## 1 Introduction

We consider the Sobolev type embedding constant $$C_{p}(\Omega)$$ from $$W^{1,q}(\Omega)$$ ($$1\leq q\leq p\leq\infty$$) to $$L^{p}(\Omega)$$. The constant $$C_{p}(\Omega)$$ satisfies

\begin{aligned} \biggl( \int_{\Omega} \bigl\vert u(x) \bigr\vert ^{p}\,dx \biggr)^{\frac {1}{p}}\leq C_{p}(\Omega) \biggl( \int_{\Omega} \bigl\vert u(x) \bigr\vert ^{q}\,dx+ \int_{\Omega} \bigl\vert \nabla u(x) \bigr\vert ^{q} \,dx \biggr)^{\frac{1}{q}} \end{aligned}
(1)

for all $$u\in W^{1,q}(\Omega)$$, where $$\Omega\subset\mathbb {R}^{N}$$ ($$N\in\mathbb{N}$$) is a bounded domain and $$\vert x \vert =\sqrt{\sum_{j=1}^{N}x_{j}^{2}}$$ for $$x=(x_{1},\ldots, x_{N})\in \mathbb{R}^{N}$$. Here, $$L^{p}(\Omega)$$ ($$1\leq p<\infty$$) is the functional space of the pth power Lebesgue integrable functions over Ω endowed with the norm $$\Vert f \Vert _{L^{p}(\Omega)}:=(\int_{\Omega } \vert f(x) \vert ^{p}\,dx)^{1/p}$$ for $$f\in L^{p}(\Omega)$$, and $$L^{\infty}(\Omega)$$ is the functional space of Lebesgue measurable functions over Ω endowed with the norm $$\Vert f \Vert _{L^{\infty}(\Omega)}=\operatorname{ess\,sup}_{x\in\Omega } \vert f(x) \vert$$ for $$f\in L^{\infty}(\Omega)$$. Moreover, $$W^{k,p}(\Omega)$$ is the kth order $$L^{p}$$-Sobolev space on Ω endowed with the norm $$\Vert f \Vert _{W^{1,p}(\Omega)}=(\int_{\Omega} \vert f(x) \vert ^{p}\,dx+\int_{\Omega} \vert \nabla f(x) \vert ^{p}\,dx)^{1/p}$$ for $$f\in W^{1,p}(\Omega)$$ if $$1\leq p<\infty$$ and $$\Vert f \Vert _{W^{1,\infty}(\Omega)}=\operatorname{ess\,sup}_{x\in \Omega} \vert f(x) \vert +\operatorname{ess\,sup}_{x\in \Omega} \vert \nabla f(x) \vert$$ for $$f\in W^{1,\infty }(\Omega)$$ if $$p=\infty$$.

Since inequality (1) has significance for studies on partial differential equations, many researchers studied this type of Sobolev inequality and an explicit value of $$C_{p}(\Omega)$$ (see, e.g., [17]) following the pioneering work by Sobolev [1]. In particular, our interest is in the applicability of this constant to verified numerical computation methods for PDEs which originate from Nakao’s [8] and Plum’s work [9]. These methods have been further developed by many researchers (see, e.g., [810] and the references therein).

The existence of $$C_{p}(\Omega)$$ for various domains Ω (e.g., domains with the cone condition, domains with the Lipschitz boundary, and the $$(\varepsilon, \delta)$$-domains) has been proven by constructing suitable extension operators from $$W^{k,p}(\Omega)$$ to $$W^{k,p}(\mathbb{R}^{N})$$ (see, e.g., [37]).

Several formulas for computing explicit values of $$C_{p}(\Omega)$$ have been proposed under suitable conditions. For example, the best constant in the classical Sobolev inequality on $$\mathbb{R}^{N}$$ was independently shown by Aubin [11] and Talenti [12]. For the case in which $$N=1$$ and $$p=\infty$$, the best constant of $$C_{p}(\Omega)$$ was proposed under some boundary conditions, e.g., the Dirichlet, the Neumann, and the periodic condition [1317]. For a square domain $$\Omega\subset\mathbb{R}^{2}$$, a tight estimate of $$C_{p}(\Omega)$$ was provided in [10]. Moreover, the best constant for the embedding $$W^{1,2}_{0}(\Omega )\hookrightarrow L^{p}(\Omega)$$ ($$p=3,4,5,6,7$$) with a square domain $$\Omega\subset\mathbb{R}^{2}$$ was very sharply estimated in [18], where $$W^{1,2}_{0}(\Omega)$$ denotes the closure of $$C^{\infty}_{0}(\Omega )$$ in $$W^{1,2}(\Omega)$$. Furthermore, we have previously proposed a formula for computing an explicit value of $$C_{p}(\Omega)$$ for (bounded and unbounded) Lipschitz domains $$\Omega\subset\mathbb{R}^{N}$$ ($$N\geq2$$) by estimating the norm of Stein’s extension operator [19]. This formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains $$\Omega_{i}$$ ($$i=1,2,3,\ldots, n$$) such that

\begin{aligned} \overline{\Omega}=\bigcup_{1\leq i\leq n} \overline{\Omega_{i}} \end{aligned}
(2)

and

\begin{aligned} \Omega_{i}\cap\Omega_{j}=\phi\quad (i\neq j), \end{aligned}
(3)

where ϕ is the empty set and Ω̅ denotes the closure of Ω (see Theorem 6.1). Although this formula is applicable to such general domains, the values computed by this formula are very large; see Section 4 for concrete values.

