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Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 299.
Published online 2017 November 29. doi:  10.1186/s13660-017-1571-0
PMCID: PMC5707241

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 W1,q(Ω) to Lp(Ω) for a domain Ω ⊂ ℝN (N ∈ ℕ), where 1 ≤ q ≤ p ≤ ∞. 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.

Keywords: Sobolev embedding constant, Hardy-Littlewood-Sobolev inequality, Young inequality

Introduction

We consider the Sobolev type embedding constant Cp(Ω) from W1,q(Ω) (1 ≤ q ≤ p ≤ ∞) to Lp(Ω). The constant Cp(Ω) satisfies

(Ω|u(x)|pdx)1pCp(Ω)(Ω|u(x)|qdx+Ω|u(x)|qdx)1q
1

for all u ∈ W1,q(Ω), where Ω ⊂ ℝN (N ∈ ℕ) is a bounded domain and |x|=j=1Nxj2 for x = (x1, …, xN) ∈ ℝN. Here, Lp(Ω) (1 ≤ p < ∞) is the functional space of the pth power Lebesgue integrable functions over Ω endowed with the norm fLp(Ω): = (∫Ω|f(x)|p dx)1/p for f ∈ Lp(Ω), and L(Ω) is the functional space of Lebesgue measurable functions over Ω endowed with the norm fL(Ω) = ess supx∈Ω|f(x)| for f ∈ L(Ω). Moreover, Wk,p(Ω) is the kth order Lp-Sobolev space on Ω endowed with the norm fW1,p(Ω) = (∫Ω|f(x)|p dx+∫Ω|∇f(x)|p dx)1/p for f ∈ W1,p(Ω) if 1 ≤ p < ∞ and fW1,∞(Ω) = ess supx∈Ω|f(x)| + ess supx∈Ω|∇f(x)| for f ∈ W1,∞(Ω) if p = ∞.

Since inequality (1) has significance for studies on partial differential equations, many researchers studied this type of Sobolev inequality and an explicit value of Cp(Ω) (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 Cp(Ω) for various domains Ω (e.g., domains with the cone condition, domains with the Lipschitz boundary, and the (εδ)-domains) has been proven by constructing suitable extension operators from Wk,p(Ω) to Wk,p(ℝN) (see, e.g., [37]).

Several formulas for computing explicit values of Cp(Ω) have been proposed under suitable conditions. For example, the best constant in the classical Sobolev inequality on N was independently shown by Aubin [11] and Talenti [12]. For the case in which N = 1 and p = ∞, the best constant of Cp(Ω) was proposed under some boundary conditions, e.g., the Dirichlet, the Neumann, and the periodic condition [1317]. For a square domain Ω ⊂ ℝ2, a tight estimate of Cp(Ω) was provided in [10]. Moreover, the best constant for the embedding W01,2(Ω)Lp(Ω) (p = 3, 4, 5, 6, 7) with a square domain Ω ⊂ ℝ2 was very sharply estimated in [18], where W01,2(Ω) denotes the closure of C0(Ω) in W1,2(Ω). Furthermore, we have previously proposed a formula for computing an explicit value of Cp(Ω) for (bounded and unbounded) Lipschitz domains Ω ⊂ ℝN (N ≥ 2) 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 Ωi (i = 1, 2, 3, …, n) such that

Ω=1inΩi
2

and

Ωi ∩ Ωjϕ (i ≠ j), 
3

where ϕ is the empty set and [Omega with overline] 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 Cp(Ω) is significantly improved by restricting each Ωi 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 Cp(Ω), we focus on the constants Dp(Ω) such that

