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Sharp numerical inclusion of the best constant for embedding H01(Ω)↪Lp(Ω) on bounded convex domain
Journal of Computational and Applied Mathematics 311 (2017) 306–313
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Sharp numerical inclusion of the best constant for embedding H01 (Ω ) ↩→ Lp (Ω ) on bounded convex domain Kazuaki Tanaka a,∗ , Kouta Sekine b , Makoto Mizuguchi a , Shin’ichi Oishi b a
Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
b
Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
article
abstract
info
Article history: Received 2 December 2015 Received in revised form 18 July 2016
In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding H01 (Ω ) ↩→ Lp (Ω ) on a bounded convex domain in R2 . We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented. © 2016 Elsevier B.V. All rights reserved.
Keywords: Computer-assisted proof Elliptic problem Embedding constant Error bounds Sobolev inequality Verified numerical computation
1. Introduction We consider the best constant for the embedding H01 (Ω ) ↩→ Lp (Ω ), i.e., the smallest constant Cp (Ω ) that satisfies
∥u∥Lp (Ω ) ≤ Cp (Ω ) ∥u∥H 1 (Ω ) , 0
∀u ∈ H01 (Ω ) ,
(1)
where Ω ⊂ Rn (n = 2, 3, . . .), 2 < p < ∞ if n = 2, and 2 < p ≤ 2n/(n − 2) if n ≥ 3. Here, Lp (Ω ) (1 ≤ p < ∞) is the functional space of pth power Lebesgue integrable functions over Ω . Moreover, assuming that H 1 (Ω ) denotes the first order L2 -Sobolev space on Ω , we define H01 (Ω ) := {u ∈ H 1 (Ω ) : u = 0 on ∂ Ω in the trace sense} with inner product (·, ·)H 1 (Ω ) := (∇·, ∇·)L2 (Ω ) and norm ∥·∥H 1 (Ω ) := ∥∇·∥L2 (Ω ) . 0
0
Such constants are important in studies on partial differential equations (PDEs). In particular, our interest is in the applicability of these constants to verified numerical computation methods for PDEs, which originate from Nakao’s [1] and Plum’s work [2] and have been further developed by many researchers. Such methods require explicit bounds for the embedding constant corresponding to a target equation at various points within them (see, e.g., [3–7]). Moreover, the precision in evaluating the embedding constants directly affects the precision of the verification results for the target equation. Occasionally, rough estimates of the embedding constants lead to failure in the verification. Therefore, accurately estimating such embedding constants is essential. It is well known that the best constant in the classical Sobolev inequality has been proposed [8,9] (see Theorem A.1). A rough upper bound of Cp (Ω ) for a bounded domain Ω ⊂ Rn can be obtained from the best constant by considering zero extension outside Ω (see Corollary A.2). Moreover, Plum [6] proposed another estimation formula that requires not the boundedness of Ω but an explicit lower bound for the minimum eigenvalue of −∆ (see Theorem A.3), where ∆ denotes
∗
Corresponding author. E-mail address: [email protected] (K. Tanaka).
http://dx.doi.org/10.1016/j.cam.2016.07.021 0377-0427/© 2016 Elsevier B.V. All rights reserved.
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
307
the usual Laplace operator. Although these formulas enable us to easily compute the upper bound of Cp (Ω ), little is known about the best constant. In this paper, we propose a numerical method for obtaining a verified sharp inclusion of the best constant Cp (Ω ) that satisfies (1) for a bounded convex domain Ω ⊂ R2 . As a verified result, we prove the following theorem by using our method through a computer-assisted technique: Theorem 1.1. For the square Ωs = (0, 1)2 , the smallest values of Cp (Ωs ) (p = 3, 4, 5, 6, 7) that satisfy (1) are enclosed as follows: C3 (Ωs ) ∈ [0.25712475017618, 0.25712475017620]; C4 (Ωs ) ∈ [0.28524446071925, 0.28524446071929]; C5 (Ωs ) ∈ [0.31058015094505, 0.31058015094512]; C6 (Ωs ) ∈ [0.33384042151102, 0.33384042151112]; C7 (Ωs ) ∈ [0.35547994288611, 0.35547994288634]. Remark 1.2. Since it follows from a simple variable transformation that 2
Cp ((a, b)2 ) = (b − a) p Cp (Ωs ),
(2)
the values in Theorem 1.1 can be directly used for all squares (a, b)2 (−∞ < a < b < ∞) by multiplying them with (b − a)2/p . Moreover, these values can be applied to deriving an explicit upper bound of Cp (Ω ) for a general domain Ω ⊂ (a, b)2 by considering zero extension outside Ω , while the precision of the upper bound depends on the shape of Ω. Hereafter, we replace the notation Cp (Ω ) with Cp+1 (Ω ) (1 < p < ∞ if n = 2, and 1 < p ≤ (n + 2)/(n − 2) if n ≥ 3) for the sake of convenience. The smallest value of Cp+1 (Ω ) can be written as Cp+1 (Ω ) =
sup
Φ (u) ,
(3)
u∈H01 (Ω )\{0}
where Φ (u) = ∥u∥Lp+1 (Ω ) / ∥u∥H 1 (Ω ) . This variational problem is still the topic of current research (see, e.g., [10,11] and the 0
references therein). The boundedness of Cp+1 (Ω ) in (3) is ensured by considering zero extension outside Ω (see Corollary A.2). In addition, it is true that the supremum Cp+1 (Ω ) in (3) can be realized by an extremal function in H01 (Ω ). A proof of this fact is sketched as follows. Let {ui } ∈ H01 (Ω ) be a sequence such that ∥ui ∥H 1 (Ω ) = 1 and ∥ui ∥Lp+1 (Ω ) → Cp+1 (Ω ) as i → ∞. The 0
Rellich–Kondrachov compactness theorem (see, e.g., [12, Theorem 7.22]) ensures that there exists a subsequence {uij } that
converges to some u∗ in Lp+1 (Ω ). Moreover, there exists a subsequence {uik } ⊂ {uij } that converges to some u′ ∈ H01 (Ω ) in
the weak topology of H01 (Ω ) because H01 (Ω ) is a Hilbert space. Since {uik } converges to u∗ in Lp+1 (Ω ), it follows that u∗ = u′ . Hence, u∗ ∈ H01 (Ω )(⊂Lp+1 (Ω )) and ∥u∗ ∥Lp+1 (Ω ) = Cp+1 (Ω ). Since |u| ∈ H 1 (Ω ) for all u ∈ H 1 (Ω ) (see, e.g., [12, Lemma 7.6]) and Φ (u∗ ) = Φ (|u∗ |), we are looking for the extremal function u∗ such that u∗ ≥ 0 (in fact, the later discussion additionally proves that u∗ > 0 in Ω ). The Euler–Lagrange equation for the variational problem is
−1u = lup u=0
in Ω , on ∂ Ω
(4)
with some positive constant l (see, e.g., [10] for a detailed proof). Since Φ is scale-invariant (i.e., Φ (ku∗ ) = Φ (u∗ ) for any k > 0), it suffices to consider the case that l = 1 for finding an extremal function u∗ of Φ (recall that we consider the case that p > 1). Moreover, the strong maximum principle ensures that nontrivial solutions u to (4) such that u ≥ 0 in Ω are positive in Ω . Therefore, in order to find the extremal function u∗ , we consider the problem of finding weak solutions to the following problem:
−1u = up u>0 u=0
in Ω , in Ω , on ∂ Ω .
