Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$L_p$-estimates of the conormal derivative problem for parabolic equations with time measurable coefficients and $A_p$-weights

Hongjie Dong, Pilgyu Jung, Doyoon Kim

TL;DR

This work develops $L_p$-type solvability and weighted mixed-norm estimates for divergence-form parabolic equations with leading coefficients that are merely measurable in time and have small spatial mean oscillations on Reifenberg-flat domains, under conormal boundary conditions. A central novelty is the use of fractional time derivatives, specifically the half-time derivative $D_t^{1/2}$, to manage time-irregularities and boundary roughness, enabling absorption of troublesome terms via small-volume boundary regions. The authors establish a priori estimates, existence, and uniqueness in $H_{p,q,oldsymbol{ullet}}^{1/2,1}$ spaces, and extend results from equal-time exponents $(p=q)$ to general $(p,q)$ via extrapolation, covering both infinite and finite time intervals and, in principle, systems. The results significantly advance the $L_p$-theory for parabolic conormal problems with rough coefficients and irregular boundaries, with potential applications to nonlinear PDE analysis and weighted regularity theory.

Abstract

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the time variable and to have small mean oscillations in the spatial variables. In deriving the boundary estimates, we overcome a regularity issue by employing half-time derivative estimates.

$L_p$-estimates of the conormal derivative problem for parabolic equations with time measurable coefficients and $A_p$-weights

TL;DR

This work develops -type solvability and weighted mixed-norm estimates for divergence-form parabolic equations with leading coefficients that are merely measurable in time and have small spatial mean oscillations on Reifenberg-flat domains, under conormal boundary conditions. A central novelty is the use of fractional time derivatives, specifically the half-time derivative , to manage time-irregularities and boundary roughness, enabling absorption of troublesome terms via small-volume boundary regions. The authors establish a priori estimates, existence, and uniqueness in spaces, and extend results from equal-time exponents to general via extrapolation, covering both infinite and finite time intervals and, in principle, systems. The results significantly advance the -theory for parabolic conormal problems with rough coefficients and irregular boundaries, with potential applications to nonlinear PDE analysis and weighted regularity theory.

Abstract

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the time variable and to have small mean oscillations in the spatial variables. In deriving the boundary estimates, we overcome a regularity issue by employing half-time derivative estimates.
Paper Structure (12 sections, 18 theorems, 261 equations, 1 figure)

This paper contains 12 sections, 18 theorems, 261 equations, 1 figure.

Table of Contents

  1. introduction
  2. Function spaces and main results
  3. Basic notation
  4. Function spaces
  5. Main results
  6. Auxiliary results
  7. Estimates on a Reifenberg-flat domain
  8. Estimates of u and Du on a Reifenberg-flat domain
  9. Half-derivative estimates for the heat equation on a Reifenberg-flat domain
  10. Proofs of main results
  11. Proof of Theorem \ref{['conormal entire time']}: Level set argument
  12. Proofs of Theorem \ref{['conormal -infty to T']} and Corollary \ref{['conormal bounded interval']}

Key Result

Theorem 2.8

Let $p,q\in(1,\infty)$, $\omega_1=\omega_1(x)\in A_p(\mathbb{R}^d)$, $\omega_2=\omega_2(t)\in A_q(\mathbb{R})$, $[\omega_1]_{A_p}+[\omega_2]_{A_q}\le K$ for some constant $K\ge 1$, $\lambda \geq 0$, $h,g_i,f\in L_{p,q,\omega}(\mathcal{Q})$ with $f \equiv 0$ if $\lambda = 0$. Denote $\omega(t,x)=\ome with the conormal derivative boundary condition on $\mathbb{R}\times\partial\Omega$, we have provi

Figures (1)

  • Figure 1:

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Corollary 2.10
  • Remark 2.11
  • ...and 36 more