(Ω|u(x)uΩ(x)|pdx)1pDp(Ω)(Ω|u(x)|qdx)1qfor all uW1,q(Ω).
4

Here, |Ω| is the measure of Ω and uΩ:Ω → ℝ is a constant function defined by Ω ∋ x ↦ uΩ(x) = |Ω|−1Ωu(y) dy. Inequality (4) is called the Sobolev-Poincaré inequality, and Dp(Ω) in (4) leads to the explicit value of Cp(Ω) (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 Dp(Ω) was shown while assuming that 1 ≤ q < N, pNq/(N − q) [23]. It was also shown that, when p ≠ Nq/(N − q), Dp(Ω) exists if and only if W1,q(Ω) is continuously embedded into Lp(Ω) [24]. Moreover, there are several formulas for obtaining an explicit value of Dp(Ω) for one-dimensional domains Ω [2527]. In the higher-dimensional cases, however, little is known about explicit values of Dp(Ω), except for some special cases (see, e.g., [28] and [29] for the cases in which pq = 1 and pq = 2, respectively).

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

Table 1
The assumptions of p , q , and N imposed on Theorems 3.1 , 3.2 , 3.3 , and 3.4

Theorems 3.1 and 3.2 are derived from the best constant in the Hardy-Littlewood-Sobolev inequality on N. Theorems 3.3 and 3.4 are derived from the best constant in Young’s inequality on N. The values of Dp(Ω) calculated by these theorems yield the explicit values of Cp(Ω) 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 Cp(Ω) from known Dp(Ω) is provided. In Section 3, we prove the four formulas (Theorems 3.1 to 3.4) for obtaining the explicit values of Dp(Ω). In Section 4, we present examples where explicit values of Cp(Ω) are estimated for certain domains.

Estimation of embedding constant Cp(Ω)

The following notation is used throughout this paper. For any bounded domain S ⊂ ℝN (N ∈ ℕ), we define dS:=supx,yS|x − y|. The closed ball centered around z ∈ ℝN with radius ρ > 0 is denoted by B(zρ): = {x ∈ ℝN∣|x − z| ≤ ρ}. For m ≥ 1, let m be Hölder’s conjugate of m, that is, m is defined by

{m=,if m=1,m=mm1,if 1<m<,m=1,if m=.

For two domains Ω ⊆ ℝN and Ω ⊆ ℝN such that Ω ⊆ Ω, we define the operator EΩ,Ω:Lp(Ω) → Lp) (1 ≤ p ≤ ∞) by

(EΩ,Ωf)(x)={f(x),xΩ,0,xΩΩ

for f ∈ Lp(Ω). Note that EΩ,Ωf ∈ Lp) satisfies

EΩ,ΩfLp) = ∥fLp(Ω).

In the following theorem, we provide a formula for obtaining an explicit value of Cp(Ω) from known Dp(Ω).

Theorem 2.1

Let Ω ⊂ ℝN (N ∈ ℕ) be a bounded domain, and let p and q satisfy 1 ≤ q ≤ p ≤ ∞. Suppose that there exists a finite number of bounded domains Ωi (i = 1, 2, 3, …, n) satisfying (2) and (3). Moreover, suppose that, for every Ωi (i = 1, 2, 3, …, n), there exist constants Dpi) such that

uuΩiLpi) ≤ Dpi)∥∇uLqi) for all u ∈ W1,qi).
5

Then (1) holds valid for

Cp(Ω)={max(1,max1inD(Ωi))(p=q=),211qmax(max1in|Ωi|1p1q,max1inDp(Ωi))(otherwise),
6

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

Proof

Let u ∈ W1,q(Ω). Since every Ωi is bounded, Hölder’s inequality states that

uΩiLp(Ωi)=|Ωi|Ωi|1u(y)dy|1Lp(Ωi)|Ωi|1+1quLq(Ωi)|Ωi|1p=|Ωi|1p1quLq(Ωi).
7

We describe the following proof separately for the case of p = ∞ and p < ∞.

When p = ∞, we have

uL(Ω)=max1inuL(Ωi)max1in(uΩiL(Ωi)+uuΩiL(Ωi)).

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

uL(Ω)max1in(|Ωi|1quLq(Ωi)+D(Ωi)uLq(Ωi))max{max1in|Ωi|1q,max1inD(Ωi)}max1in(uLq(Ωi)+uLq(Ωi)).