In this paper, we report that the accuracy of the estimation of $$C_{p}(\Omega)$$ is significantly improved by restricting each $$\Omega _{i}$$ to bounded convex domain. Since any bounded convex domain is a Lipschitz domain (see, e.g., [20]), the present class of Ω is somewhat special compared with the class treated in [19]. Nevertheless, the formulas presented in this paper still have applicability to various domains. To obtain a sharper estimation of $$C_{p}(\Omega)$$, we focus on the constants $$D_{p}(\Omega)$$ such that

\begin{aligned} \biggl( \int_{\Omega} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}}\leq D_{p}(\Omega) \biggl( \int_{\Omega} \bigl\vert \nabla u(x) \bigr\vert ^{q} \,dx \biggr)^{\frac{1}{q}}\quad \mbox{for all } u\in W^{1,q}(\Omega). \end{aligned}
(4)

Here, $$\vert \Omega \vert$$ is the measure of Ω and $$u_{\Omega}:\Omega\to\mathbb{R}$$ is a constant function defined by $$\Omega\ni x\mapsto u_{\Omega}(x)= \vert \Omega \vert ^{-1}\int_{\Omega}u(y)\,dy$$. Inequality (4) is called the Sobolev-Poincaré inequality, and $$D_{p}(\Omega)$$ in (4) leads to the explicit value of $$C_{p}(\Omega)$$ (see Theorem 2.1). Inequality (4) has also been studied by many researchers (see, e.g., [2124]). For example, for a John domain Ω, the existence of $$D_{p}(\Omega)$$ was shown while assuming that $$1\leq q< N$$, $$p=Nq/(N-q)$$ [23]. It was also shown that, when $$p\neq Nq/(N-q)$$, $$D_{p}(\Omega)$$ exists if and only if $$W^{1,q}(\Omega)$$ is continuously embedded into $$L^{p}(\Omega)$$ [24]. Moreover, there are several formulas for obtaining an explicit value of $$D_{p}(\Omega)$$ for one-dimensional domains Ω [2527]. In the higher-dimensional cases, however, little is known about explicit values of $$D_{p}(\Omega)$$, except for some special cases (see, e.g., [28] and [29] for the cases in which $$p=q=1$$ and $$p=q=2$$, respectively).

We propose four theorems (Theorem 3.1 to 3.4) for obtaining explicit values of $$D_{p}(\Omega)$$ on a bounded convex domain Ω. Each theorem can be used under the corresponding conditions listed in Table 1.

Theorems 3.1 and 3.2 are derived from the best constant in the Hardy-Littlewood-Sobolev inequality on $$\mathbb{R}^{N}$$. Theorems 3.3 and 3.4 are derived from the best constant in Young’s inequality on $$\mathbb{R}^{N}$$. The values of $$D_{p}(\Omega)$$ calculated by these theorems yield the explicit values of $$C_{p}(\Omega)$$ combined with Theorem 2.1.

The remainder of this paper is organized as follows. In Section 2, we propose Theorem 2.1 in which a formula for deriving an explicit value of $$C_{p}(\Omega)$$ from known $$D_{p}(\Omega)$$ is provided. In Section 3, we prove the four formulas (Theorems 3.1 to 3.4) for obtaining the explicit values of $$D_{p}(\Omega)$$. In Section 4, we present examples where explicit values of $$C_{p}(\Omega)$$ are estimated for certain domains.

## 2 Estimation of embedding constant $$C_{p}(\Omega)$$

The following notation is used throughout this paper. For any bounded domain $$S\subset\mathbb{R}^{N}$$ ($$N\in\mathbb{N}$$), we define $$d_{S}$$:=$$\sup_{x,y\in S} \vert x-y \vert$$. The closed ball centered around $$z\in\mathbb{R}^{N}$$ with radius $$\rho >0$$ is denoted by $$B(z,\rho):=\{x\in\mathbb{R}^{N}\mid \vert x-z \vert \leq\rho\}$$. For $$m\geq1$$, let $$m'$$ be Hölder’s conjugate of m, that is, $$m'$$ is defined by

\begin{aligned} \textstyle\begin{cases} m'=\infty,& \mbox{if } m=1,\\ m'=\frac{m}{m-1},&\mbox{if } 1< m< \infty,\\ m'=1,&\mbox{if } m=\infty. \end{cases}\displaystyle \end{aligned}

For two domains $$\Omega\subseteq\mathbb{R}^{N}$$ and $$\Omega'\subseteq \mathbb{R}^{N}$$ such that $$\Omega\subseteq\Omega'$$, we define the operator $$E_{\Omega,\Omega'}:L^{p}(\Omega)\to L^{p}(\Omega')$$ ($$1\leq p\leq\infty$$) by

\begin{aligned} (E_{\Omega,\Omega'}f ) (x)= \textstyle\begin{cases} f(x),&x\in\Omega,\\ 0,&x\in\Omega'\setminus\Omega \end{cases}\displaystyle \end{aligned}

for $$f\in L^{p}(\Omega)$$. Note that $$E_{\Omega,\Omega'}f\in L^{p}(\Omega')$$ satisfies

$$\Vert E_{\Omega,\Omega'}f \Vert _{L^{p}(\Omega')}= \Vert f \Vert _{L^{p}(\Omega)}.$$

In the following theorem, we provide a formula for obtaining an explicit value of $$C_{p}(\Omega)$$ from known $$D_{p}(\Omega)$$.

