arXiv subjects

Junior da Silva Bessa

Publications and source records attributed to Junior da Silva Bessa.

13 recordsLinked to original sources

arXiv2026

Optimal $C^{1,α}$ regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

In this paper we establish optimal $C^{1,α}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here relies on first deriving uniform boundary Hölder estimates for perturbed models with oblique boundary data in ``almost $C^{1}$-flat'' domains. Building upon these estimates, the desired regularity is obtained through a compactness and stability framework for viscosity solutions. As a byproduct of our analysis, we determine the optimal Hölder exponent for solutions when the governing operator is quasiconvex or quasiconcave. In addition, we establish an improved regularity result along vanishing points of the source term.

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arXiv2026

Geometric approaches for improved regularity in fully nonlinear parabolic models

In this paper, we derive improved estimates for a class of fully nonlinear parabolic equations with continuous drift and admissible source terms of the form $$ \partial_{t}u - F(D^2u,x,t) + \langle B(x,t), Du\rangle = f(x, t, u^{+}, u^{-}) \quad \text{in}\quad Q_1. $$ Our analysis reveals two distinct regimes. In the first, $f=f(x,t)$ exhibits $θ$-Hölder decay ($θ\in(0,1)$), yielding improved gradient regularity at vanishing points via perturbative methods and geometric iteration, as well as nondegeneracy with explicit growth rates under a suitable structural condition. In the second, $f(\cdot,u^+,u^-)=(u^+)^γ-(u^-)^γ$ with $γ\in(0,1)$ (corresponding to an evolutionary semilinear two-phase model), we obtain enhanced regularity at branching points by combining a robust blow-up analysis with local derivative estimates for linear equations. Our results remain relevant even in linear settings with merely continuous data, linking to classical free boundary problems arising in mathematical physics and related areas.

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arXiv2026

Bifurcation phenomena in one-phase quasilinear equations with nonstandard growth

The bifurcation phenomenon in a singularly perturbed one-phase free boundary problem governed by the g-Laplacian is established under prescribed boundary conditions. This bifurcation is characterized by the existence of a third solution, obtained via the Mountain Pass Lemma, when the boundary data decreases below a suitable threshold. Moreover, we analyze the asymptotic behavior of the associated evolution problem, proving convergence to stable stationary solutions and showing that the Mountain Pass solution is unstable in this dynamical sense.

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arXiv2026

Schauder estimates for a class of fully nonlinear elliptic PDEs: a geometric tangential approach

In this paper, we establish local and global Schauder estimates for classical solutions of a class of fully nonlinear elliptic partial differential equations, which are not necessarily convex or concave, under suitable Hölder continuity assumptions on the data. Our approach is based on a robust blow-up argument, combined with geometric tangential analysis and compactness techniques. This strategy is strongly influenced by methodologies developed in the contemporary theory of nonlinear elliptic PDEs.

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arXiv2026

A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities

In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type \[ F(x,D^{2}u)+\mathscr{H}(x,Du)=\mathscr{F}(x,u^{+},u^{-}) \quad \text{in } B_1 \subset \mathbb{R}^{n}, \] where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely \[ \mathscr{H}(x,ξ):= \langle \mathcal{B}(x),ξ\rangle+\varrho(x)|ξ|^σ, \quad 0< σ\leq2, \,\,\, σ\neq 1, \] and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type \[ \mathscr{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)\big[(u^{+})^{m}-(u^{-})^{m}\big] \quad \text{for} \quad 0<m<1. \] Under suitable structural assumptions on the coefficients and the operator $F$, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order ($σ\neq 1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.

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arXiv2026

Fully Nonlinear Evolution equations: From Critical Integrability to Schauder Estimates

We investigate the sharp regularity up to the boundary for viscosity solutions of fully nonlinear parabolic equations with oblique derivative boundary conditions given by \begin{equation*} \left\{ \begin{array}{rclcl} F(D^2u,x,t) - u_{t} &=& f(x,t) & \text{in} & Q_{1}^{+}, \\ β(x,t) \cdot Du &=& g(x,t) & \text{on} & Q_{1}^{*}. \end{array} \right. \end{equation*} The regularity theory is developed according to the integrability of the source term, the smoothness of the boundary data, and the oscillation of the coefficients of the operator $F$, using a compactness method combined with polynomial approximation. In the borderline case $f\in L^{n+2}$, we obtain Log-Lipschitz continuity of solutions up to the boundary. Under stronger integrability, specifically when $f\in L^{p}$ for some $p>n+2$, we establish optimal $C^{1+α',\,\frac{1+α'}{2}}$ boundary estimates. At a higher regularity level, we prove Schauder-type estimates under appropriate assumptions on the operator $F$ and the boundary data. As a byproduct, we obtain parabolic $C^{1,\mathrm{Log\text{-}Lip}}$ regularity in the critical borderline case where the source term belongs to BMO space.

