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

TL;DR

This paper develops advanced regularity estimates for solutions to fully nonlinear parabolic PDEs with oblique boundary conditions, employing weighted Orlicz-Sobolev and variable exponent Morrey spaces, and explores their applications.

Contribution

It introduces new global weighted Orlicz-Sobolev and variable exponent Morrey-Sobolev estimates for fully nonlinear parabolic equations with oblique boundary conditions, extending regularity theory.

Findings

Established weighted Orlicz-Sobolev estimates for viscosity solutions.

Proved regularity results for obstacle problems related to these PDEs.

Derived variable exponent Morrey-Sobolev estimates using extrapolation techniques.

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} & \Omega_{\mathrm{T}}, \\ \beta \cdot Du + \gamma u &=& g(x,t) & \text{on} & \mathrm{S}_{\mathrm{T}}, \\ u(x, 0) &=& 0 & \text{on} & \Omega_{0}, \end{array} \right. \end{equation*} where \(\Omega_{\mathrm{T}} = \Omega \times (0,\mathrm{T})\) denotes the parabolic cylinder with spatial base \(\Omega\) (a bounded domain in \(\mathbb{R}^{n}\), \(n \geq 2\)) and temporal height \(\mathrm{T} > 0\), \(\mathrm{S}_{\mathrm{T}} = \partial \Omega \times (0,\mathrm{T})\), and \(\Omega_{0} = \Omega \times \{0\}\). Additionally, \(f\) represents the source term of the parabolic equation, while the boundary data are given by \(\beta\), \(\gamma\), 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.