# On the Regularity Problem for Parabolic Operators and the Role of Half-Time Derivative

## TL;DR

This paper proves a new regularity result for parabolic equations involving half-time derivatives, enabling the study of solutions on complex time-varying domains.

## Contribution

The novelty lies in establishing regularity for parabolic PDEs with half-time derivatives on non-smooth domains.

## Key findings

- Solutions to the parabolic equation maintain regularity under specific boundary conditions involving half-time derivatives.
- The result allows the formulation of the parabolic Regularity problem on a broad class of time-varying domains.
- Both the half-time derivative of the solution and its Hilbert transform are shown to belong to L^p spaces.

## Abstract

In this paper we present the following result on regularity of solutions of the second order parabolic equation \documentclass[12pt]{minimal}
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\begin{document}$$\partial _t u - {{\,\textrm{div}\,}}(A \nabla u)+B\cdot \nabla u=0$$\end{document}∂tu-div(A∇u)+B·∇u=0 on cylindrical domains of the form \documentclass[12pt]{minimal}
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\begin{document}$$\Omega ={\mathcal {O}}\times {\mathbb {R}}$$\end{document}Ω=O×R where \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}\subset {\mathbb {R}}^n$$\end{document}O⊂Rn is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is \documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document}n-1-Ahlfors regular. Let u be a solution of such PDE in \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}Ω and the non-tangential maximal function of its gradient in spatial directions \documentclass[12pt]{minimal}
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\begin{document}$$\tilde{N}(\nabla u)$$\end{document}N~(∇u) belongs to \documentclass[12pt]{minimal}
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\begin{document}$$L^p(\partial \Omega )$$\end{document}Lp(∂Ω) for some \documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}p>1. Furthermore, assume that for \documentclass[12pt]{minimal}
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\begin{document}$$u|_{\partial \Omega }=f$$\end{document}u|∂Ω=f we have that \documentclass[12pt]{minimal}
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\begin{document}$$D^{1/2}_tf\in L^p(\partial \Omega )$$\end{document}Dt1/2f∈Lp(∂Ω). Then both \documentclass[12pt]{minimal}
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\begin{document}$$\tilde{N}(D^{1/2}_t u)$$\end{document}N~(Dt1/2u) and \documentclass[12pt]{minimal}
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\begin{document}$$\tilde{N}(D^{1/2}_tH_t u)$$\end{document}N~(Dt1/2Htu) also belong to \documentclass[12pt]{minimal}
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\begin{document}$$L^p(\partial \Omega )$$\end{document}Lp(∂Ω), where \documentclass[12pt]{minimal}
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\begin{document}$$D^{1/2}_t$$\end{document}Dt1/2 and \documentclass[12pt]{minimal}
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\begin{document}$$H_t$$\end{document}Ht are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the \documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}Lp parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.