(5)
This problem has a unique solution if Ω ⊂ R2 is bounded and convex [13]. Therefore, we can obtain an inclusion of Cp+1 (Ω ) as ∥u∗ ∥Lp+1 (Ω ) / ∥u∗ ∥H 1 (Ω ) by enclosing the solution u∗ to (5) with verification. 0
Numerous numerical methods for verifying a solution to semilinear elliptic boundary value problems exist (e.g., [3–7] along with related works [14,15]). Such methods enable a concrete ball containing exact solutions to elliptic equations to be obtained; this is typically in the sense of the norms ∥·∥H 1 (Ω ) and ∥·∥L∞ (Ω ) , where L∞ (Ω ) is the functional space of Lebesgue 0
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measurable functions over Ω with the norm ∥u∥L∞ (Ω ) := ess sup{|u (x)| | x ∈ Ω } for u ∈ L∞ (Ω ). In order to verify a solution to (5), we first verify a solution to the following problem: in Ω , on ∂ Ω ,
−1u = |u|p−1 u u=0
(6)
by combining the methods described in [5,15]. Then, we prove the positiveness of the verified solution. In exceptional cases, when p is even, we directly consider problem (5). Indeed, the maximum principle ensures the positiveness of the solutions to the problem −1u = up in Ω , u = 0 on ∂ Ω a priori (note that −1u ≥ 0 when p is even), except for the trivial solution u ≡ 0 in Ω . The remainder of this paper is organized as follows: In Sections 2 and 3, we propose methods for proving the positiveness of the solution to (6) and estimating the embedding constant Cp+1 (Ω ), respectively. In Section 4, we present some numerical examples that yield Theorem 1.1. 2. Method for verifying positiveness In this section, we propose a sufficient condition for the positiveness of the solution to (6), which will be summarized in Theorem 2.2. This theorem allows us to check for positiveness numerically. Throughout this paper, for simplicity, we omit the expression ‘‘almost everywhere’’ for Lebesgue measurable functions. For example, we employ the notation u > 0 in place of u(x) > 0 a.e. x ∈ Ω . We introduce the following lemma, which is required to prove Theorem 2.2. Lemma 2.1. Let Ω be a bounded domain in Rn (n = 1, 2, 3, . . .). All nontrivial weak solutions 0 ̸≡ u ∈ H01 (Ω ) to (5) satisfy ess sup{u (x)p−1 | x ∈ Ω } ≥ λ1 , where λ1 > 0 is the first eigenvalue of the following problem:
(∇ u, ∇v)L2 (Ω ) = λ (u, v)L2 (Ω ) ,
∀v ∈ H01 (Ω ) .
(7)
Proof. Let φ1 ≥ 0 (φ1 ̸≡ 0) be the first eigenfunction corresponding to λ1 , which satisfies
Ω
up (x) φ1 (x) dx = λ1
Ω
u (x) φ1 (x) dx.
We have
Ω
up (x) φ1 (x) dx =
{u (x)}p−1 {u (x) φ1 (x)} dx Ω ≤ ess sup{u (x)p−1 | x ∈ Ω } u (x) φ1 (x) dx Ω 1 p−1 = λ− ess sup { u x | x ∈ Ω } up (x) φ1 (x) dx. ( ) 1 Ω
The positiveness of
Ω
u (x) φ1 (x) dx implies ess sup{u (x)p−1 | x ∈ Ω } ≥ λ1 . p
Using Lemma 2.1, we can prove the following theorem, which provides a sufficient condition for the positiveness of solutions to (6). Theorem 2.2. Let Ω be a bounded domain in Rn (n = 1, 2, 3, . . .). If a solution u ∈ C 2 (Ω ) ∩ C Ω to (6) is positive in a
nonempty subdomain Ω ′ ⊂ Ω and sup{u− (x)p−1 | x ∈ Ω } < λ1 (Ω ), then u > 0 in the original domain Ω ; that is, u is also a solution to (5). Here, λ1 (Ω ) > 0 is the first eigenvalue of the problem (7) and u− is defined by u− (x) :=
−u ( x ) , 0,
u (x) < 0, u (x) ≥ 0.
Proof. Assume that u is not always positive in Ω . The strong maximum principle ensures that u is also not always nonnegative in Ω . In other words, there exists a nonempty subdomain Ω ′′ ⊂ Ω \ Ω ′ such that u < 0 in Ω ′′ and u = 0 on ∂ Ω ′′ . Therefore, the restricted function v := −u|Ω ′′ can be regarded as a solution to
−1v = v p v>0 v = 0
in Ω ′′ , in Ω ′′ , on ∂ Ω ′′ .