This implies that Theorem 2.1 holds for the case of p = ∞ and q = ∞.

For q < ∞, we have

uL(Ω)max{max1in|Ωi|1q,max1inD(Ωi)}(1in(uLq(Ωi)+uLq(Ωi))q)1q211qmax{max1in|Ωi|1q,max1inD(Ωi)}uW1,q(Ω),

where the last inequality follows from (s+t)q ≤ 2q−1(sqtq) for st ≥ 0.

When p < ∞, we have

uLp(Ω)=(1inΩi|u(y)|pdy)1p=(1inuLp(Ωi)p)1p(1in(uΩiLp(Ωi)+uuΩiLp(Ωi))p)1p.

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

uLp(Ω)(1in(|Ωi|1p1quLq(Ωi)+Dp(Ωi)uLq(Ωi))p)1p(1in(|Ωi|1p1quLq(Ωi)+Dp(Ωi)uLq(Ωi))q)1q211q(1in(|Ωi|qp1uLq(Ωi)q+Dp(Ωi)quLq(Ωi)q))1q.

Therefore, we obtain

uLp(Ω)211qmax{max1in|Ωi|1p1q,max1inDi(Ωi)}uW1,q(Ω).

Estimation of Dpi)

Let Γ be the gamma function, that is, Γ(x)=0tx1etdt for x > 0. For f ∈ Lr(ℝN) and g ∈ Ls(ℝN) (1 ≤ rs ≤ ∞), let f ∗ g:ℝN → ℝ be the convolution of f and g defined by

(f ∗ g)(x): = ∫Nf(x − y)g(y) dy( = ∫Nf(x)g(x − y) dy).

In the following three lemmas, we recall some known results required to obtain explicit values of Dpi) in (5) for bounded convex domains Ωi.

Lemma 3.1

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

Let Ω ⊂ ℝN (N ∈ ℕ) be a bounded convex domain. For u ∈ W1,1(Ω) and any point x ∈ Ω, we have

|u(x)uΩ(x)|dΩNN|Ω|Ω|xy|1N|u(y)|dy.

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 Dpi).

Lemma 3.2

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

For λ > 0, we put hλ(x): = |x|λ. If 0 < λ < N,

hλgL2Nλ(RN)Cλ,NgL2N2Nλ(RN)for all gL2N2Nλ(RN)
8

holds valid for

Cλ,N=πλ2Γ(N2λ2)Γ(Nλ2)(Γ(N2)Γ(N))1+λN,
9

where this is the best constant in (8).

Moreover, if N < 2λ < 2N,

hλgL2N2λN(RN)C˜λ,NgL2(RN)for all gL2(RN)
10

holds valid for

C˜λ,N=πλ2Γ(N2λ2)Γ(λ2)Γ(λN2)Γ(3N2λ)(Γ(N2)Γ(N))1+λN,
11

where this is the best constant in (10).

Lemma 3.3

(Young’s inequality [33])

Suppose that 1 ≤ trs ≤ ∞ and 1/t = 1/r + 1/s − 1 ≥ 0. For f ∈ Lr(ℝN) and g ∈ Ls(ℝN), we have

fgLt(ℝN) ≤ (ArAsAt)NfLr(ℝN)gLs(ℝN)
12

with

Am={m2m1(m1)11m(1<m<),1(m=1,).

The constant (ArAsAt)N is the best constant in (12).

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

Theorem 3.1

Let Ω ⊂ ℝN (N ∈ ℕ) be a bounded convex domain. Assume that p ∈ ℝ satisfies 2 < p ≤ 2N/(N − 1) if N ≥ 2 and 2 < p < ∞ if N = 1. For q ∈ ℝ such that q ≥ p/(p − 1), we have

uuΩLp(Ω) ≤ Dp(Ω)∥∇uLq(Ω) for all u ∈ W1,q(Ω)

with

Dp(Ω)=dΩ1+2NpπNpN|Ω|1p+1qΓ(p22pN)Γ(p1pN)(Γ(N)Γ(N2))p2p.

