem with variable exponents. Calc. Var. Partial Differ. Equ. 61, 29 (2022).