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
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From Lemma 2.1, we have that sup u− (x)p−1 ≥ sup v (x)p−1 ≥ λ1 Ω ′′ ,
x∈Ω
x∈Ω ′′
where λ1 Ω ′′ is the first eigenvalue of (7), with the notational replacement Ω = Ω ′′ . Since the inclusion Ω ′′ ⊂ Ω ensures that all functions in H01 Ω ′′ can be regarded as functions in H01 (Ω ) by considering the zero extension outside Ω ′′ , the ′′ inequality λ1 Ω ≥ λ1 (Ω ) follows. This contradicts the assumption sup{u− (x)p−1 | x ∈ Ω } < λ1 (Ω ). Remark 2.3. (i) For each h ∈ L2 (Ω ), the problem
in Ω , on ∂ Ω
−1u = h u=0
has a unique solution u ∈ H 2 (Ω ), such as when Ω is a bounded convex domain with a piecewise C 2 boundary (see, e.g., [16, Section 3.3]). Therefore, the so-called bootstrap argument ensures that a weak solution u ∈ H01 (Ω ) to (6) on such a domain Ω is in C ∞ (Ω ) (⊂C 2 (Ω )). (ii) The first eigenvalue of the problem (7) can be numerically estimated by, e.g., the method in [17], which allows us to concretely evaluate the eigenvalues of (7) on polygonal domains. 3. Method for estimating the embedding constant In this section, we propose a method for estimating the embedding constant Cp+1 (Ω ) defined in (3). Hereafter, B (x, r ; ∥ · ∥) denotes the closed ball whose center is x and whose radius is r ≥ 0 in the sense of the norm ∥ · ∥. The following theorem provides an explicit estimation of the embedding constant from a verified solution to (5). Theorem 3.1. Let Ω ⊂ R2 be a bounded convex domain. If there exists a solution to (5) in a closed ball B(ˆu, r ; ∥ · ∥H 1 (Ω ) ) with uˆ ∈ H01 (Ω ) satisfying uˆ H 1 (Ω ) > 2r, then the embedding constant Cp+1 (Ω ) defined in (3) is estimated as
0
0
uˆ p+1 uˆ p+1 L L (Ω ) (Ω ) ≤ Cp+1 (Ω ) ≤ . uˆ 1 uˆ 1 − 2r H (Ω ) H (Ω ) 0
0
Proof. It is clear that uˆ Lp+1 (Ω ) / uˆ H 1 (Ω ) is a lower bound of Cp+1 (Ω ). A solution to (5) is unique when Ω ⊂ R2 is bounded
0
and convex [13]. Therefore, the ratio ∥u∥Lp+1 (Ω ) / ∥u∥H 1 (Ω ) is maximized by the solution u to (5). By writing the solution to 0
(5) as uˆ + r v with v ∈ H01 (Ω ) , ∥v∥H 1 (Ω ) ≤ 1, we have that 0
uˆ + r v p+1 uˆ p+1 + rCp+1 (Ω ) L L (Ω ) (Ω ) Cp+1 (Ω ) = ≤ . uˆ + r v 1 uˆ 1 − r H (Ω ) H (Ω ) 0
0
In other words, it follows that
uˆ
− 2r Cp+1 (Ω ) ≤ uˆ Lp+1 (Ω ) . Hence, when uˆ H 1 (Ω ) > 2r , uˆ Lp+1 (Ω ) /(uˆ H 1 (Ω ) − 2r ) becomes an upper bound of Cp+1 (Ω ). H01 (Ω )
0
0
Remark 3.2. Theorem 3.1 can be naturally applied to the case of n ≥ 3 under suitable conditions. For example, if Ω is a convex symmetric domain in Rn (n ≥ 3) and 1 < p < (n + 2)/(n − 2), (5) has only one solution [18,19]. 4. Numerical example In this section, we present some numerical examples where the best values of the embedding constants on the square domain Ωs = (0, 1)2 are estimated to yield Theorem 1.1. The upper bounds for the embedding constants on the L-shaped domain ΩL = (0, 2)2 \ [1, 2]2 through the application of Theorem 1.1 are also presented. All computations were carried out on a computer with Intel Xeon E7-4830 2.20 GHz × 4 processors, 2 TB RAM, CentOS 6.6, and MATLAB 2012b. All rounding errors were strictly estimated by using toolboxes for the verified numerical computations: the INTLAB version 9 [20] and KV library version 0.4.16 [21]. Therefore, the accuracy of all results was guaranteed mathematically.
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Table 1 Verification results for the cases p = 2, 3, 4, 5, and 6 on Ωs = (0, 1)2 . p
N
r1
r2
sup (u− (x))p−1
Cp+1 (Ωs )
2
100
1.289071378e−12
3.643032347e−12
–
0.25712475017620 18 0.285244460719295
3
150
6.636469152e−13
4.363745213e−12
1.904227228e−23
4
150
5.841283013e−13
2.002871834e−11
–
0.31058015094512 05
5
150
7.088374332e−13
1.724519836e−10
8.844489601e−40
0.33384042151112 02
6
200
1.310581865e−12
4.769644376e−08
–
0.35547994288634 11
Table 2 Estimates of Cp (Ωs ) derived by our method, Corollary A.2, and Theorem A.3. Cp (Ωs )
Our method
Corollary A.2
Theorem A.3
C3 (Ωs )
0.25712475017620 18
0.27991104681668
0.32964899322075
C4 (Ωs )
0.285244460719295
0.31830988618380
0.39894228040144
C5 (Ωs )
0.31058015094512 05
0.35780388458051
0.48909030972535
C6 (Ωs )
0.33384042151112 02
0.39585399866620
0.55266945714001
C7 (Ωs )
0.35547994288634 11
0.43211185419351
0.63763213907292
We consider the cases where p = 2, 3, 4, 5, and 6, which correspond to the critical point problems for embedding constants Cp+1 (Ω ). We computed approximate solutions uˆ to (6), which are displayed in Fig. 1, with Legendre polynomials, i.e., we constructed uˆ as uˆ =
N
ui,j φi φj ,
ui,j ∈ R,
(8)
i,j=1
where each φi is defined by
φn (x) =
1 n(n + 1)
x(1 − x)
dPn dx
(x),
n = 1, 2, 3, . . .
(9)
with the Legendre polynomials Pn defined by Pn =
(−1)n n!
d dx
n
xn (1 − x)n ,
n = 0, 1, 2, . . . .
(10)
We used [22, Theorem 2.3] to obtain a concrete interpolation error bound for the basis {φi }Ni=1 . Other important properties of the basis are also discussed in [23]. We then proved the existence of solutions u to (6) in an H01 -ball B(ˆu, r1 ; ∥ · ∥H 1 (Ωs ) ) 0
and an L∞ -ball B(ˆu, r2 ; ∥ · ∥L∞ (Ωs ) ), both centered around the approximations uˆ ; this was done by combining the methods described in [5] and [15]. On the basis of the ‘‘Existence and Enclosure Theorem’’ on p. 154 in [5], we obtained the H01 -ball. The required constants δ and K and function G in the theorem were computed as follows:
• δ was computed as δ = C2 (Ωs )∥1uˆ + uˆ p ∥L2 (Ωs ) ; • K was computed by the method described in [15]; • The method described in [24, Section 3.4] was used for the explicit selection of G. By using [5, 4.1], we also obtained the L∞ -ball. Note that the verified solution has the regularity to be in Theorem 2 C (Ωs )∩C Ωs regardless of the regularity of the approximations uˆ owing to the argument in Remark 2.3(i). Table 1 presents the verification results, which ensure the positiveness of the verified solutions to (6) centered around uˆ for cases p = 3 and 5 owing to the condition sup{(u− (x))p−1 | x ∈ Ωs } < λ1 (=2π ). Here, the upper bounds of sup{(u− (x))p−1 | x ∈ Ωs } were
p−1 calculated by min{ˆu (x) | x ∈ Ωs } + r2 with verification. The last column in the table presents intervals containing
Cp+1 (Ωs ), e.g., 1.23789 456 represents the interval [1.23456, 1.23789]. These intervals yield the results in Theorem 1.1. Table 2 compares the lower and upper bounds derived by our method, the upper bounds derived by Corollary A.2, and the upper bounds derived by Plum’s formula [6] (Theorem A.3). In addition, we applied the results of Theorem 1.1 to estimate the upper bounds of the embedding constants on ΩL . Since ΩL ⊂ (0, 2)2 , which is the smallest square that encloses ΩL , Cp (ΩL ) is bounded by 22/p Cp (Ωs ) owing to the discussion in Remark 1.2. Table 3 compares 22/p Cp (Ωs ) derived by our method, the upper bounds for Cp (ΩL ) derived by Corollary A.2, and the upper bounds for Cp (ΩL ) derived by Theorem A.3. Theorem A.3 requires an explicit value for the minimum eigenvalue of −∆. Therefore, we employed the result of λ1 ≥ 9.5585 presented in [25, Table 5.1].
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
311
Fig. 1. Approximate solutions to (6) in Ωs for p = 2, 3, 4, 5, and 6.