Proof

Let u ∈ W1,q(Ω). Since p ≤ 2N/(N − 1) and 1 − N + (2N/p) ≥ 0, it follows that |xz|1N+2NpdΩ1N+2Np for xz ∈ Ω. Lemma 3.1 implies that, for a fixed x ∈ Ω,

|u(x)uΩ(x)|dΩNN|Ω|Ω|xz|1N+2Np|xz|2Np|u(z)|dzdΩ1+2NpN|Ω|Ω|xz|2Np|u(z)|dzdΩ1+2NpN|Ω|RN|xz|2Np(EΩ,RN|u|)(z)dz.

Therefore,

uuΩLp(Ω)dΩ1+2NpN|Ω|(Ω(RN|xz|2Np(EΩ,RN|u|)(z)dz)pdx)1pdΩ1+2NpN|Ω|(RN(RN|xz|2Np(EΩ,RN|u|)(z)dz)pdx)1p.

Since q ≥ p/(p − 1) and Ω is bounded, we have |∇u| ∈ Lp/(p−1)(Ω). Therefore, Lemma 3.2 ensures

uuΩLp(Ω)dΩ1+2NpN|Ω|C2Np,NEΩ,RN|u|Lpp1(RN)=dΩ1+2NpN|Ω|C2Np,NuLpp1(Ω),

where C2Np,N is defined in (9) with λ = 2N/p. Since q ≥ p/(p − 1), Hölder’s inequality moreover implies

uuΩLp(Ω)dΩ1+2NpN|Ω|1p+1qC2Np,NuLq(Ω).

Theorem 3.2

Let Ω ⊂ ℝN (N ≥ 2) be a bounded convex domain. Assume that 2 < p ≤ 2N/(N − 2) if N ≥ 3 and 2 < p < ∞ if N = 2. For all u ∈ W1,2(Ω), we have

uuΩLp(Ω) ≤ Dp(Ω)∥∇uL2(Ω)

with

Dp(Ω)=dΩ1+p+22pNπp+24pNN|Ω|Γ(p24pN)Γ(p+24pN)Γ(Np)Γ(p1pN)(Γ(N)Γ(N2))p22p.

Proof

Let u ∈ W1,2(Ω). Since p ≤ 2N/(N − 2), it follows that |xz|1N+(p+2)N/(2p)dΩ1N+(p+2)N/(2p) for xz ∈ Ω. Lemma 3.1 leads to

|u(x)uΩ(x)|dΩNN|Ω|Ω|xz|1N+p+22pN|xz|p+22pN|u(z)|dzdΩ1+p+22pNN|Ω|Ω|xz|p+22pN|u(z)|dzdΩ1+p+22pNN|Ω|RN|xz|p+22pN(EΩ,RN|u|)(z)dz.

Therefore,

uuΩLp(Ω)dΩ1+p+22pNN|Ω|(Ω(RN|xz|p+22pN(EΩ,RN|u|)(z)dz)pdx)1pdΩ1+p+22pNN|Ω|(RN(RN|xz|p+22pN(EΩ,RN|u|)(z)dz)pdx)1p.

From (10), it follows that

uuΩLp(Ω)dΩ1+p+22pNN|Ω|C˜p+22pN,NEΩ,RN|u|L2(RN)=dΩ1+p+22pNN|Ω|C˜p+22pN,NuL2(Ω),

where C˜p+22pN,N is defined in (11) with λ = (p + 2)N/(2p).

Theorem 3.3

Let Ω ⊂ ℝN (N ∈ ℕ) be a bounded convex domain. Suppose that 1 ≤ q ≤ p < qN/(N − q) if N > q, and 1 ≤ q ≤ p < ∞ if Nq. Then we have

uuΩLp(Ω) ≤ Dp(Ω)∥∇uLq(Ω) for all u ∈ W1,q(Ω)
13

with

Dp(Ω)=dΩNN|Ω|(ArAqAp)N|x|1NLr(V),

where Ωx: = {x − yy ∈ Ω} for x ∈ Ω, V: = ⋃x∈ΩΩx, and rqp/((q − 1)pq).

