Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

Abstract

This paper is concerned with an explicit value of the embedding constant from $W^{1,q}(\Omega)$ to $L^{p}(\Omega)$ for a domain $\Omega\subset\mathbb{R}^{N}$ ($N\in\mathbb{N}$), where $1\leq q\leq p\leq\infty$. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.

Appendices

Appendix 1: Embedding constant $C_{p}(\Omega)$ on dividable domains

Theorem 6.1 provides an estimation of the embedding constant $C_{p}(\Omega)$ for a domain Ω that can be divided into domains $\Omega_{i}$ (such as convex domains and Lipschitz domains) satisfying (2) and (3).

Theorem 6.1

Let $\Omega\subset\mathbb{R}^{N}$ ($N\in\mathbb{N}$) be a domain that can be divided into a finite number of domains $\Omega _{i}$ ($i=1,2,3,\ldots, n$) satisfying (2) and (3). Assume that, for every $\Omega_{i}$ ($i=1,2,3,\ldots, n$), there exists a constant $C_{p}(\Omega_{i})$ such that $\Vert u \Vert _{L^{p}(\Omega_{i})}\leq C_{p}(\Omega_{i}) \Vert u \Vert _{W^{1,q}(\Omega_{i})}$ for all $u\in W^{1,q}(\Omega_{i})$. Then (1) holds valid for

$$C_{p}(\Omega)=M_{p,q}\max_{1\leq i\leq n} C_{p}(\Omega_{i}), $$

where

$$\begin{aligned} M_{p,q}= \textstyle\begin{cases} 1&(p\geq q),\\ n^{\frac{1}{p}-\frac{1}{q}}&(p< q). \end{cases}\displaystyle \end{aligned}$$

Proof

We consider both the cases in which $p<\infty$ and $p=\infty$.

When $p<\infty$, it follows that

$$\begin{aligned} \begin{aligned} \Vert u \Vert _{L^{p}(\Omega)}&= \biggl(\sum _{1\leq i\leq n} \Vert u \Vert _{L^{p}(\Omega_{i})}^{p} \biggr)^{1/p} \\ &\leq \biggl(\sum_{1\leq i\leq n}C_{p}( \Omega_{i})^{p} \Vert u \Vert _{W^{1,q}(\Omega_{i})}^{p} \biggr)^{1/p} \\ &\leq\max_{1\leq i\leq n} C_{p}( \Omega_{i}) \biggl(\sum_{1\leq i\leq n} \Vert u \Vert _{W^{1,q}(\Omega_{i})}^{p} \biggr)^{1/p} \\ &\leq M_{p,q}\max_{1\leq i\leq n} C_{p}( \Omega_{i}) \Vert u \Vert _{W^{1,q}(\Omega)}. \end{aligned} \end{aligned}$$

Note that $\vert x \vert _{p}\leq M_{p,q} \vert x \vert _{q}$ holds for $x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$ (see [19, Lemma A.1] for a detailed proof), where we denote

$$\begin{aligned} \vert x \vert _{p}= \textstyle\begin{cases} \displaystyle \biggl(\sum_{1\leq i\leq n} \vert x_{i} \vert ^{p} \biggr)^{\frac{1}{p}}&(1\leq p< \infty),\\ \displaystyle\max_{1\leq i\leq n} \vert x_{i} \vert &(p=\infty). \end{cases}\displaystyle \end{aligned}$$

When $p=\infty$,

$$\begin{aligned} \Vert u \Vert _{L^{\infty}(\Omega)}&= \max_{1\leq i\leq n} \Vert u \Vert _{L^{\infty}(\Omega_{i})} \\ &\leq\max_{1\leq i\leq n} C_{p}( \Omega_{i}) \Vert u \Vert _{W^{1,q}(\Omega_{i})} \\ &\leq\max_{1\leq i\leq n} C_{p}( \Omega_{i})\max_{1\leq i\leq n} \Vert u \Vert _{W^{1,q}(\Omega_{i})}. \end{aligned}$$

Since $M_{\infty,q}=1$, we have

$$\begin{aligned} \Vert u \Vert _{L^{\infty}(\Omega)}&\leq\max_{1\leq i\leq n} C_{p}(\Omega_{i}) \Vert u \Vert _{W^{1,q}(\Omega)}. \end{aligned}$$

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Appendix 2: A proof of Lemma 3.1

This section provides a proof of Lemma 3.1 based on [31, Lemma 7.16].