5. Conclusion In this paper, we proposed a numerical method for obtaining a sharp inclusion of the best constant Cp (Ω ) that satisfies (1) for a bounded convex domain Ω . We obtained inclusions of Cp (Ω ) on the basis of Theorem 3.1. Since Theorem 3.1 requires a concrete inclusion of the extremal function corresponding to target Cp (Ω ), we obtained an inclusion of the function by
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Table 3 Same as Table 2 but for ΩL = (0, 2)2 \ [1, 2]2 . Cp (ΩL )
Our method (22/p Cp (Ωs ))
Corollary A.2
Theorem A.3
C3 (ΩL ) C4 (ΩL ) C5 (ΩL ) C6 (ΩL ) C7 (ΩL )
0.51424950035240 0.45279735701391 0.43922666167046 0.44050507711968 0.44787666285793
0.40370158699565 0.41891936927236 0.44572736933656 0.47539569585243 0.50554097277928
0.41978967493887 0.47823908300428 0.56542767015609 0.62367087563741 0.70723155088841
computing a solution to problem (5) with verification. In order to verify the solutions to (5), we first verified solutions to (6) and then proved their positiveness by using Theorem 2.2, except for the case where the positiveness of the solutions was ensured a priori. The accuracy of all results, e.g., those in Theorem 1.1, was mathematically guaranteed by using toolboxes for verified numerical computations [20,21]. In future work, we would like to present numerical inclusions for the best value of the embedding constant for other cases, e.g., when p is a fraction or Ω is a more complicated polygonal domain. Acknowledgments The authors express their sincere thanks to Professor Y. Watanabe (Kyushu University, Fukuoka, Japan) for referring them to useful literature on previously known results for the basis constructed by Legendre polynomials. They also express their profound gratitude to an editor and two anonymous referees for their highly insightful comments and suggestions. This work was supported by CREST, Japan Science and Technology Agency. The first author (K.T.) is supported by the Waseda Research Institute for Science and Engineering, the Grant-in-Aid for Young Scientists (Early Bird Program). The second author (K.S.) is supported by JSPS KAKENHI Grant Number 16K17651. Appendix. Simple bounds for the embedding constant The following theorem provides the best constant in the classical Sobolev inequality. Theorem A.1 (T. Aubin [8] and G. Talenti [9]). Let u be any function in W 1,q (Rn ) (n = 2, 3, . . .). Moreover, let q be any real number such that 1 < q < n, and let p = nq/ (n − q). Then,
Rn
|u(x)|p dx
1p
≤ Tp Rn
1q |∇ u(x)|q2 dx
holds for
Tp = π
− 21
− 1q
n
q−1
1 n Γ 1 + 2n Γ (n) , Γ n Γ 1 + n − n q q
1− 1q
n−q
1/2
where |∇ u|2 = (∂ u/∂ x1 )2 + (∂ u/∂ x2 )2 + · · · + (∂ u/∂ xn )2
(A.1)
, and Γ denotes the Gamma function.
The following corollary obtained from Theorem A.1 provides a simple bound for the embedding constant from H01 (Ω ) to Lp (Ω ) for a bounded domain Ω . Corollary A.2. Let Ω ⊂ Rn (n = 2, 3, . . .) be a bounded domain. Let p be a real number such that p ∈ (n/(n − 1), 2n/(n − 2)] if n ≥ 3 and p ∈ (n/(n − 1), ∞) if n = 2. In addition, set q = np/(n + p). Then, (1) holds for Cp (Ω ) = |Ω |
2−q 2q
Tp ,
where |Ω | is the measure of Ω and Tp is the constant in (A.1). Proof. By zero extension outside Ω , we may regard u ∈ H01 (Ω ) as an element u ∈ H01 (Rn ). Therefore, from Theorem A.1,
∥u∥Lp (Ω ) ≤ Tp
Ω
1q
q 2
|∇ u (x)| dx
.
(A.2)
Hölder’s inequality gives
Ω
q 2
|∇ u (x)| dx ≤
q· 2q
Ω
= |Ω |
|∇ u (x)|2 dx
2−q 2
Ω
2q
|∇ u (x)|22 dx
1 Ω
2q
,
2 2−q
2−2 q dx
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
313
that is,
Rn
1q 2−q |∇ u (x)|q2 dx ≤ |Ω | 2q ∥∇ u∥L2 (Ω ) .
(A.3)
From (A.2) and (A.3), it follows that
∥u∥Lp (Ω ) ≤ |Ω |
2−q 2q
Tp ∥∇ u∥L2 (Ω ) .
Using the following theorem, an upper bound of the embedding constant can be obtained when the minimal point of the spectrum of −∆ on H01 (Ω ) is explicitly estimated, where H01 (Ω ) is endowed with the inner product
(·, ·)H 1 (Ω ) := (∇·, ∇·)L2 (Ω ) + σ (·, ·)L2 (Ω )
(A.4)
0
with a nonnegative number σ . Recall that we selected σ = 0 when computing the numerical values in the last columns of Tables 2 and 3. Theorem A.3 (M. Plum [6]). Let ρ ∈ [0, ∞) denote the minimal point of the spectrum of −∆ on H01 (Ω ) endowed with the inner product (A.4), where σ is selected so that σ > 0 when ρ = 0. (a) Let n = 2 and p ∈ [2, ∞). With the largest integer ν satisfying ν ≤ p/2, (1) holds for Cp ( Ω ) =
3 21 + 2ν− p p p 1
2
2
2
1 p 2p p − p ρ+ σ , − 1 ··· −ν+2
2
2
where 2 2 − 1 · · · 2 − ν + 2 = 1 if ν = 1. (b) Let n ≥ 3 and p ∈ [2, 2n/(n − 2)]. With s := n(p−1 − 2−1 + n−1 ) ∈ [0, 1], (1) holds for
p p
p
Cp ( Ω ) =
√
n−1
n (n − 2)
1−s
s sρ + σ
2s
.