Proof

First, we prove I:=|x|1NLr(V)r<. Let ρ = 2dΩ so that V ⊂ B(0, ρ). We have

pq(1N)(q1)p+q+N1=pq(1N)+Np(q1)+Nq(q1)p+q1=Nq(Nq)p(q1)p+q1>1.

Therefore,

I=V|x|pq(1N)(q1)p+qdxB(0,ρ)|x|pq(1N)(q1)p+qdx=J0ρρpq(1N)(q1)p+q+N1dρ<,

where J is defined by

J={2(N=1),2π(N=2),2π[0,π]N2i=1N2(sinθi)Ni1dθ1dθN2(N3).

Next, we show (13). For x ∈ Ω, it follows from Lemma 3.1 that

|u(x)uΩ(x)|dΩNN|Ω|Ω|xy|1N|u(y)|dy=dΩNN|Ω|Ωx|y|1N|u(xy)|dydΩNN|Ω|V|y|1N(EΩ,V|u|)(xy)dy.

Since EV,ℝNEΩ,VEΩ,ℝN,

|u(x)uΩ(x)|dΩNN|Ω|RN(EV,RNψ)(y)(EΩ,RN|u|)(xy)dy,
14

where ψ(y) = |y|1−N for y ∈ V. We denote f(x) = (EV,ℝNψ)(x) and g(x) = (EΩ,ℝN|∇u|)(x). Lemma 3.3 and (14) give

uuΩLp(Ω)dΩNN|Ω|fgLp(Ω)dΩNN|Ω|fgLp(RN)dΩNN|Ω|(ArAqAp)NfLr(RN)gLq(RN)=dΩNN|Ω|(ArAqAp)NI1ruLq(Ω).

Theorem 3.4

Let Ω ⊂ ℝN (N ∈ ℕ) be a bounded convex domain, and let q > N. Then we have

uuΩL(Ω) ≤ D(Ω)∥∇uLq(Ω) for all u ∈ W1,q(Ω)
15

with

D(Ω)=dΩNN|Ω||x|1NLq(V),

where V is defined in Theorem  3.3.

Proof

First, we show I:=|x|1NLq(V)q<. Let ρ = 2dΩ so that V ⊂ B(0, ρ). We have

q(1N)+N1=q(1N)+N(q1)q11=qNq11>1.

Therefore,

I=V|x|q(1N)dxB(0,ρ)|x|q(1N)dx=J0ρρq(1N)+N1dρ<,

where J is defined in the proof of Theorem 3.3.

Next, we prove (15). Let r=qq1(1), f(x) = (EV,ℝNψ)(x), and g(x) = (EΩ,ℝN|∇u|)(x), where ψ is denoted in the proof of Theorem 3.3. From Lemma 3.3 and (14), for u ∈ W1,q(Ω), it follows that

uuΩL(Ω)dΩNN|Ω|fgL(Ω)dΩNN|Ω|fgL(RN)dΩNN|Ω|fLq(RN)gLq(RN)=dΩNN|Ω|I1quLq(Ω).

Explicit values of Cp(Ω) for certain domains

In this section, we present numerical examples where explicit values of Cp(Ω) 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 Cp(Ω)’s.

First, we select domains Ωi (1 ≤ i ≤ n) satisfying (2) and (3). For all domains Ωi (1 ≤ i ≤ n), we then compute the values of Dpi) using Theorems 3.1, 3.2, 3.3, and 3.4. Next, explicit values of Cp(Ω) are computed through Theorem 2.1.

Estimation on a square domain

For the first example, we select the case in which Ω = (0,1)2. For n = 1, 4, 16, 64, … , we define each Ωi (1 ≤ i ≤ n) as a square with side length 1/n; see Figure 1 for the cases in which n = 4 and n = 16. For this division of Ω, Theorem 2.1 states that

Cp(Ω)=211qmax(n(1p1q),max1inDp(Ωi)).