### Theorem 2.1

Let $$\Omega\subset\mathbb{R}^{N}$$ ($$N\in\mathbb{N}$$) be a bounded domain, and let p and q satisfy $$1\leq q\leq p\leq\infty$$. Suppose that there exists a finite number of bounded domains $$\Omega _{i}$$ ($$i=1,2,3,\ldots, n$$) satisfying (2) and (3). Moreover, suppose that, for every $$\Omega_{i}$$ ($$i=1,2,3,\ldots, n$$), there exist constants $$D_{p}(\Omega_{i})$$ such that

\begin{aligned} \Vert u-u_{\Omega_{i}} \Vert _{L^{p}(\Omega_{i})}\leq D_{p}(\Omega_{i}) \Vert \nabla u \Vert _{L^{q}(\Omega_{i})}\quad \textit{for all } u\in W^{1,q}(\Omega_{i}). \end{aligned}
(5)

Then (1) holds valid for

\begin{aligned} C_{p}(\Omega)= \textstyle\begin{cases} \displaystyle\max \Bigl(1, \max_{1\leq i\leq n}D_{\infty}(\Omega_{i}) \Bigr) & (p=q=\infty),\\ \displaystyle2^{1-\frac{1}{q}}\max \Bigl(\max_{1\leq i\leq n} \vert \Omega _{i} \vert ^{\frac{1}{p}-\frac{1}{q}}, \max_{1\leq i\leq n}D_{p}(\Omega_{i}) \Bigr) & (\textit{otherwise}), \end{cases}\displaystyle \end{aligned}
(6)

where this formula is understood with $$1/\infty=0$$ when $$p=\infty$$ and/or $$q=\infty$$.

### Proof

Let $$u\in W^{1,q}(\Omega)$$. Since every $$\Omega_{i}$$ is bounded, Hölder’s inequality states that

\begin{aligned} \Vert u_{\Omega_{i}} \Vert _{L^{p}(\Omega_{i})}&= \biggl\vert \int_{\Omega_{i}} \vert \Omega_{i} \vert ^{-1}u(y)\,dy \biggr\vert \Vert 1 \Vert _{L^{p}(\Omega_{i})} \\ &\leq \vert \Omega_{i} \vert ^{-1+\frac{1}{q'}} \Vert u \Vert _{L^{q}(\Omega_{i})} \vert \Omega_{i} \vert ^{\frac{1}{p}} \\ &= \vert \Omega_{i} \vert ^{\frac{1}{p}-\frac{1}{q}} \Vert u \Vert _{L^{q}(\Omega_{i})}. \end{aligned}
(7)

We describe the following proof separately for the case of $$p=\infty$$ and $$p<\infty$$.

When $$p=\infty$$, we have

\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} \bigl( \Vert u_{\Omega_{i}} \Vert _{L^{\infty}(\Omega_{i})}+ \Vert u-u_{\Omega_{i}} \Vert _{L^{\infty}(\Omega_{i})} \bigr). \end{aligned}

From (5) and (7), it follows that

\begin{aligned} & \Vert u \Vert _{L^{\infty}(\Omega)} \\ &\quad \leq\max_{1\leq i\leq n} \bigl( \vert \Omega_{i} \vert ^{-\frac{1}{q}} \Vert u \Vert _{L^{q}(\Omega_{i})}+D_{\infty }(\Omega_{i}) \Vert \nabla u \Vert _{L^{q}(\Omega_{i})} \bigr) \\ &\quad \leq\max \Bigl\{ \max_{1\leq i\leq n} \vert \Omega_{i} \vert ^{-\frac{1}{q}}, \max_{1\leq i\leq n}D_{\infty}(\Omega _{i}) \Bigr\} \max_{1\leq i\leq n} \bigl( \Vert u \Vert _{L^{q}(\Omega_{i})}+ \Vert \nabla u \Vert _{L^{q}(\Omega _{i})} \bigr). \end{aligned}

This implies that Theorem 2.1 holds for the case of $$p=\infty$$ and $$q=\infty$$.

For $$q<\infty$$, we have

\begin{aligned} & \Vert u \Vert _{L^{\infty}(\Omega)} \\ &\quad \leq\max \Bigl\{ \max_{1\leq i\leq n} \vert \Omega_{i} \vert ^{-\frac{1}{q}}, \max_{1\leq i\leq n}D_{\infty}(\Omega _{i}) \Bigr\} \biggl(\sum_{1\leq i\leq n} \bigl( \Vert u \Vert _{L^{q}(\Omega_{i})}+ \Vert \nabla u \Vert _{L^{q}(\Omega _{i})} \bigr)^{q} \biggr)^{\frac{1}{q}} \\ &\quad \leq2^{1-\frac{1}{q}}\max \Bigl\{ \max_{1\leq i\leq n} \vert \Omega_{i} \vert ^{-\frac{1}{q}}, \max_{1\leq i\leq n}D_{\infty }( \Omega_{i}) \Bigr\} \Vert u \Vert _{W^{1,q}(\Omega)}, \end{aligned}

where the last inequality follows from $$(s+t)^{q}\leq2^{q-1}(s^{q}+t^{q})$$ for $$s,t\geq0$$.