References [1] M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems, Japan J. Appl. Math. 5 (2) (1988) 313–332. [2] M. Plum, Computer-assisted existence proofs for two-point boundary value problems, Computing 46 (1) (1991) 19–34. [3] M.T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim. 22 (3–4) (2001) 321–356. [4] M.T. Nakao, Y. Watanabe, Numerical verification methods for solutions of semilinear elliptic boundary value problems, Nonlinear Theory Appl. IEICE 2 (1) (2011) 2–31. [5] M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl. 324 (1) (2001) 147–187. [6] M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresber. Deutsch. Math. Verein. 110 (1) (2008) 19–54. [7] A. Takayasu, X. Liu, S. Oishi, Remarks on computable a priori error estimates for finite element solutions of elliptic problems, Nonlinear Theory Appl. IEICE 5 (1) (2014) 53–63. [8] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (4) (1976) 573–598. [9] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1) (1976) 353–372. [10] T. Carroll, J. Ratzkin, Interpolating between torsional rigidity and principal frequency, J. Math. Anal. Appl. 379 (2) (2011) 818–826. [11] S. Juhnke, J. Ratzkin, A numerical investigation of level sets of extremal sobolev functions, Involve 8 (5) (2015) 787–799. [12] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer Science & Business Media, 2001. [13] C.-S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2 , Manuscripta Math. 84 (1) (1994) 13–19. [14] M.T. Nakao, K. Hashimoto, Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing 75 (1) (2005) 1–14. [15] K. Tanaka, A. Takayasu, X. Liu, S. Oishi, Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation, Japan J. Indust. Appl. Math. 31 (3) (2014) 665–679. [16] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 69, SIAM, 2011. [17] X. Liu, S. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal. 51 (3) (2013) 1634–1654. [18] M. Grossi, et al., A uniqueness result for a semilinear elliptic equation in symmetric domains, Adv. Differential Equations 5 (1–3) (2000) 193–212. [19] F. Pacella, Uniqueness of positive solutions of semilinear elliptic equations and related eigenvalue problems, Milan J. Math. 73 (1) (2005) 221–236. [20] S. Rump, INTLAB - INTerval LABoratory, in: T. Csendes (Ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77–104. http://www.ti3.tuhh.de/rump/. [21] M. Kashiwagi, KV library, 2015. http://verifiedby.me/kv/. [22] S. Kimura, N. Yamamoto, On explicit bounds in the error for the H01 -projection into piecewise polynomial spaces, Bull. Inform. Cybernet. 31 (2) (1999) 109–115. [23] T. Kinoshita, Y. Watanabe, M.T. Nakao, Recurrence relations of orthogonal polynomials in H01 and H02 , Nonlinear Theory Appl. IEICE 6 (3) (2015) 404–409. [24] M. Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan J. Indust. Appl. Math. 26 (2–3) (2009) 419–442. [25] X. Liu, T. Okayama, S. Oishi, High-precision eigenvalue bound for the Laplacian with singularities, in: Computer Mathematics, Springer, 2014, pp. 311–323.
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Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Sharp numerical inclusion of the best constant for embedding H01 (Ω ) ↩→ Lp (Ω ) on bounded convex domain Kazuaki Tanaka a,∗ , Kouta Sekine b , Makoto Mizuguchi a , Shin’ichi Oishi b a
Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
b
Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
article
abstract
info
Article history: Received 2 December 2015 Received in revised form 18 July 2016
In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding H01 (Ω ) ↩→ Lp (Ω ) on a bounded convex domain in R2 . We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented. © 2016 Elsevier B.V. All rights reserved.
Keywords: Computer-assisted proof Elliptic problem Embedding constant Error bounds Sobolev inequality Verified numerical computation
1. Introduction We consider the best constant for the embedding H01 (Ω ) ↩→ Lp (Ω ), i.e., the smallest constant Cp (Ω ) that satisfies
∥u∥Lp (Ω ) ≤ Cp (Ω ) ∥u∥H 1 (Ω ) , 0
∀u ∈ H01 (Ω ) ,
(1)
where Ω ⊂ Rn (n = 2, 3, . . .), 2 < p < ∞ if n = 2, and 2 < p ≤ 2n/(n − 2) if n ≥ 3. Here, Lp (Ω ) (1 ≤ p < ∞) is the functional space of pth power Lebesgue integrable functions over Ω . Moreover, assuming that H 1 (Ω ) denotes the first order L2 -Sobolev space on Ω , we define H01 (Ω ) := {u ∈ H 1 (Ω ) : u = 0 on ∂ Ω in the trace sense} with inner product (·, ·)H 1 (Ω ) := (∇·, ∇·)L2 (Ω ) and norm ∥·∥H 1 (Ω ) := ∥∇·∥L2 (Ω ) . 0
0
Such constants are important in studies on partial differential equations (PDEs). In particular, our interest is in the applicability of these constants to verified numerical computation methods for PDEs, which originate from Nakao’s [1] and Plum’s work [2] and have been further developed by many researchers. Such methods require explicit bounds for the embedding constant corresponding to a target equation at various points within them (see, e.g., [3–7]). Moreover, the precision in evaluating the embedding constants directly affects the precision of the verification results for the target equation. Occasionally, rough estimates of the embedding constants lead to failure in the verification. Therefore, accurately estimating such embedding constants is essential. It is well known that the best constant in the classical Sobolev inequality has been proposed [8,9] (see Theorem A.1). A rough upper bound of Cp (Ω ) for a bounded domain Ω ⊂ Rn can be obtained from the best constant by considering zero extension outside Ω (see Corollary A.2). Moreover, Plum [6] proposed another estimation formula that requires not the boundedness of Ω but an explicit lower bound for the minimum eigenvalue of −∆ (see Theorem A.3), where ∆ denotes
∗
Corresponding author. E-mail address: [email protected] (K. Tanaka).
http://dx.doi.org/10.1016/j.cam.2016.07.021 0377-0427/© 2016 Elsevier B.V. All rights reserved.
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
307
the usual Laplace operator. Although these formulas enable us to easily compute the upper bound of Cp (Ω ), little is known about the best constant. In this paper, we propose a numerical method for obtaining a verified sharp inclusion of the best constant Cp (Ω ) that satisfies (1) for a bounded convex domain Ω ⊂ R2 . As a verified result, we prove the following theorem by using our method through a computer-assisted technique: Theorem 1.1. For the square Ωs = (0, 1)2 , the smallest values of Cp (Ωs ) (p = 3, 4, 5, 6, 7) that satisfy (1) are enclosed as follows: C3 (Ωs ) ∈ [0.25712475017618, 0.25712475017620]; C4 (Ωs ) ∈ [0.28524446071925, 0.28524446071929]; C5 (Ωs ) ∈ [0.31058015094505, 0.31058015094512]; C6 (Ωs ) ∈ [0.33384042151102, 0.33384042151112]; C7 (Ωs ) ∈ [0.35547994288611, 0.35547994288634]. Remark 1.2. Since it follows from a simple variable transformation that 2
Cp ((a, b)2 ) = (b − a) p Cp (Ωs ),
(2)
the values in Theorem 1.1 can be directly used for all squares (a, b)2 (−∞ < a < b < ∞) by multiplying them with (b − a)2/p . Moreover, these values can be applied to deriving an explicit upper bound of Cp (Ω ) for a general domain Ω ⊂ (a, b)2 by considering zero extension outside Ω , while the precision of the upper bound depends on the shape of Ω. Hereafter, we replace the notation Cp (Ω ) with Cp+1 (Ω ) (1 < p < ∞ if n = 2, and 1 < p ≤ (n + 2)/(n − 2) if n ≥ 3) for the sake of convenience. The smallest value of Cp+1 (Ω ) can be written as Cp+1 (Ω ) =
sup
Φ (u) ,
(3)
u∈H01 (Ω )\{0}
where Φ (u) = ∥u∥Lp+1 (Ω ) / ∥u∥H 1 (Ω ) . This variational problem is still the topic of current research (see, e.g., [10,11] and the 0
references therein). The boundedness of Cp+1 (Ω ) in (3) is ensured by considering zero extension outside Ω (see Corollary A.2). In addition, it is true that the supremum Cp+1 (Ω ) in (3) can be realized by an extremal function in H01 (Ω ). A proof of this fact is sketched as follows. Let {ui } ∈ H01 (Ω ) be a sequence such that ∥ui ∥H 1 (Ω ) = 1 and ∥ui ∥Lp+1 (Ω ) → Cp+1 (Ω ) as i → ∞. The 0
Rellich–Kondrachov compactness theorem (see, e.g., [12, Theorem 7.22]) ensures that there exists a subsequence {uij } that
converges to some u∗ in Lp+1 (Ω ). Moreover, there exists a subsequence {uik } ⊂ {uij } that converges to some u′ ∈ H01 (Ω ) in
the weak topology of H01 (Ω ) because H01 (Ω ) is a Hilbert space. Since {uik } converges to u∗ in Lp+1 (Ω ), it follows that u∗ = u′ . Hence, u∗ ∈ H01 (Ω )(⊂Lp+1 (Ω )) and ∥u∗ ∥Lp+1 (Ω ) = Cp+1 (Ω ). Since |u| ∈ H 1 (Ω ) for all u ∈ H 1 (Ω ) (see, e.g., [12, Lemma 7.6]) and Φ (u∗ ) = Φ (|u∗ |), we are looking for the extremal function u∗ such that u∗ ≥ 0 (in fact, the later discussion additionally proves that u∗ > 0 in Ω ). The Euler–Lagrange equation for the variational problem is
−1u = lup u=0
in Ω , on ∂ Ω
(4)
with some positive constant l (see, e.g., [10] for a detailed proof). Since Φ is scale-invariant (i.e., Φ (ku∗ ) = Φ (u∗ ) for any k > 0), it suffices to consider the case that l = 1 for finding an extremal function u∗ of Φ (recall that we consider the case that p > 1). Moreover, the strong maximum principle ensures that nontrivial solutions u to (4) such that u ≥ 0 in Ω are positive in Ω . Therefore, in order to find the extremal function u∗ , we consider the problem of finding weak solutions to the following problem:
−1u = up u>0 u=0
in Ω , in Ω , on ∂ Ω .
