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Thialita M. Nascimento

Publications and source records attributed to Thialita M. Nascimento.

10 recordsLinked to original sources
arXiv2026

Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions

We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\emph{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $κ=Λ/λ$, then the optimal exponent in the $W^{2,ε}$ regularity theory in the plane is exactly $$ ε_2(κ)=\frac{2}{κ+1}. $$ The same mechanism reaches the known upper obstruction in dimension three throughout the full range $1\le κ\le 4$ and therefore gives $$ ε_3(κ)=\frac{3}{2κ+1} \qquad(1\leκ\le4). $$ Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature---data erased by the classical pointwise reduction---and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.

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arXiv2026

Uniform Lipschitz regularity for two-phase singularly perturbed fully nonlinear elliptic equations

We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$ F(D^2u_\varepsilon) = \fracα{\varepsilon} β\left(\frac{u_\varepsilon}{\varepsilon}\right) \qquad\text{in }B_1\subset\mathbb R^n, $$ where $F$ is uniformly elliptic and $β\in C_c (-1,1)$ is nonnegative. We prove the scale-sharp estimate $$ \|\nabla u_\varepsilon\|_{L^\infty(B_{1/2})} \leq C\left( \|u_\varepsilon\|_{L^\infty(B_1)}+\sqrtα\right), $$ with $C$ depending only on the dimension, the ellipticity constants, and $β$, and independent of $\varepsilon$ and $α$. This removes a longstanding compactness obstruction in the analysis of fully nonlinear two-phase singular perturbations. The difficulty is structural: at positive $\varepsilon$ there is neither a free boundary nor a prescribed transmission law, while the general fully nonlinear setting provides no monotonicity formula capable of controlling the interaction of the two phases. The proof develops a diffuse counterpart of the De Silva--Savin decay-versus-Lipschitz alternative. Exact planar transitions furnish the local comparison geometry, and curved-test compactness carries this geometry across collapsing reaction layers. An intrinsic transition-region estimate reduces the problem to linear growth from buffered level boundaries. The resulting dyadic continuation is closed by a large-slope stopping argument: bounded accumulated slopes yield the desired growth directly, whereas unbounded slopes force the effective reaction strength to vanish after normalization and lead to a contradiction. The estimate is quantitatively optimal and supplies the scale-invariant compactness framework required for the subsequent sharp-interface analysis.

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arXiv2026

Geometric Estimates for Solutions of Semilinear Equations with Singular Potentials

In this work, we study local minimizers of elliptic functionals with strong absorption terms and unbounded, sign-changing sources. These problems naturally interpolate between two classical free boundary problems: Bernoulli-type (cavity) and obstacle-type. While previous studies have focused on bounded and strictly positive sources, we extend sharp regularity and nondegeneracy estimates to the unbounded, sign-changing setting, providing a comprehensive analysis of how the underlying nonlinearity interacts with minimal integrability assumptions on the source.

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arXiv2025

Optimal regularity for degenerate elliptic equations with Hamiltonian terms

We establish optimal, quantitative Höder estimates for the gradient of solutions to a class of degenerate elliptic equations with Hamiltonian terms. The presence of such lower-order terms introduces additional challenges, particularly in regimes where the gradient is either very small or very large. Our approach adapts perturbative techniques to capture the interplay between the degeneracy rate and the Hamiltonian's growth. Our results naturally extend the regularity theory developed by Araujo-Ricarte-Teixeira (Calc. Var. 53:605-625, 2015) and Birindelli-Demengel (Nonlinear Differ. Equ. Appl. 23:41, 2016) to a more general setting.

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arXiv2025

Interior regularity estimates for fully nonlinear equations with arbitrary nonhomogeneous degeneracy laws

In this paper, we study regularity estimates for a class of degenerate, fully nonlinear elliptic equations with arbitrary nonhomogeneous degeneracy laws. We establish that viscosity solutions are locally continuously differentiable under suitable conditions on the degeneracy laws. Our proof employs improvement of flatness techniques alongside an alternative recursive algorithm for renormalizing the approximating solutions, linking our model to the homogeneous, fully nonlinear, uniformly elliptic equation.

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arXiv2024

Improved regularity estimates for Hardy-Hénon-type equations driven by the $\infty$-Laplacian

In this work, we establish sharp and improved regularity estimates for viscosity solutions of Hardy-Hénon-type equations with possibly singular weights and strong absorption governed by the $\infty$-Laplacian $$ Δ_{\infty} u(x) = |x|^αu_+^m(x) \quad \text{in} \quad B_1, $$ under suitable assumptions on the data. In this setting, we derive an explicit regularity exponent that depends only on universal parameters. Additionally, we prove non-degeneracy properties, providing further geometric insights into the nature of these solutions. Our regularity estimates not only improve but also extend, to some extent, the previously obtained results for zero-obstacle and dead-core problems driven by the $\infty$-Laplacian. As an application of our findings, we also address some Liouville-type results for this class of equations.

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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.

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arXiv2022

On the sharp Hessian integrability conjecture in the plane

We prove that if $u\in C^0(B_1)$ satisfies $F(x,D^2u) \le 0$ in $B_1\subset \mathbb{R}^2$, in the viscosity sense, for some fully nonlinear $(λ, Λ)$-elliptic operator, then $u \in W^{2,\varepsilon}(B_{1/2})$, with appropriate estimates, for a sharp exponent $ \varepsilon = \varepsilon(λ, Λ)$ verifying $$ \frac{1.629}{\fracΛλ + 1} < \varepsilon(λ, Λ) \le \frac{2}{\fracΛλ + 1}, $$ uniformly as $\fracλΛ \to 0$. This is closely related to the Armstrong-Silvestre-Smart conjecture, raised in [Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184], where the upper bound is postulated to be the optimal one.

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arXiv2022

New regularity estimates for fully nonlinear elliptic equations

We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability $\varepsilon = \varepsilon(λ, Λ, n)$ in the celebrated $W^{2, \varepsilon}$-regularity estimate satisfies $$\frac{ \left (1+ \frac{2}{3}\left(1- \fracλΛ \right )\right )^{n-1}}{\ln n^4} \cdot \left( \fracλΛ \right) ^{n-1} \le \varepsilon \le \frac{nλ}{(n-1)Λ+λ}, $$ where $n\ge 3$ is the dimension and $0< λ< Λ$ are the ellipticity constants. In particular, $\left( \fracΛλ \right) ^{n-1} \varepsilon(λ, Λ, n)$ blows-up, as $n\to\infty$; previous results yielded fast decay of such a quantity. The upper estimate improves the one obtained by Armstrong, Silvestre, and Smart in arXiv:1103.3677

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