When $$p<\infty$$, we have

\begin{aligned} \Vert u \Vert _{L^{p}(\Omega)}&= \biggl(\sum _{1\leq i\leq n} \int_{\Omega_{i}} \bigl\vert u(y) \bigr\vert ^{p}\,dy \biggr)^{\frac {1}{p}} \\ &= \biggl(\sum_{1\leq i\leq n} \Vert u \Vert _{L^{p}(\Omega _{i})}^{p} \biggr)^{\frac{1}{p}} \\ &\leq \biggl(\sum_{1\leq i\leq n} \bigl( \Vert u_{\Omega _{i}} \Vert _{L^{p}(\Omega_{i})}+ \Vert u-u_{\Omega_{i}} \Vert _{L^{p}(\Omega_{i})} \bigr)^{p} \biggr)^{\frac{1}{p}}. \end{aligned}

From (5) and (7), it follows that

\begin{aligned} \Vert u \Vert _{L^{p}(\Omega)}&\leq \biggl(\sum _{1\leq i\leq n} \bigl( \vert \Omega_{i} \vert ^{\frac{1}{p}-\frac {1}{q}} \Vert u \Vert _{L^{q}(\Omega_{i})}+D_{p}(\Omega _{i}) \Vert \nabla u \Vert _{L^{q}(\Omega_{i})} \bigr)^{p} \biggr)^{\frac{1}{p}} \\ &\leq \biggl(\sum_{1\leq i\leq n} \bigl( \vert \Omega_{i} \vert ^{\frac{1}{p}-\frac{1}{q}} \Vert u \Vert _{L^{q}(\Omega_{i})}+D_{p}(\Omega_{i}) \Vert \nabla u \Vert _{L^{q}(\Omega_{i})} \bigr)^{q} \biggr)^{\frac{1}{q}} \\ &\leq2^{1-\frac{1}{q}} \biggl(\sum_{1\leq i\leq n} \bigl( \vert \Omega_{i} \vert ^{\frac{q}{p}-1} \Vert u \Vert _{L^{q}(\Omega_{i})}^{q}+D_{p}(\Omega_{i})^{q} \Vert \nabla u \Vert _{L^{q}(\Omega_{i})}^{q} \bigr) \biggr)^{\frac{1}{q}}. \end{aligned}

Therefore, we obtain

\begin{aligned} \Vert u \Vert _{L^{p}(\Omega)}\leq2^{1-\frac{1}{q}}\max \Bigl\{ \max_{1\leq i\leq n} \vert \Omega_{i} \vert ^{\frac {1}{p}-\frac{1}{q}}, \max_{1\leq i\leq n}D_{i}( \Omega_{i}) \Bigr\} \Vert u \Vert _{W^{1,q}(\Omega)}. \end{aligned}

□

## 3 Estimation of $$D_{p}(\Omega_{i})$$

Let Γ be the gamma function, that is, $$\Gamma(x)=\int _{0}^{\infty}t^{x-1}e^{-t}\,dt$$ for $$x>0$$. For $$f\in L^{r}(\mathbb{R}^{N})$$ and $$g\in L^{s}(\mathbb{R}^{N})$$ ($$1\leq r,s\leq\infty$$), let $$f*g: \mathbb{R}^{N}\to\mathbb{R}$$ be the convolution of f and g defined by

\begin{aligned} (f*g) (x):= \int_{\mathbb{R}^{N}}f(x-y)g(y)\,dy \biggl(= \int_{\mathbb {R}^{N}}f(x)g(x-y)\,dy \biggr). \end{aligned}

In the following three lemmas, we recall some known results required to obtain explicit values of $$D_{p}(\Omega_{i})$$ in (5) for bounded convex domains $$\Omega_{i}$$.

### Lemma 3.1

(see, e.g., [30, 31])

Let $$\Omega\subset\mathbb{R}^{N}$$ ($$N\in\mathbb{N}$$) be a bounded convex domain. For $$u\in W^{1,1}(\Omega)$$ and any point $$x\in\Omega$$, we have

\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert \leq\frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-y \vert ^{1-N} \bigl\vert \nabla u(y) \bigr\vert \,dy. \end{aligned}

A proof of Lemma 3.1 is provided in Appendix 2 because Lemma 3.1 plays an especially important role in obtaining the explicit values of $$D_{p}(\Omega_{i})$$.

### Lemma 3.2

(Hardy-Littlewood-Sobolev’s inequality [32])

For $$\lambda>0$$, we put $$h_{\lambda}(x):= \vert x \vert ^{-\lambda}$$. If $$0<\lambda<N$$,

\begin{aligned} \Vert h_{\lambda}*g \Vert _{L^{\frac{2N}{\lambda }}(\mathbb{R}^{N})}\leq C_{\lambda, N} \Vert g \Vert _{L^{\frac{2N}{2N-\lambda}}(\mathbb{R}^{N})}\quad \textit{for all } g\in L^{\frac{2N}{2N-\lambda}}\bigl(\mathbb{R}^{N}\bigr) \end{aligned}
(8)

holds valid for

\begin{aligned} C_{\lambda, N}=\pi^{\frac{\lambda}{2}}\frac{\Gamma(\frac {N}{2}-\frac{\lambda}{2})}{\Gamma(N-\frac{\lambda}{2})} \biggl( \frac{\Gamma(\frac{N}{2})}{\Gamma(N)} \biggr)^{-1+\frac{\lambda}{N}}, \end{aligned}
(9)

where this is the best constant in (8).