Proof of Lemma 3.1

Since $C^{\infty}(\Omega)\cap W^{1,1}(\Omega)$ is densely defined in $W^{1,1}(\Omega)$, it suffices to prove Lemma 3.1 for $u\in C^{1}(\Omega)$. Since Ω is convex, we have, for $x, y\in\Omega$,

$$\begin{aligned} u(x)-u(y)=- \int_{0}^{ \vert x-y \vert }\partial _{r}u(x+r\omega) \,dr, \end{aligned}$$

where $\omega=(y-x)/ \vert y-x \vert$ and $\partial _{r}u(x+r\omega)=\frac{\partial}{\partial r}u(x+r\omega)$. Integrating with respect to y over Ω, we obtain

$$\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &= \vert \Omega \vert ^{-1} \biggl\vert \int_{\Omega} \int_{0}^{\vert x-y\vert }\partial_{r}u(x+r\omega)\,dr \,dy \biggr\vert \\ &\leq \vert \Omega \vert ^{-1} \int_{\Omega} \int _{0}^{ \vert x-y \vert } \bigl\vert \partial_{r}u(x+r \omega ) \bigr\vert \,dr\,dy \\ &\leq \vert \Omega \vert ^{-1} \int_{\Omega} \int _{0}^{\infty} \bigl\vert (E_{\Omega,\mathbb{R}^{N}} \partial _{r}u) (x+r\omega) \bigr\vert \,dr\,dy \\ &\leq \vert \Omega \vert ^{-1} \int_{B(x,d_{\Omega})} \int _{0}^{\infty} \bigl\vert (E_{\Omega,\mathbb{R}^{N}} \partial _{r}u) (x+r\omega) \bigr\vert \,dr\,dy \\ &= \vert \Omega \vert ^{-1} \int_{0}^{d_{\Omega}} \int _{ \vert \omega \vert =1} \int_{0}^{\infty} \bigl\vert (E_{\Omega,\mathbb{R}^{N}} \partial_{r}u) (x+r\omega) \bigr\vert \rho ^{N-1}\,dr \,d \omega \,d\rho \\ &= \vert \Omega \vert ^{-1} \int_{0}^{\infty} \int_{ \vert \omega \vert =1} \int_{0}^{d_{\Omega}} \bigl\vert (E_{\Omega,\mathbb{R}^{N}} \partial_{r}u) (x+r\omega) \bigr\vert \rho ^{N-1}\,d\rho \,d \omega \,dr \\ &= \frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int _{0}^{\infty} \int_{ \vert \omega \vert =1} \bigl\vert (E_{\Omega,\mathbb{R}^{N}}\partial_{r}u) (x+r\omega) \bigr\vert \,d\omega \,dr \\ &= \frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int_{ \vert \omega \vert =1} \int_{0}^{\infty} \bigl\vert (E_{\Omega ,\mathbb{R}^{N}} \partial_{r}u) (x+r\omega) \bigr\vert r^{1-N}r^{N-1}drd \omega \\ &= \frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int _{\mathbb{R}^{N}} \bigl\vert (E_{\Omega,\mathbb{R}^{N}}\partial _{r}u) (y) \bigr\vert \vert x-y \vert ^{1-N}\,dy \\ &= \frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int _{\Omega} \bigl\vert \partial_{r}u(y) \bigr\vert \vert x-y \vert ^{1-N}\,dy. \end{aligned}$$

Therefore, a proof of Lemma 3.1 is completed. □