arXiv subjects

Ginaldo de Santana Sá

Publications and source records attributed to Ginaldo de Santana Sá.

3 recordsLinked to original sources

arXiv2026

Gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure

We establish regularity results for viscosity solutions to a class of quasilinear parabolic equations exhibiting nonhomogeneous degeneracy or singularity (a double phase regime) of the form \[ u_t - \big(|Du|^{\mathfrak{p}} + \mathfrak{a}(x,t)|Du|^{\mathfrak{q}}\big)Δ_p^{\mathrm N} u = f(x,t) \quad \text{in } Q_1, \] where $-1 < \mathfrak{p} < 0$, $\mathfrak{p} \leq \mathfrak{q}$, and $\mathfrak{a}, f : Q_1 \to \mathbb{R}$ are prescribed functions. Using the Jensen--Ishii method, we prove Lipschitz regularity for appropriately translated solutions. Moreover, combining this approach with intrinsic scaling techniques, we establish interior Hölder continuity estimates for the gradient. Our results extend recent work of Fang and Zhang on the homogeneous case via a different approach.

math.AP↗

arXiv2025

Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature

In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this class is given by \[ \partial_{t} u = \left(|\nabla u|^{\mathfrak{p}} + \mathfrak{a}(x, t)|\nabla u|^{\mathfrak{q}}\right)Δ_{p}^{\mathrm{N}} u + f(x, t) \quad \text{in} \quad Q_1 = B_1 \times (-1, 0], \] where $p \in (1, \infty)$, $\mathfrak{p}, \mathfrak{q} \in [0, \infty)$, and $\mathfrak{a}, f: Q_1 \to \mathbb{R}$ are suitably defined functions. Our approach is based on geometric tangential methods, incorporating a refined oscillation mechanism, compactness arguments, ``alternative methods,'' and scaling techniques. Furthermore, we derive pointwise estimates in settings exhibiting singular-degenerate or doubly singular signatures. To some extent, our regularity estimates refine and extend previous results from \cite{FZ23} through distinct methodological advancements. Finally, we explore connections between our findings and fundamental nonlinear models in the theory of quasilinear PDEs, which may be of independent interest.

math.AP↗

arXiv2022

Infinity Laplacian equations with singular absorptions

In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal $C^{1,α}$ regularity along the free boundary. We also show existence of solutions, nondegeneracy properties and fine geometric estimates for the free boundary.

math.AP↗