[Paper Review] Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains

Francesco Di Plinio, Roger Témam|arXiv (Cornell University)|Oct 21, 2013

Advanced Harmonic Analysis Research15 references5 citations

TL;DR

This paper establishes sharp $ L^p $ and endpoint estimates for the solution operator of the Dirichlet problem on polygonal domains in $ \mathbb{R}^2 $, extending Grisvard's shift theorem to domains with corner singularities up to aperture $ \alpha_j \leq \pi/2 $. It proves $ \|G_\Omega f\|_{W^{2,p}(\Omega)} \leq Cp\|f\|_{L^p(\Omega)} $ for $ 2 \leq p < \infty $, and $ \|D^2 G_\Omega f\|_{\mathrm{Exp}L^1(\Omega)} \leq C\|f\|_{L^\infty(\Omega)} $, enabling the extension of Yudovich's theory for the 2D Euler equations to such domains.

ABSTRACT

Let Omega be a bounded, simply connected domain with boundary of class C^{1,1} except at finitely many points S_j where the boundary is locally a corner of aperture alpha_j&lt;=pi/2. Improving on results of Grisvard, we show that the solution Gf to the Dirichlet problem on Omega with data f in L^infinity(Omega) and homogeneous boundary conditions has exponentially integrable second derivatives. The proof uses sharp L^p bounds for singular integrals on power weighted spaces inspired by the work of Buckley. Our results allow for the extension of the Yudovich theory of existence, uniqueness and regularity of weak solutions to the Euler equations on Omega x (0,T) to polygonal domains Omega as above.

Motivation & Objective

• To extend Grisvard's shift theorem to polygonal domains with corner singularities where $ \partial\Omega \in \mathcal{C}^{1,1} $ except at finitely many corners with aperture $ \alpha_j \leq \pi/2 $.
• To establish $ L^p $-boundedness of second derivatives of the Green operator with linear growth in $ p $, even in the presence of corner singularities.
• To derive an endpoint estimate in the Orlicz space $ \mathrm{Exp}L^1 $ for $ L^\infty $ data, replacing the failure of $ L^\infty $-boundedness of Calderón-Zygmund operators.
• To apply these estimates to extend Yudovich's existence, uniqueness, and regularity theory for the 2D Euler equations to polygonal domains.
• To develop sharp weighted $ L^p $ bounds for singular integrals on power-weighted spaces, inspired by Buckley's work, to handle non-smooth boundaries.

Proposed method

• Use of sharp $ L^p $ bounds for singular integrals on power-weighted $ L^p $ spaces, leveraging the $ A_p $-Muckenhoupt weights theory.
• Application of the Calderón-Zygmund decomposition and maximal function estimates in weighted $ L^p $ spaces to control the second derivatives of the Green operator.
• Adaptation of Rubio de Francia's extrapolation and weighted norm inequalities to handle the singular behavior near corners.
• Use of the John-Nirenberg inequality and $ \mathrm{Exp}L^1 $-type estimates to derive the endpoint $ L^\infty $ bound via extrapolation from $ L^p $ bounds with linear $ p $-dependence.
• Proof of weighted $ L^p $ bounds for the second derivatives of the Green function via dyadic decomposition and testing over annuli $ B_j $, using the $ A_p $ characteristic $[w]_{A_p}$.
• Establishment of a key $ L^p $ operator norm bound $ \|T\|_{L^p(w)} \lesssim p $ for singular integral operators on power weights $ w(x) = |x|^{2\delta(p-1)} $, with $ \delta \in (0,1) $.

Experimental results

Research questions

• RQ1 Can Grisvard's shift theorem be extended to polygonal domains with corner singularities where the boundary is only $ \mathcal{C}^{1,1} $ except at finitely many points with $ \alpha_j \leq \pi/2 $?
• RQ2 Does the $ L^p $-bound for the second derivatives of the Green operator on such domains still exhibit linear growth in $ p $, even when the kernel is not a Calderón-Zygmund kernel?
• RQ3 Is there a substitute endpoint estimate in $ \mathrm{Exp}L^1 $ for $ L^\infty $ data when $ L^\infty $-boundedness fails due to corner singularities?
• RQ4 Can Yudovich's theory for the 2D Euler equations be extended to polygonal domains, given the lack of $ C^{1,1} $ regularity?
• RQ5 What weighted $ L^p $ estimates for singular integrals on power weights are sufficient to control the second derivatives of the Green function on non-smooth domains?

Key findings

• The solution operator $ G_\Omega $ satisfies $ \|G_\Omega f\|_{W^{2,p}(\Omega)} \leq Cp\|f\|_{L^p(\Omega)} $ for all $ 2 \leq p < \infty $, with $ C $ independent of $ p $, on polygonal domains with corner apertures $ \alpha_j \leq \pi/2 $.
• The endpoint estimate $ \|D^2 G_\Omega f\|_{\mathrm{Exp}L^1(\Omega)} \leq C\|f\|_{L^\infty(\Omega)} $ holds, providing a substitute for the failure of $ L^\infty $-boundedness of second derivatives.
• The $ L^p $ operator norm of the second derivatives of the Green function grows linearly in $ p $, consistent with extrapolation from $ L^p $ bounds with $ \|T\|_{L^p} \lesssim p $.
• The proof relies on sharp weighted $ L^p $ bounds for singular integrals on power weights $ w(x) = |x|^{2\delta(p-1)} $, with the $ A_p $ characteristic $[w]_{A_p} \sim (1-\delta)^{-1} $, enabling control of the operator norm.
• The $ L^p $-boundedness result is extended to $ p \in (1,\infty) \setminus \{p_{\alpha_j} = \frac{2\alpha_j}{2\alpha_j - \pi} \mid \alpha_j > \pi/2\} $, with a gap in the range when $ \alpha_j > \pi/2 $, but the main result holds for $ p \geq 2 $.
• The Yudovich theory for the 2D Euler equations is extended to polygonal domains with $ \alpha_j \leq \pi/2 $, ensuring existence, uniqueness, and regularity of weak solutions on $ \Omega \times (0,T) $.