arXiv subjects

Ginaldo Sá

Publications and source records attributed to Ginaldo Sá.

4 recordsLinked to original sources

arXiv2025

Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications

In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$ -Ψ(x, |\nabla u|)Δ_p^{\mathrm{N}}u(x)+\mathscr{H}(x,\nabla u)=f(x) \quad \mathrm{in} \quad Ω, \quad \mathrm{for} \,\,\,1<p< \infty, $$ under suitable assumptions on the data of the problem. Particularly, our results are relevant for a class of quasi-linear models with Hamiltonian terms. Additionally, we address non-degeneracy estimates for such solutions and present a couple of applications.

math.AP↗

arXiv2025

Geometric regularity estimates for quasi-linear elliptic models in non-divergence form with strong absorption

In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under strong absorption conditions of the form: \[ |\nabla u(x)|^γ Δ_p^{\mathrm{N}} u(x) = f(x, u) \quad \text{in} \quad B_1, \] where $γ> -1$, $p \in (1, \infty)$, and the mapping $u \mapsto f(x, u) \lesssim \mathfrak{a}(x) u_{+}^m$ (with $m \in [0, γ+ 1)$) does not decay sufficiently fast at the origin. This condition allows for the emergence of plateau regions, i.e., a priori unknown subsets where the non-negative solution vanishes identically. We establish improved geometric $\mathrm{C}^κ_{\text{loc}}$ regularity along the set $\mathscr{F}_0 = \partial \{u > 0\} \cap B_1$ (the free boundary of the model) for a sharp value of $κ\gg 1$, which is explicitly determined in terms of the structural parameters. Additionally, we derive non-degeneracy results and other measure-theoretic properties. Furthermore, we prove a sharp Liouville theorem for entire solutions exhibiting controlled growth at infinity.

math.AP↗

arXiv2024

Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications

In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the $C^{p^{\prime}}$ conjecture for the $p$-Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the $p$-Laplacian with regular coefficients: $$ \mathrm{div}\left( |\nabla u|^{p-2}\mathfrak{A}(|x|) \nabla u\right) = |x|^αu_+^m(x) \quad \text{in} \quad B_1 $$ under suitable assumptions on the data, with possibly singular weight $\mathfrak{h}(|x|) = |x|^α$, which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models.

math.AP↗

arXiv2024

Higher regularity of solutions to fully nonlinear elliptic equations

We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent regularity limitations dictated by the heterogeneity of the problem. We explore various scenarios, revealing a plethora of improved regularity estimates. Notably, depending on the model's parameters, we establish estimates that transcend the natural regularity regime of the model, from $C^{0,α_0}$ to $C^{1,α_1}$ and further to $C^{2,α_2}$, with the potential for even higher estimates.

math.AP↗