(5)
This problem has a unique solution if Ω ⊂ R2 is bounded and convex [13]. Therefore, we can obtain an inclusion of Cp+1 (Ω ) as ∥u∗ ∥Lp+1 (Ω ) / ∥u∗ ∥H 1 (Ω ) by enclosing the solution u∗ to (5) with verification. 0
Numerous numerical methods for verifying a solution to semilinear elliptic boundary value problems exist (e.g., [3–7] along with related works [14,15]). Such methods enable a concrete ball containing exact solutions to elliptic equations to be obtained; this is typically in the sense of the norms ∥·∥H 1 (Ω ) and ∥·∥L∞ (Ω ) , where L∞ (Ω ) is the functional space of Lebesgue 0
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measurable functions over Ω with the norm ∥u∥L∞ (Ω ) := ess sup{|u (x)| | x ∈ Ω } for u ∈ L∞ (Ω ). In order to verify a solution to (5), we first verify a solution to the following problem: in Ω , on ∂ Ω ,
−1u = |u|p−1 u u=0
(6)
by combining the methods described in [5,15]. Then, we prove the positiveness of the verified solution. In exceptional cases, when p is even, we directly consider problem (5). Indeed, the maximum principle ensures the positiveness of the solutions to the problem −1u = up in Ω , u = 0 on ∂ Ω a priori (note that −1u ≥ 0 when p is even), except for the trivial solution u ≡ 0 in Ω . The remainder of this paper is organized as follows: In Sections 2 and 3, we propose methods for proving the positiveness of the solution to (6) and estimating the embedding constant Cp+1 (Ω ), respectively. In Section 4, we present some numerical examples that yield Theorem 1.1. 2. Method for verifying positiveness In this section, we propose a sufficient condition for the positiveness of the solution to (6), which will be summarized in Theorem 2.2. This theorem allows us to check for positiveness numerically. Throughout this paper, for simplicity, we omit the expression ‘‘almost everywhere’’ for Lebesgue measurable functions. For example, we employ the notation u > 0 in place of u(x) > 0 a.e. x ∈ Ω . We introduce the following lemma, which is required to prove Theorem 2.2. Lemma 2.1. Let Ω be a bounded domain in Rn (n = 1, 2, 3, . . .). All nontrivial weak solutions 0 ̸≡ u ∈ H01 (Ω ) to (5) satisfy ess sup{u (x)p−1 | x ∈ Ω } ≥ λ1 , where λ1 > 0 is the first eigenvalue of the following problem:
(∇ u, ∇v)L2 (Ω ) = λ (u, v)L2 (Ω ) ,
∀v ∈ H01 (Ω ) .
(7)
Proof. Let φ1 ≥ 0 (φ1 ̸≡ 0) be the first eigenfunction corresponding to λ1 , which satisfies
Ω
up (x) φ1 (x) dx = λ1
Ω
u (x) φ1 (x) dx.
We have
Ω
up (x) φ1 (x) dx =
{u (x)}p−1 {u (x) φ1 (x)} dx Ω ≤ ess sup{u (x)p−1 | x ∈ Ω } u (x) φ1 (x) dx Ω 1 p−1 = λ− ess sup { u x | x ∈ Ω } up (x) φ1 (x) dx. ( ) 1 Ω
The positiveness of
Ω
u (x) φ1 (x) dx implies ess sup{u (x)p−1 | x ∈ Ω } ≥ λ1 . p
Using Lemma 2.1, we can prove the following theorem, which provides a sufficient condition for the positiveness of solutions to (6). Theorem 2.2. Let Ω be a bounded domain in Rn (n = 1, 2, 3, . . .). If a solution u ∈ C 2 (Ω ) ∩ C Ω to (6) is positive in a
nonempty subdomain Ω ′ ⊂ Ω and sup{u− (x)p−1 | x ∈ Ω } < λ1 (Ω ), then u > 0 in the original domain Ω ; that is, u is also a solution to (5). Here, λ1 (Ω ) > 0 is the first eigenvalue of the problem (7) and u− is defined by u− (x) :=
−u ( x ) , 0,
u (x) < 0, u (x) ≥ 0.
Proof. Assume that u is not always positive in Ω . The strong maximum principle ensures that u is also not always nonnegative in Ω . In other words, there exists a nonempty subdomain Ω ′′ ⊂ Ω \ Ω ′ such that u < 0 in Ω ′′ and u = 0 on ∂ Ω ′′ . Therefore, the restricted function v := −u|Ω ′′ can be regarded as a solution to
−1v = v p v>0 v = 0
in Ω ′′ , in Ω ′′ , on ∂ Ω ′′ .