In this case, V (in Theorems 3.3 and 3.4) becomes a square with side length 2/n (see Figure 2). Note that ∥|x|1−NLr(V) = ∫V|x|β dx, where βqp(1 − N)/((q − 1)pq) if p < ∞ and βq(1 − N) if p = ∞.

Figure 1
Ωi for the cases in which n = 4 (the left-hand side) and n = 16 (the right-hand side).
Figure 2
The domain V in Theorems 3.3 and 3.4 .

Table 2 compares upper bounds for Cp(Ω) 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].

Figure 3
Computed values of Cp(Ω) for Ω = (0,1)2 and 3 ≤ p ≤ 80 .
Table 2
Computed values of Cp(Ω) for Ω = (0,1)2 and q = 2 . The numbers of division n are shown in the corresponding parentheses. Theorem 3.1 cannot be used for p > 4 when N = 2

Theorems 3.1, 3.2, 3.3, and [10, Lemma 2.3] provide sharper estimates of Cp(Ω) than [19, Corollary D.1] for all p’s. The estimates derived by Theorem 3.2 and Theorem 3.3 for 32 ≤ p ≤ 80 are sharper than the estimates obtained by [10, Lemma 2.3].

We also show the values of C(Ω) computed by Theorem 3.4 for 3 ≤ q ≤ 10 in Table 3.

Table 3
Computed values of C(Ω) for a square domain Ω and 3 ≤ q ≤ 10 . The numbers of division n are shown in the corresponding parentheses

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 (1/2,3/2). For n = 1, 4, 16, 64, … , we define each Ωi (1 ≤ i ≤ n) as a regular triangle with side length 1/n; see Figure 4 for the case in which n = 4 and n = 16. For this division of Ω, Theorem 2.1 states that

Cp(Ω)=211qmax((4n3)(1p1q),max1inDp(Ωi)).

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

Figure 4
Ωi when n = 4 (the left-hand side) and n = 16 (the right-hand side).
Figure 5
The domain V in Theorems 3.3 and 3.4 .

Table 4 compares upper bounds of Cp(Ω) 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 ≤ p ≤ 80, Theorem 3.3 provides the sharpest estimates.

Figure 6
Computed values of Cp(Ω) for a regular triangle domain Ω and 3 ≤ p ≤ 80 .
Table 4
Computed values of Cp(Ω) for a regular triangle domain Ω and q = 2 . The numbers of division n are shown in the corresponding parentheses. Theorem 3.1 cannot be used for p > 4 when N = 2

We also show the values of C(Ω) computed by Theorem 3.4 for 3 ≤ q ≤ 10 in Table 5.

Table 5
Computed values of C(Ω) for a regular triangle domain Ω and 3 ≤ q ≤ 10 . The numbers of division n are shown in the corresponding parentheses

Remark 4.1

The values of Cp(Ω) derived from Theorem 3.1 to 3.4 (provided in Tables Tables11 to to5)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).

Figure 7
Some examples of domains Ω that are composed of unit squares and triangles with side length 1.

Estimation on a cube domain

For the third example, we select the case in which Ω = (0,1)3. For n = 1, 8, 64, 512, … , we define each Ωi (1 ≤ i ≤ n) as a cube with side length 1/n3. For this division of Ω, Theorem 2.1 states that

Cp(Ω)=211qmax(n(1p1q),max1inDp(Ωi)).

In this case, V is also a cube with the side length 2/n3.

Table 6 compares upper bounds of Cp(Ω) 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(Ω) computed by Theorem 3.4 for 4 ≤ q ≤ 10 in Table 7.

Table 6
Computed values of Cp(Ω) for a cube domain Ω and q = 2 . The numbers of division n are shown in the corresponding parentheses. Theorem 3.1 for p > 3 cannot be used when N = 3 . Theorem 3.2 can be used for p = 6 only when N = 3
Table 7
Computed values of C(Ω) for a cube domain Ω and 4 ≤ q ≤ 10 . The numbers of division n are shown in the corresponding parentheses

Conclusion

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

Acknowledgements

This work was supported by CREST, Japan Science and Technology Agency. The second author (KT) was supported by JSPS Grant-in-Aid for Research Activity Start-up Grant Number JP17H07188 and Mizuho Foundation for the Promotion of Sciences. The third author (KS) was supported by JSPS KAKENHI Grant Number 16K17651. We thank the editors and reviewers for giving useful comments to improve the contents of this manuscript.

