[Paper Review] Second Order Operators Subject to Dirichlet Boundary Conditions in Weighted Triebel-Lizorkin Spaces: Parabolic Problems

Nick Lindemulder|arXiv (Cornell University)|Dec 13, 2018

Advanced Harmonic Analysis Research35 references6 citations

TL;DR

This paper establishes $L_q$-maximal regularity for second-order parabolic PDEs with inhomogeneous Dirichlet boundary conditions in weighted Triebel-Lizorkin spaces on smooth domains. By employing power weights in time and space, it achieves optimal regularity for initial-boundary data without requiring compatibility conditions, enabling treatment of rough boundary data through a smoothing effect.

ABSTRACT

In this paper we consider second order parabolic partial differential equations subject to the Dirichlet boundary condition on smooth domains. We establish weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces for such parabolic problems with inhomogeneous boundary data. The weights that we consider are power weights in time and space, and yield flexibility in the optimal regularity of the initial-boundary data, allow to avoid compatibility conditions at the boundary and provide a smoothing effect. In particular, we can treat rough inhomogeneous boundary data.

Motivation & Objective

• To establish $L_q$-maximal regularity for second-order parabolic problems with inhomogeneous Dirichlet boundary conditions in weighted Triebel-Lizorkin spaces.
• To extend maximal regularity theory to rough initial-boundary data by using power weights in time and space.
• To eliminate the need for compatibility conditions at the boundary by leveraging weighted function spaces.
• To provide a framework that ensures isomorphism between data and solution spaces in mixed-norm weighted settings.
• To generalize previous results on $L_p$-maximal regularity to the broader class of Triebel-Lizorkin spaces with variable smoothness and weights.

Proposed method

• Utilizes weighted Triebel-Lizorkin spaces $F^s_{p,r}(ullet, w_ u^ ext{dist})$ with power weights $w_ u^ ext{dist}(x) = \text{dist}(x,\partial\mathscr{O})^\nu$ to model variable smoothness and integrability.
• Applies $H^\infty$-calculus techniques for second-order elliptic operators to derive maximal regularity results.
• Employs mixed-norm weighted $L_q$-spaces in time with weight $v_\mu(t) = t^\mu$, $\mu \in (-1, q-1)$, to control time-regularity and initial data behavior.
• Establishes trace operators $\mathrm{Tr}_{\partial\mathscr{O}}$ as bounded linear maps from solution spaces to trace spaces, enabling boundary data control.
• Derives embedding results showing that solutions gain regularity in higher-order weighted Sobolev spaces via microscopic improvements.
• Uses the independence of the spectral parameter $\lambda_0$ across different smoothness and weight parameters to unify estimates across scales.

Experimental results

Research questions

• RQ1 Can $L_q$-maximal regularity be established for parabolic problems with inhomogeneous Dirichlet boundary conditions in weighted Triebel-Lizorkin spaces?
• RQ2 How do power weights in time and space affect the regularity of initial-boundary data and the solution space?
• RQ3 To what extent can compatibility conditions at the boundary be avoided in maximal regularity theory using weighted spaces?
• RQ4 What is the precise trace space to which solutions belong, and how does it relate to the data space?
• RQ5 Can the solution space be embedded into higher-order weighted Sobolev spaces, indicating a smoothing effect?

Key findings

• The paper establishes $L_q$-maximal regularity for parabolic problems in weighted Triebel-Lizorkin spaces with power weights, ensuring a topological isomorphism between data and solution spaces.
• For the heat equation, the trace operator $u \mapsto u|_{\partial\mathscr{O}}$ defines an isomorphism from the solution space to ${{}_0B^\delta_{p,p}}(J; L_p(\partial\mathscr{O})) \cap L_p(J; B^{2\delta}_{p,p}(\partial\mathscr{O}))$ with $\delta = 1 - \frac{1+\gamma}{2p} \in (0,1)$.
• The solution space embeds into $\bigcap_{\mu > -1} \left[ W^1_q(J, v_\mu; F^{s + \frac{\mu - \gamma}{p}}_{p,1,\mu}(\mathscr{O})) \cap L_q(J, v_\mu; F^{s + \frac{\mu - \gamma}{p} + 2}_{p,1,\mu}(\mathscr{O})) \right]$, showing improved regularity across weights.
• Solutions gain regularity in higher-order weighted Sobolev spaces: $\bigcap_{k \in \mathbb{N}} \left[ W^1_q(J, v_\mu; W^{k}_p(\mathscr{O}, w_{\gamma + (k-s)p}^{\partial\mathscr{O}})) \cap L_q(J, v_\mu; W^{k+2}_p(\mathscr{O}, w_{\gamma + (k-s)p}^{\partial\mathscr{O}})) \right]$, indicating a smoothing effect.
• The results hold for both infinite time intervals $J = \mathbb{R}$ and finite intervals $J = (0,T)$, with $\lambda_0 = -\infty$ possible in the finite case.
• The framework allows for rough inhomogeneous boundary data by avoiding compatibility conditions through appropriate weight selection.