Moreover, if $$N<2\lambda<2N$$,

\begin{aligned} \Vert h_{\lambda}*g \Vert _{L^{\frac{2N}{2\lambda -N}}(\mathbb{R}^{N})}\leq\tilde{C}_{\lambda, N} \Vert g \Vert _{L^{2}(\mathbb{R}^{N})}\quad \textit{for all } g\in L^{2}\bigl(\mathbb{R}^{N}\bigr) \end{aligned}
(10)

holds valid for

\begin{aligned} \tilde{C}_{\lambda, N}=\pi^{\frac{\lambda}{2}}\frac{\Gamma(\frac {N}{2}-\frac{\lambda}{2})}{\Gamma(\frac{\lambda}{2})} \sqrt{\frac {\Gamma(\lambda-\frac{N}{2})}{\Gamma(\frac{3N}{2}-\lambda)}} \biggl(\frac{\Gamma(\frac{N}{2})}{\Gamma(N)} \biggr)^{-1+\frac{\lambda}{N}}, \end{aligned}
(11)

where this is the best constant in (10).

### Lemma 3.3

(Young’s inequality [33])

Suppose that $$1\leq t,r,s\leq\infty$$ and $$1/t=1/r+1/s-1\geq0$$. For $$f\in L^{r}(\mathbb{R}^{N})$$ and $$g\in L^{s}(\mathbb{R}^{N})$$, we have

\begin{aligned} \Vert f*g \Vert _{L^{t}(\mathbb{R}^{N})}\leq (A_{r}A_{s}A_{t'})^{N} \Vert f \Vert _{L^{r}(\mathbb {R}^{N})} \Vert g \Vert _{L^{s}(\mathbb{R}^{N})} \end{aligned}
(12)

with

\begin{aligned} A_{m}= \textstyle\begin{cases} \sqrt{m^{\frac{2}{m}-1}(m-1)^{1-\frac{1}{m}}}&(1< m< \infty),\\ 1&(m=1, \infty). \end{cases}\displaystyle \end{aligned}

The constant $$(A_{r}A_{s}A_{t'})^{N}$$ is the best constant in (12).

The following Theorems 3.1, 3.2, 3.3, and 3.4 provide estimations of $$D_{p}(\Omega)$$ for a bounded convex domain Ω, where p, q, and N are imposed on the assumptions listed in Table 1.

### Theorem 3.1

Let $$\Omega\subset\mathbb{R}^{N}$$ ($$N\in\mathbb{N}$$) be a bounded convex domain. Assume that $$p\in\mathbb{R}$$ satisfies $$2< p\leq 2N/(N-1)$$ if $$N\geq2$$ and $$2< p<\infty$$ if $$N=1$$. For $$q\in\mathbb{R}$$ such that $$q\geq p/(p-1)$$, we have

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}\leq D_{p}(\Omega ) \Vert \nabla u \Vert _{L^{q}(\Omega)} \quad \textit{for all } u\in W^{1,q}(\Omega) \end{aligned}

with

\begin{aligned} D_{p}(\Omega)=\frac{d_{\Omega}^{1+\frac{2N}{p}}\pi^{\frac {N}{p}}}{N \vert \Omega \vert ^{\frac{1}{p}+\frac {1}{q}}}\frac{\Gamma(\frac{p-2}{2p}N)}{\Gamma(\frac {p-1}{p}N)} \biggl( \frac{\Gamma(N)}{\Gamma(\frac{N}{2})} \biggr)^{\frac{p-2}{p}}. \end{aligned}

### Proof

Let $$u\in W^{1,q}(\Omega)$$. Since $$p\leq2N/(N-1)$$ and $$1-N+(2N/p)\geq 0$$, it follows that $$\vert x-z \vert ^{1-N+\frac {2N}{p}}\leq d_{\Omega}^{1-N+\frac{2N}{p}}$$ for $$x, z\in\Omega$$. Lemma 3.1 implies that, for a fixed $$x\in\Omega$$,

\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq \frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{1-N+\frac{2N}{p}} \vert x-z \vert ^{-\frac{2N}{p}} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{-\frac {2N}{p}} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac {2N}{p}} \bigl(E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz. \end{aligned}

Therefore,

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \biggl( \int_{\Omega} \biggl( \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac{2N}{p}} \bigl(E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}} \\ &\leq\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \biggl( \int_{\mathbb{R}^{N}} \biggl( \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac{2N}{p}} \bigl(E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}

Since $$q\geq p/(p-1)$$ and Ω is bounded, we have $$\vert \nabla u \vert \in L^{p/(p-1)}(\Omega)$$. Therefore, Lemma 3.2 ensures