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
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From Lemma 2.1, we have that sup u− (x)p−1 ≥ sup v (x)p−1 ≥ λ1 Ω ′′ ,
x∈Ω
x∈Ω ′′
where λ1 Ω ′′ is the first eigenvalue of (7), with the notational replacement Ω = Ω ′′ . Since the inclusion Ω ′′ ⊂ Ω ensures that all functions in H01 Ω ′′ can be regarded as functions in H01 (Ω ) by considering the zero extension outside Ω ′′ , the ′′ inequality λ1 Ω ≥ λ1 (Ω ) follows. This contradicts the assumption sup{u− (x)p−1 | x ∈ Ω } < λ1 (Ω ). Remark 2.3. (i) For each h ∈ L2 (Ω ), the problem
in Ω , on ∂ Ω
−1u = h u=0
has a unique solution u ∈ H 2 (Ω ), such as when Ω is a bounded convex domain with a piecewise C 2 boundary (see, e.g., [16, Section 3.3]). Therefore, the so-called bootstrap argument ensures that a weak solution u ∈ H01 (Ω ) to (6) on such a domain Ω is in C ∞ (Ω ) (⊂C 2 (Ω )). (ii) The first eigenvalue of the problem (7) can be numerically estimated by, e.g., the method in [17], which allows us to concretely evaluate the eigenvalues of (7) on polygonal domains. 3. Method for estimating the embedding constant In this section, we propose a method for estimating the embedding constant Cp+1 (Ω ) defined in (3). Hereafter, B (x, r ; ∥ · ∥) denotes the closed ball whose center is x and whose radius is r ≥ 0 in the sense of the norm ∥ · ∥. The following theorem provides an explicit estimation of the embedding constant from a verified solution to (5). Theorem 3.1. Let Ω ⊂ R2 be a bounded convex domain. If there exists a solution to (5) in a closed ball B(ˆu, r ; ∥ · ∥H 1 (Ω ) ) with uˆ ∈ H01 (Ω ) satisfying uˆ H 1 (Ω ) > 2r, then the embedding constant Cp+1 (Ω ) defined in (3) is estimated as
0
0
uˆ p+1 uˆ p+1 L L (Ω ) (Ω ) ≤ Cp+1 (Ω ) ≤ . uˆ 1 uˆ 1 − 2r H (Ω ) H (Ω ) 0
0
Proof. It is clear that uˆ Lp+1 (Ω ) / uˆ H 1 (Ω ) is a lower bound of Cp+1 (Ω ). A solution to (5) is unique when Ω ⊂ R2 is bounded
0
and convex [13]. Therefore, the ratio ∥u∥Lp+1 (Ω ) / ∥u∥H 1 (Ω ) is maximized by the solution u to (5). By writing the solution to 0
(5) as uˆ + r v with v ∈ H01 (Ω ) , ∥v∥H 1 (Ω ) ≤ 1, we have that 0
uˆ + r v p+1 uˆ p+1 + rCp+1 (Ω ) L L (Ω ) (Ω ) Cp+1 (Ω ) = ≤ . uˆ + r v 1 uˆ 1 − r H (Ω ) H (Ω ) 0
0
In other words, it follows that
uˆ
− 2r Cp+1 (Ω ) ≤ uˆ Lp+1 (Ω ) . Hence, when uˆ H 1 (Ω ) > 2r , uˆ Lp+1 (Ω ) /(uˆ H 1 (Ω ) − 2r ) becomes an upper bound of Cp+1 (Ω ). H01 (Ω )
0
0
Remark 3.2. Theorem 3.1 can be naturally applied to the case of n ≥ 3 under suitable conditions. For example, if Ω is a convex symmetric domain in Rn (n ≥ 3) and 1 < p < (n + 2)/(n − 2), (5) has only one solution [18,19]. 4. Numerical example In this section, we present some numerical examples where the best values of the embedding constants on the square domain Ωs = (0, 1)2 are estimated to yield Theorem 1.1. The upper bounds for the embedding constants on the L-shaped domain ΩL = (0, 2)2 \ [1, 2]2 through the application of Theorem 1.1 are also presented. All computations were carried out on a computer with Intel Xeon E7-4830 2.20 GHz × 4 processors, 2 TB RAM, CentOS 6.6, and MATLAB 2012b. All rounding errors were strictly estimated by using toolboxes for the verified numerical computations: the INTLAB version 9 [20] and KV library version 0.4.16 [21]. Therefore, the accuracy of all results was guaranteed mathematically.
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Table 1 Verification results for the cases p = 2, 3, 4, 5, and 6 on Ωs = (0, 1)2 . p
N
r1
r2
sup (u− (x))p−1
Cp+1 (Ωs )
2
100
1.289071378e−12
3.643032347e−12
–
0.25712475017620 18 0.285244460719295
3
150
6.636469152e−13
4.363745213e−12
1.904227228e−23
4
150
5.841283013e−13
2.002871834e−11
–
0.31058015094512 05
5
150
7.088374332e−13
1.724519836e−10
8.844489601e−40
0.33384042151112 02
6
200
1.310581865e−12
4.769644376e−08
–
0.35547994288634 11
Table 2 Estimates of Cp (Ωs ) derived by our method, Corollary A.2, and Theorem A.3. Cp (Ωs )
Our method
Corollary A.2
Theorem A.3
C3 (Ωs )
0.25712475017620 18
0.27991104681668
0.32964899322075
C4 (Ωs )
0.285244460719295
0.31830988618380
0.39894228040144
C5 (Ωs )
0.31058015094512 05
0.35780388458051
0.48909030972535
C6 (Ωs )
0.33384042151112 02
0.39585399866620
0.55266945714001
C7 (Ωs )
0.35547994288634 11
0.43211185419351
0.63763213907292
We consider the cases where p = 2, 3, 4, 5, and 6, which correspond to the critical point problems for embedding constants Cp+1 (Ω ). We computed approximate solutions uˆ to (6), which are displayed in Fig. 1, with Legendre polynomials, i.e., we constructed uˆ as uˆ =
N
ui,j φi φj ,
ui,j ∈ R,
(8)
i,j=1
where each φi is defined by
φn (x) =
1 n(n + 1)
x(1 − x)
dPn dx
(x),
n = 1, 2, 3, . . .
(9)
with the Legendre polynomials Pn defined by Pn =
(−1)n n!
d dx
n
xn (1 − x)n ,
n = 0, 1, 2, . . . .