Appendix 1: Embedding constant Cp(Ω) on dividable domains

Theorem 6.1 provides an estimation of the embedding constant Cp(Ω) for a domain Ω that can be divided into domains Ωi (such as convex domains and Lipschitz domains) satisfying (2) and (3).

Theorem 6.1

Let Ω ⊂ ℝN (N ∈ ℕ) be a domain that can be divided into a finite number of domains Ωi (i = 1, 2, 3, …, n) satisfying (2) and (3). Assume that, for every Ωi (i = 1, 2, 3, …, n), there exists a constant Cpi) such that uLpi) ≤ Cpi)∥uW1,qi) for all u ∈ W1,qi). Then (1) holds valid for

Cp(Ω)=Mp,qmax1inCp(Ωi),

where

Mp,q={1(pq),n1p1q(p<q).

Proof

We consider both the cases in which p < ∞ and p = ∞.

When p < ∞, it follows that

uLp(Ω)=(1inuLp(Ωi)p)1/p(1inCp(Ωi)puW1,q(Ωi)p)1/pmax1inCp(Ωi)(1inuW1,q(Ωi)p)1/pMp,qmax1inCp(Ωi)uW1,q(Ω).

Note that |x|p ≤ Mp,q|x|q holds for x = (x1x2, …, xn) ∈ ℝn (see [19, Lemma A.1] for a detailed proof), where we denote

|x|p={(1in|xi|p)1p(1p<),max1in|xi|(p=).

When p = ∞,

uL(Ω)=max1inuL(Ωi)max1inCp(Ωi)uW1,q(Ωi)max1inCp(Ωi)max1inuW1,q(Ωi).

Since M∞,q = 1, we have

uL(Ω)max1inCp(Ωi)uW1,q(Ω).

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(Ω) ∩ W1,1(Ω) is densely defined in W1,1(Ω), it suffices to prove Lemma 3.1 for u ∈ C1(Ω). Since Ω is convex, we have, for xy ∈ Ω,

u(x)u(y)=0|xy|ru(x+rω)dr,

where ω = (y − x)/|y − x| and ru(x+rω)=ru(x+rω). Integrating with respect to y over Ω, we obtain

|u(x)uΩ(x)|=|Ω|1|Ω0|xy|ru(x+rω)drdy||Ω|1Ω0|xy||ru(x+rω)|drdy|Ω|1Ω0|(EΩ,RNru)(x+rω)|drdy|Ω|1B(x,dΩ)0|(EΩ,RNru)(x+rω)|drdy=|Ω|10dΩ|ω|=10|(EΩ,RNru)(x+rω)|ρN1drdωdρ=|Ω|10|ω|=10dΩ|(EΩ,RNru)(x+rω)|ρN1dρdωdr=dΩNN|Ω|0|ω|=1|(EΩ,RNru)(x+rω)|dωdr=dΩNN|Ω||ω|=10|(EΩ,RNru)(x+rω)|r1NrN1drdω=dΩNN|Ω|RN|(EΩ,RNru)(y)||xy|1Ndy=dΩNN|Ω|Ω|ru(y)||xy|1Ndy.

Therefore, a proof of Lemma 3.1 is completed.

Authors’ contributions

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Notes

Competing interests

The authors declare that they have no competing interests.

Footnotes

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Contributor Information

Makoto Mizuguchi, pj.adesaw.inoa@htam.otokam.

Kazuaki Tanaka, pj.adesaw.inoa@akanat.k.

Kouta Sekine, gro.daini@321enikes.

Shin’ichi Oishi, pj.adesaw@ihsio.

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