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert }C_{\frac{2N}{p}, N} \bigl\Vert E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr\Vert _{L^{\frac{p}{p-1}}(\mathbb{R}^{N})} \\ &=\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert }C_{\frac{2N}{p}, N} \Vert \nabla u \Vert _{L^{\frac {p}{p-1}}(\Omega)}, \end{aligned}

where $$C_{\frac{2N}{p}, N}$$ is defined in (9) with $$\lambda=2N/p$$. Since $$q\geq p/(p-1)$$, Hölder’s inequality moreover implies

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert ^{\frac{1}{p}+\frac{1}{q}}}C_{\frac{2N}{p}, N} \Vert \nabla u \Vert _{L^{q}(\Omega)}. \end{aligned}

□

### Theorem 3.2

Let $$\Omega\subset\mathbb{R}^{N}$$ ($$N\geq2$$) be a bounded convex domain. Assume that $$2< p\leq2N/(N-2)$$ if $$N\geq3$$ and $$2< p<\infty$$ if $$N=2$$. For all $$u\in W^{1,2}(\Omega)$$, we have

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}\leq D_{p}(\Omega ) \Vert \nabla u \Vert _{L^{2}(\Omega)} \end{aligned}

with

\begin{aligned} D_{p}(\Omega)=\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}\pi^{\frac {p+2}{4p}N}}{N \vert \Omega \vert }\frac{\Gamma(\frac {p-2}{4p}N)}{\Gamma(\frac{p+2}{4p}N)}\sqrt{ \frac{\Gamma(\frac {N}{p})}{\Gamma(\frac{p-1}{p}N)}} \biggl(\frac{\Gamma(N)}{\Gamma (\frac{N}{2})} \biggr)^{\frac{p-2}{2p}}. \end{aligned}

### Proof

Let $$u\in W^{1,2}(\Omega)$$. Since $$p\leq2N/(N-2)$$, it follows that $$\vert x-z \vert ^{1-N+(p+2)N/(2p)}\leq d_{\Omega }^{1-N+(p+2)N/(2p)}$$ for $$x, z\in\Omega$$. Lemma 3.1 leads to

\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq \frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{1-N+\frac{p+2}{2p}N} \vert x-z \vert ^{-\frac{p+2}{2p}N} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{-\frac {p+2}{2p}N} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac {p+2}{2p}N} \bigl(E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz. \end{aligned}

Therefore,

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \biggl( \int_{\Omega} \biggl( \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac{p+2}{2p}N} \bigl(E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}} \\ &\leq\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \biggl( \int_{\mathbb{R}^{N}} \biggl( \int_{\mathbb {R}^{N}} \vert x-z \vert ^{-\frac{p+2}{2p}N} \bigl(E_{\Omega ,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}

From (10), it follows that

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert }\tilde{C}_{\frac{p+2}{2p}N, N} \bigl\Vert E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr\Vert _{L^{2}(\mathbb {R}^{N})} \\ &=\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert }\tilde{C}_{\frac{p+2}{2p}N, N} \Vert \nabla u \Vert _{L^{2}(\Omega)}, \end{aligned}

where $$\tilde{C}_{\frac{p+2}{2p}N, N}$$ is defined in (11) with $$\lambda=(p+2)N/(2p)$$. □

### Theorem 3.3

Let $$\Omega\subset\mathbb{R}^{N}$$ ($$N\in\mathbb{N}$$) be a bounded convex domain. Suppose that $$1\leq q\leq p< qN/(N-q)$$ if $$N>q$$, and $$1\leq q\leq p<\infty$$ if $$N=q$$. Then we have

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}\leq D_{p}(\Omega ) \Vert \nabla u \Vert _{L^{q}(\Omega)}\quad \textit{for all } u\in W^{1,q}(\Omega) \end{aligned}
(13)

with

\begin{aligned} D_{p}(\Omega)=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }(A_{r}A_{q}A_{p'})^{N} \bigl\Vert \vert x \vert ^{1-N} \bigr\Vert _{L^{r}(V)}, \end{aligned}

where $$\Omega_{x}:=\{x-y\mid y\in\Omega\}$$ for $$x\in\Omega$$, $$V:=\bigcup_{x\in\Omega}\Omega_{x}$$, and $$r=qp/((q-1)p+q)$$.

### Proof

First, we prove $$I:= \Vert \vert x \vert ^{1-N} \Vert _{L^{r}(V)}^{r}<\infty$$. Let $$\rho=2d_{\Omega}$$ so that $$V\subset B(0,\rho)$$. We have

\begin{aligned} \frac{pq(1-N)}{(q-1)p+q}+N-1&=\frac{pq(1-N)+Np(q-1)+Nq}{(q-1)p+q}-1 \\ &=\frac{Nq-(N-q)p}{(q-1)p+q}-1>-1. \end{aligned}

Therefore,

\begin{aligned} I&= \int_{V} \vert x \vert ^{\frac{pq(1-N)}{(q-1)p+q}}\,dx \leq \int_{B(0,\rho)} \vert x \vert ^{\frac{pq(1-N)}{(q-1)p+q}}\,dx =J \int_{0}^{\rho}\rho^{\frac{pq(1-N)}{(q-1)p+q}+N-1}\,d\rho < \infty, \end{aligned}

where J is defined by

$$J= \textstyle\begin{cases} 2&(N=1),\\ 2\pi&(N=2),\\ 2\pi\int_{[0,\pi]^{N-2}}\prod_{i=1}^{N-2}(\sin\theta_{i})^{N-i-1} \,d\theta_{1}\cdots \,d\theta_{N-2}&(N\geq3). \end{cases}$$