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arXiv2026

Fully Nonlinear Elliptic Grad--Mercier Equations in Weighted Orlicz Spaces

In this article, we study the existence and global regularity results for the fully nonlinear elliptic Grad--Mercier type equations with oblique boundary conditions in the context of weighted Orlicz spaces. Our approach employs an asymptotic analysis in which global regularity is transferred from a limit profile, namely, the recession operator associated with the governing operator, using topological and stability methods. In addition to the main regularity result, we derive global weighted Orlicz estimates for the Hessian and establish global Morrey-type estimates for the problem. This article extends the results established by Caffarelli--Tomasetti (Comm. Pure Appl. Math. 76 (3): 604--615, 2023), Zhang et al. (Nonlinearity 39 (2): 025011, 2026), and Bessa (J. Funct. Anal. 286 (4): 110295, 2024).

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arXiv2026

Sharp regularity for degenerate fully nonlinear equations with oblique boundary conditions and Hamiltonian terms

We prove optimal boundary $C^{1,α}$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^γF(D^2 u) + \varrho(x)|Du|^σ = f(x) & \text{in } Ω,\\ β(x)\cdot Du+ζ(x)u = g(x) & \text{on } \partial Ω, \end{cases} \] where $γ>0$ and $0<σ\le 1+γ$. We develop a compactness framework for affine translations, linking the size of the translation to the Hamiltonian structure. This is combined with a boundary improvement-of-flatness argument adapted to oblique boundary data, yielding the optimal boundary regularity.

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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.

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arXiv2026

Schauder estimates for flat solutions to a class of fully nonlinear elliptic PDEs with Dini continuous data: a geometric tangential approach

In this manuscript, we establish local Schauder estimates for flat viscosity solutions, that is, solutions with sufficiently small norms, to a class of fully nonlinear elliptic partial differential equations of the form \[ F(D^{2} u, x) + \langle \mathfrak{B}(x), D u \rangle = f(x) \quad \text{in} \quad \mathrm{B}_1 \subset \mathbb{R}^{n}, \] where the operator $F$ is differentiable, though not necessarily convex or concave. In addition, we impose suitable Dini-type continuity assumptions on the data. Our methodology is based on geometric tangential techniques, combined with compactness and perturbative arguments. This approach is strongly motivated by recent advances in the theory of nonlinear elliptic equations and free boundary problems. As a byproduct of our analysis, we also obtain an Evans-Krylov type estimate. Our results can be viewed as an extension of the work by dos Prazeres and Teixeira (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 485-500), now within the framework of linear drift terms and Dini continuity assumptions. Finally, we apply our results to characterize the nodal sets of flat viscosity solutions of non-convex, fully nonlinear, uniformly elliptic PDEs.

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arXiv2026

Regularity to Thin Obstacle Problem in Orlicz spaces

In this work, we establish regularity results for minimizers of the energy functional associated with the thin obstacle problem in Orlicz spaces. More precisely, we prove the Lipschitz continuity and the Hölder continuity of the gradient of minimizers. The analysis is based on techniques from De Giorgi's classical regularity theory. As a byproduct of our results, we also provide a characterization of the structure of the nodal sets of the minimizers.

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arXiv2025

Regularity estimates for quasilinear elliptic PDEs in non-divergence form with Hamiltonian terms and applications

In this manuscript, we investigate regularity estimates for a class of quasilinear elliptic equations in the non-divergence form that may exhibit degenerate behavior at critical points of their gradient. The prototype equation under consideration is $$ |\nabla u(x)|^θ\left( Δ_{p}^{\mathrm{N}} u(x) + \langle\mathfrak{B}(x), \nabla u\rangle \right) + \varrho(x) |\nabla u(x)|^σ = f(x) \quad \text{in} \quad B_1, $$ where $θ> 0$, $σ\in (θ, θ+ 1)$, and $p \in (1, \infty)$. The coefficients $\mathfrak{B}$ and $\varrho$ are bounded continuous functions, and the source term $f \in \mathrm{C}^{0}(B_1) \cap L^{\infty}(B_1)$. We establish interior $\mathrm{C}^{1,α}_{\text{loc}}$ regularity for some $α\in (0,1)$, along with sharp quantitative estimates at critical points of existing solutions. Additionally, we prove a non-degeneracy property and establish both a Strong Maximum Principle and a Hopf-type lemma. In the final part, we apply our analytical framework to study existence, uniqueness, improved regularity, and non-degeneracy estimates for Hénon-type models in the non-divergence form. These models incorporate strong absorption terms and linear/sublinear Hamiltonian terms and are of independent mathematical interest. Our results partially extend (for the non-variational quasilinear setting) the recent work by the second author in collaboration with Nornberg [Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 202, 40 pp.], where sharp quantitative estimates were established for the fully nonlinear uniformly elliptic setting with Hamiltonian terms.

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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.

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