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Weighted Orlicz-Sobolev and variable exponent Morrey regularity for fully nonlinear parabolic PDEs with oblique boundary conditions and applications

Junior da S. Bessa, João Vitor da Silva, Maria N. B. Frederico, Gleydson C. Ricarte

TL;DR

This work develops global regularity theory for viscosity solutions to fully nonlinear parabolic equations with oblique boundary conditions by embedding the problem in weighted Orlicz-Sobolev and variable-exponent Morrey spaces. Central to the approach is the recession operator $F^{\sharp}$ and a caloric approximation scheme that yields Calderón–Zygmund-type control via tangential analysis, culminating in global $W^{2,\Upsilon}_{\omega}$ and $W^{2,\varsigma(\cdot),\varrho(\cdot)}$ estimates. The authors establish density results, obstacle-problem regularity, and $\text{Orlicz}$-BMO bounds for the Hessian and $u_t$, together with a variable-exponent Morrey theory and Campanato-type equivalence to parabolic Hölder spaces. These results extend prior convexity-reliant parabolic regularity to oblique boundary problems and nonconvex fully nonlinear operators, providing new tools for analysis in weighted and variable-exponent frameworks with broad applications.

Abstract

In this manuscript, we establish global weighted Orlicz-Sobolev and variable exponent Morrey-Sobolev estimates for viscosity solutions to fully nonlinear parabolic equations subject to oblique boundary conditions on a portion of the boundary, within the following framework: \begin{equation*} \left\{ \begin{array}{rclcl} F(D^2u,Du,u,x,t) - u_{t} &=& f(x,t) & \text{in} & Ω_{\mathrm{T}}, \\ β\cdot Du + γu &=& g(x,t) & \text{on} & \mathrm{S}_{\mathrm{T}}, \\ u(x, 0) &=& 0 & \text{on} & Ω_{0}, \end{array} \right. \end{equation*} where \(Ω_{\mathrm{T}} = Ω\times (0,\mathrm{T})\) denotes the parabolic cylinder with spatial base \(Ω\) (a bounded domain in \(\mathbb{R}^{n}\), \(n \geq 2\)) and temporal height \(\mathrm{T} > 0\), \(\mathrm{S}_{\mathrm{T}} = \partial Ω\times (0,\mathrm{T})\), and \(Ω_{0} = Ω\times \{0\}\). Additionally, \(f\) represents the source term of the parabolic equation, while the boundary data are given by \(β\), \(γ\), and \(g\). Our first main result is a global weighted Orlicz-Sobolev estimate for the solution, obtained under asymptotic structural conditions on the differential operator and appropriate assumptions on the boundary data, assuming that the source term belongs to the corresponding weighted Orlicz space. Leveraging these estimates, we demonstrate several applications, including a density result within the fundamental class of parabolic equations, regularity results for the related obstacle problem, and weighted Orlicz-BMO estimates for both the Hessian and the time derivative of the solution. Lastly, we derive variable exponent Morrey-Sobolev estimates for the problem via an extrapolation technique, which are of independent mathematical interest.

Weighted Orlicz-Sobolev and variable exponent Morrey regularity for fully nonlinear parabolic PDEs with oblique boundary conditions and applications

TL;DR

This work develops global regularity theory for viscosity solutions to fully nonlinear parabolic equations with oblique boundary conditions by embedding the problem in weighted Orlicz-Sobolev and variable-exponent Morrey spaces. Central to the approach is the recession operator and a caloric approximation scheme that yields Calderón–Zygmund-type control via tangential analysis, culminating in global and estimates. The authors establish density results, obstacle-problem regularity, and -BMO bounds for the Hessian and , together with a variable-exponent Morrey theory and Campanato-type equivalence to parabolic Hölder spaces. These results extend prior convexity-reliant parabolic regularity to oblique boundary problems and nonconvex fully nonlinear operators, providing new tools for analysis in weighted and variable-exponent frameworks with broad applications.

Abstract

In this manuscript, we establish global weighted Orlicz-Sobolev and variable exponent Morrey-Sobolev estimates for viscosity solutions to fully nonlinear parabolic equations subject to oblique boundary conditions on a portion of the boundary, within the following framework: \begin{equation*} \left\{ \begin{array}{rclcl} F(D^2u,Du,u,x,t) - u_{t} &=& f(x,t) & \text{in} & Ω_{\mathrm{T}}, \\ β\cdot Du + γu &=& g(x,t) & \text{on} & \mathrm{S}_{\mathrm{T}}, \\ u(x, 0) &=& 0 & \text{on} & Ω_{0}, \end{array} \right. \end{equation*} where \(Ω_{\mathrm{T}} = Ω\times (0,\mathrm{T})\) denotes the parabolic cylinder with spatial base (a bounded domain in , ) and temporal height , \(\mathrm{S}_{\mathrm{T}} = \partial Ω\times (0,\mathrm{T})\), and . Additionally, represents the source term of the parabolic equation, while the boundary data are given by , , and . Our first main result is a global weighted Orlicz-Sobolev estimate for the solution, obtained under asymptotic structural conditions on the differential operator and appropriate assumptions on the boundary data, assuming that the source term belongs to the corresponding weighted Orlicz space. Leveraging these estimates, we demonstrate several applications, including a density result within the fundamental class of parabolic equations, regularity results for the related obstacle problem, and weighted Orlicz-BMO estimates for both the Hessian and the time derivative of the solution. Lastly, we derive variable exponent Morrey-Sobolev estimates for the problem via an extrapolation technique, which are of independent mathematical interest.
Paper Structure (14 sections, 33 theorems, 252 equations)

This paper contains 14 sections, 33 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Structural assumptions and further information
  3. Main Theorems
  4. State-of-the-Art: regularity theory for nonlinear parabolic models
  5. Preliminaries
  6. Some Basic Functional Spaces
  7. Key tools in viscosity solutions theory
  8. Caloric Approximation and Decay of the Sets $\mathrm{A}_{t}$
  9. Weighted Orlicz-Sobolev Estimates
  10. Some Applications
  11. Density results in a suitable class
  12. Calderón-Zygmund Type Estimates for Solutions of the Obstacle Problem
  13. Weighted Orlicz-BMO Estimates
  14. Variable Exponent Morrey Estimates

Key Result

Lemma 1.5

Let $\Phi$ be an $\mathrm{N}$-function satisfying $\Phi \in \Delta_2 \cap \nabla_2$, let $\omega \in \mathfrak{A}_{i(\Phi)}$, and suppose $\Omega \subset \mathbb{R}^n$ is bounded. Then, there exists a constant $p_0 \in (1, i(\Phi))$, depending only on $i(\Phi)$ and $\omega$, such that $L^\Phi_{\omeg where $\mathrm{C}' = \mathrm{C}'(n, i(\Phi), \omega) > 0$ is a constant independent of $g$.

Theorems & Definitions (70)

  • Definition 1.1: Recession operator
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • Theorem 1.7: Global Weighted Orlicz Regularity
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10: Global Variable Exponent Morrey Regularity
  • Theorem 1.11: Local $L^p$ Estimates - Lieberman1996 and Kry08
  • ...and 60 more