Next, we show (13). For $$x\in\Omega$$, it follows from Lemma 3.1 that

\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq\frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-y \vert ^{1-N} \bigl\vert \nabla u(y) \bigr\vert \,dy \\ &=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int_{\Omega _{x}} \vert y \vert ^{1-N} \bigl\vert \nabla u(x-y) \bigr\vert \,dy \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int _{V} \vert y \vert ^{1-N} \bigl(E_{\Omega,V} \vert \nabla u \vert \bigr) (x-y)\,dy. \end{aligned}

Since $$E_{V,\mathbb{R}^{N}}E_{\Omega,V}=E_{\Omega,\mathbb{R}^{N}}$$,

\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq \frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\mathbb{R}^{N}} (E_{V,\mathbb{R}^{N}}\psi ) (y) \bigl(E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr) (x-y)\,dy, \end{aligned}
(14)

where $$\psi(y)= \vert y \vert ^{1-N}$$ for $$y\in V$$. We denote $$f(x)= (E_{V,\mathbb{R}^{N}}\psi )(x)$$ and $$g(x)= (E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert )(x)$$. Lemma 3.3 and (14) give

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{p}(\Omega)} \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{p}(\mathbb{R}^{N})} \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }(A_{r}A_{q}A_{p'})^{N} \Vert f \Vert _{L^{r}(\mathbb {R}^{N})} \Vert g \Vert _{L^{q}(\mathbb{R}^{N})} \\ &=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }(A_{r}A_{q}A_{p'})^{N} I^{\frac{1}{r}} \Vert \nabla u \Vert _{L^{q}(\Omega)}. \end{aligned}

□

### Theorem 3.4

Let $$\Omega\subset\mathbb{R}^{N}$$ ($$N\in\mathbb{N}$$) be a bounded convex domain, and let $$q>N$$. Then we have

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{\infty}(\Omega)}\leq D_{\infty}(\Omega) \Vert \nabla u \Vert _{L^{q}(\Omega )}\quad \textit{for all } u\in W^{1,q}(\Omega) \end{aligned}
(15)

with

\begin{aligned} D_{\infty}(\Omega)=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \bigl\Vert \vert x \vert ^{1-N} \bigr\Vert _{L^{q'}(V)}, \end{aligned}

where V is defined in Theorem  3.3.

### Proof

First, we show $$I:= \Vert \vert x \vert ^{1-N} \Vert _{L^{q'}(V)}^{q'}<\infty$$. Let $$\rho=2d_{\Omega}$$ so that $$V\subset B(0,\rho)$$. We have

\begin{aligned} q'(1-N)+N-1=\frac{q(1-N)+N(q-1)}{q-1}-1=\frac{q-N}{q-1}-1>-1. \end{aligned}

Therefore,

\begin{aligned} I= \int_{V} \vert x \vert ^{q'(1-N)}\,dx\leq \int_{B(0,\rho )} \vert x \vert ^{q'(1-N)}\,dx =J \int_{0}^{\rho}\rho^{q'(1-N)+N-1}\,d\rho < \infty, \end{aligned}

where J is defined in the proof of Theorem 3.3.

Next, we prove (15). Let $$r=\frac{q}{q-1}(\geq1)$$, $$f(x)= (E_{V,\mathbb{R}^{N}}\psi )(x)$$, and $$g(x)= (E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert )(x)$$, where ψ is denoted in the proof of Theorem 3.3. From Lemma 3.3 and (14), for $$u\in W^{1,q}(\Omega)$$, it follows that

\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{\infty}(\Omega)}&\leq\frac {d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{\infty}(\Omega)} \leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{\infty}(\mathbb{R}^{N})} \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f \Vert _{L^{q'}(\mathbb{R}^{N})} \Vert g \Vert _{L^{q}(\mathbb{R}^{N})} =\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }I^{\frac {1}{q'}} \Vert \nabla u \Vert _{L^{q}(\Omega)}. \end{aligned}

□

## 4 Explicit values of $$C_{p}(\Omega)$$ for certain domains

In this section, we present numerical examples where explicit values of $$C_{p}(\Omega)$$ on a square and a triangle domain are computed using Theorems 2.1, 3.1, 3.2, 3.3, and 3.4. All computations were performed on a computer with Intel Xeon E5-2687W @ 3.10 GHz, 512 GB RAM, CentOS 7, and MATLAB 2017a. All rounding errors were strictly estimated using the interval toolbox INTLAB version 10.1 [34]. Therefore, all values in the following tables are mathematically guaranteed to be upper bounds of the corresponding $$C_{p}(\Omega)$$’s.

First, we select domains $$\Omega_{i}$$ ($$1\leq i\leq n$$) satisfying (2) and (3). For all domains $$\Omega_{i}$$ ($$1\leq i\leq n$$), we then compute the values of $$D_{p}(\Omega_{i})$$ using Theorems 3.1, 3.2, 3.3, and 3.4. Next, explicit values of $$C_{p}(\Omega)$$ are computed through Theorem 2.1.

### 4.1 Estimation on a square domain

For the first example, we select the case in which $$\Omega=(0,1)^{2}$$. For $$n=1,4,16, 64, \ldots$$ , we define each $$\Omega_{i}$$ ($$1\leq i\leq n$$) as a square with side length $$1/\sqrt{n}$$; see Figure 1 for the cases in which $$n=4$$ and $$n=16$$. For this division of Ω, Theorem 2.1 states that

\begin{aligned} C_{p}(\Omega)=2^{1-\frac{1}{q}}\max \Bigl(n^{- (\frac {1}{p}-\frac{1}{q} )}, \max _{1\leq i\leq n}D_{p}(\Omega_{i}) \Bigr). \end{aligned}

In this case, V (in Theorems 3.3 and 3.4) becomes a square with side length $$2/\sqrt{n}$$ (see Figure 2). Note that $$\Vert \vert x \vert ^{1-N} \Vert _{L^{r}(V)}=\int_{V} \vert x \vert ^{\beta}\,dx$$, where $$\beta =qp(1-N)/((q-1)p+q)$$ if $$p<\infty$$ and $$\beta=q'(1-N)$$ if $$p=\infty$$.

Table 2 compares upper bounds for $$C_{p}(\Omega)$$ computed by Theorems 3.1, 3.2, 3.3, [10, Lemma 2.3], and [19, Corollary D.1] with $$q=2$$; the numbers of division n are shown in the corresponding parentheses. Moreover, these values are plotted in Figure 3, except for the values derived from [19, Corollary D.1].

Theorems 3.1, 3.2, 3.3, and [10, Lemma 2.3] provide sharper estimates of $$C_{p}(\Omega)$$ than [19, Corollary D.1] for all p’s. The estimates derived by Theorem 3.2 and Theorem 3.3 for $$32\leq p\leq80$$ are sharper than the estimates obtained by [10, Lemma 2.3].

We also show the values of $$C_{\infty}(\Omega)$$ computed by Theorem 3.4 for $$3\leq q\leq10$$ in Table 3.

### 4.2 Estimation on a triangle domain

For the second example, we select the case in which Ω is a regular triangle with the vertices $$(0,0)$$, $$(1,0)$$, and $$(1/2,\sqrt {3}/2)$$. For $$n=1,4,16,64,\ldots$$ , we define each $$\Omega_{i}$$ ($$1\leq i\leq n$$) as a regular triangle with side length $$1/\sqrt{n}$$; see Figure 4 for the case in which $$n=4$$ and $$n=16$$. For this division of Ω, Theorem 2.1 states that

$$C_{p}(\Omega)=2^{1-\frac{1}{q}}\max \biggl( \biggl(\frac{4n}{\sqrt {3}} \biggr)^{- (\frac{1}{p}-\frac{1}{q} )}, \max_{1\leq i\leq n} D_{p}( \Omega_{i}) \biggr).$$

In this case, V is the regular hexagon displayed in Figure 5.

Table 4 compares upper bounds of $$C_{p}(\Omega)$$ computed by Theorems 3.1, 3.2, 3.3, and [19, Corollary D.1] with $$q=2$$; the numbers of division n are shown in the corresponding parentheses. Moreover, these values are plotted in Figure 6. The estimate computed by Theorem 3.1 is sharpest when $$p=4$$. However, for the other p satisfying $$3\leq p\leq80$$, Theorem 3.3 provides the sharpest estimates.

We also show the values of $$C_{\infty}(\Omega)$$ computed by Theorem 3.4 for $$3\leq q\leq10$$ in Table 5.

### Remark 4.1

The values of $$C_{p}(\Omega)$$ derived from Theorem 3.1 to 3.4 (provided in Tables 1 to 5) can be directly used for any domain that is composed of unit squares and triangles with side length 1 (see Figure 7 for some examples).

### 4.3 Estimation on a cube domain

For the third example, we select the case in which $$\Omega=(0,1)^{3}$$. For $$n=1,8,64,512,\ldots$$ , we define each $$\Omega_{i}$$ ($$1\leq i\leq n$$) as a cube with side length $$1/\sqrt[3]{n}$$. For this division of Ω, Theorem 2.1 states that

$$C_{p}(\Omega)=2^{1-\frac{1}{q}}\max \Bigl(n^{- (\frac {1}{p}-\frac{1}{q} )}, \max _{1\leq i\leq n} D_{p}(\Omega _{i}) \Bigr).$$

In this case, V is also a cube with the side length $$2/\sqrt[3]{n}$$.

Table 6 compares upper bounds of $$C_{p}(\Omega)$$ computed by Theorems 3.1, 3.2, 3.3, and [19, Corollary D.1] with $$q=2$$; the numbers of division n are shown in the corresponding parentheses. The minimum value for each p is written in bold. We also show the values of $$C_{\infty}(\Omega)$$ computed by Theorem 3.4 for $$4\leq q\leq10$$ in Table 7.

## 5 Conclusion

We proposed several theorems that provide explicit values of Sobolev type embedding constant $$C_{p}(\Omega)$$ satisfying (1) for a domain Ω that can be divided into a finite number of bounded convex domains. These theorems give sharper estimates of $$C_{p}(\Omega)$$ than the previous estimates derived by the method in [19]. This accuracy improvement leads to much applicability of the estimates of $$C_{p}(\Omega)$$ to verified numerical computations for PDEs.