(10)
We used [22, Theorem 2.3] to obtain a concrete interpolation error bound for the basis {φi }Ni=1 . Other important properties of the basis are also discussed in [23]. We then proved the existence of solutions u to (6) in an H01 -ball B(ˆu, r1 ; ∥ · ∥H 1 (Ωs ) ) 0
and an L∞ -ball B(ˆu, r2 ; ∥ · ∥L∞ (Ωs ) ), both centered around the approximations uˆ ; this was done by combining the methods described in [5] and [15]. On the basis of the ‘‘Existence and Enclosure Theorem’’ on p. 154 in [5], we obtained the H01 -ball. The required constants δ and K and function G in the theorem were computed as follows:
• δ was computed as δ = C2 (Ωs )∥1uˆ + uˆ p ∥L2 (Ωs ) ; • K was computed by the method described in [15]; • The method described in [24, Section 3.4] was used for the explicit selection of G. By using [5, 4.1], we also obtained the L∞ -ball. Note that the verified solution has the regularity to be in Theorem 2 C (Ωs )∩C Ωs regardless of the regularity of the approximations uˆ owing to the argument in Remark 2.3(i). Table 1 presents the verification results, which ensure the positiveness of the verified solutions to (6) centered around uˆ for cases p = 3 and 5 owing to the condition sup{(u− (x))p−1 | x ∈ Ωs } < λ1 (=2π ). Here, the upper bounds of sup{(u− (x))p−1 | x ∈ Ωs } were
p−1 calculated by min{ˆu (x) | x ∈ Ωs } + r2 with verification. The last column in the table presents intervals containing
Cp+1 (Ωs ), e.g., 1.23789 456 represents the interval [1.23456, 1.23789]. These intervals yield the results in Theorem 1.1. Table 2 compares the lower and upper bounds derived by our method, the upper bounds derived by Corollary A.2, and the upper bounds derived by Plum’s formula [6] (Theorem A.3). In addition, we applied the results of Theorem 1.1 to estimate the upper bounds of the embedding constants on ΩL . Since ΩL ⊂ (0, 2)2 , which is the smallest square that encloses ΩL , Cp (ΩL ) is bounded by 22/p Cp (Ωs ) owing to the discussion in Remark 1.2. Table 3 compares 22/p Cp (Ωs ) derived by our method, the upper bounds for Cp (ΩL ) derived by Corollary A.2, and the upper bounds for Cp (ΩL ) derived by Theorem A.3. Theorem A.3 requires an explicit value for the minimum eigenvalue of −∆. Therefore, we employed the result of λ1 ≥ 9.5585 presented in [25, Table 5.1].
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
311
Fig. 1. Approximate solutions to (6) in Ωs for p = 2, 3, 4, 5, and 6.
5. Conclusion In this paper, we proposed a numerical method for obtaining a sharp inclusion of the best constant Cp (Ω ) that satisfies (1) for a bounded convex domain Ω . We obtained inclusions of Cp (Ω ) on the basis of Theorem 3.1. Since Theorem 3.1 requires a concrete inclusion of the extremal function corresponding to target Cp (Ω ), we obtained an inclusion of the function by
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Table 3 Same as Table 2 but for ΩL = (0, 2)2 \ [1, 2]2 . Cp (ΩL )
Our method (22/p Cp (Ωs ))
Corollary A.2
Theorem A.3
C3 (ΩL ) C4 (ΩL ) C5 (ΩL ) C6 (ΩL ) C7 (ΩL )
0.51424950035240 0.45279735701391 0.43922666167046 0.44050507711968 0.44787666285793
0.40370158699565 0.41891936927236 0.44572736933656 0.47539569585243 0.50554097277928
0.41978967493887 0.47823908300428 0.56542767015609 0.62367087563741 0.70723155088841
computing a solution to problem (5) with verification. In order to verify the solutions to (5), we first verified solutions to (6) and then proved their positiveness by using Theorem 2.2, except for the case where the positiveness of the solutions was ensured a priori. The accuracy of all results, e.g., those in Theorem 1.1, was mathematically guaranteed by using toolboxes for verified numerical computations [20,21]. In future work, we would like to present numerical inclusions for the best value of the embedding constant for other cases, e.g., when p is a fraction or Ω is a more complicated polygonal domain. Acknowledgments The authors express their sincere thanks to Professor Y. Watanabe (Kyushu University, Fukuoka, Japan) for referring them to useful literature on previously known results for the basis constructed by Legendre polynomials. They also express their profound gratitude to an editor and two anonymous referees for their highly insightful comments and suggestions. This work was supported by CREST, Japan Science and Technology Agency. The first author (K.T.) is supported by the Waseda Research Institute for Science and Engineering, the Grant-in-Aid for Young Scientists (Early Bird Program). The second author (K.S.) is supported by JSPS KAKENHI Grant Number 16K17651. Appendix. Simple bounds for the embedding constant The following theorem provides the best constant in the classical Sobolev inequality. Theorem A.1 (T. Aubin [8] and G. Talenti [9]). Let u be any function in W 1,q (Rn ) (n = 2, 3, . . .). Moreover, let q be any real number such that 1 < q < n, and let p = nq/ (n − q). Then,
Rn
|u(x)|p dx
1p
≤ Tp Rn
1q |∇ u(x)|q2 dx
holds for
Tp = π
− 21
− 1q
n
q−1
1 n Γ 1 + 2n Γ (n) , Γ n Γ 1 + n − n q q
1− 1q
n−q
1/2
where |∇ u|2 = (∂ u/∂ x1 )2 + (∂ u/∂ x2 )2 + · · · + (∂ u/∂ xn )2
(A.1)
, and Γ denotes the Gamma function.
The following corollary obtained from Theorem A.1 provides a simple bound for the embedding constant from H01 (Ω ) to Lp (Ω ) for a bounded domain Ω . Corollary A.2. Let Ω ⊂ Rn (n = 2, 3, . . .) be a bounded domain. Let p be a real number such that p ∈ (n/(n − 1), 2n/(n − 2)] if n ≥ 3 and p ∈ (n/(n − 1), ∞) if n = 2. In addition, set q = np/(n + p). Then, (1) holds for Cp (Ω ) = |Ω |
2−q 2q
Tp ,
where |Ω | is the measure of Ω and Tp is the constant in (A.1). Proof. By zero extension outside Ω , we may regard u ∈ H01 (Ω ) as an element u ∈ H01 (Rn ). Therefore, from Theorem A.1,
∥u∥Lp (Ω ) ≤ Tp
Ω
1q
q 2
|∇ u (x)| dx
.
(A.2)
Hölder’s inequality gives
Ω
q 2
|∇ u (x)| dx ≤
q· 2q
Ω
= |Ω |
|∇ u (x)|2 dx
2−q 2
Ω
2q
|∇ u (x)|22 dx
1 Ω
2q
,
2 2−q
2−2 q dx
K. Tanaka et al. / Journal of Computational and Applied Mathematics 311 (2017) 306–313
313
that is,
Rn
1q 2−q |∇ u (x)|q2 dx ≤ |Ω | 2q ∥∇ u∥L2 (Ω ) .
(A.3)
From (A.2) and (A.3), it follows that
∥u∥Lp (Ω ) ≤ |Ω |
2−q 2q
Tp ∥∇ u∥L2 (Ω ) .
Using the following theorem, an upper bound of the embedding constant can be obtained when the minimal point of the spectrum of −∆ on H01 (Ω ) is explicitly estimated, where H01 (Ω ) is endowed with the inner product
(·, ·)H 1 (Ω ) := (∇·, ∇·)L2 (Ω ) + σ (·, ·)L2 (Ω )
(A.4)
0
with a nonnegative number σ . Recall that we selected σ = 0 when computing the numerical values in the last columns of Tables 2 and 3. Theorem A.3 (M. Plum [6]). Let ρ ∈ [0, ∞) denote the minimal point of the spectrum of −∆ on H01 (Ω ) endowed with the inner product (A.4), where σ is selected so that σ > 0 when ρ = 0. (a) Let n = 2 and p ∈ [2, ∞). With the largest integer ν satisfying ν ≤ p/2, (1) holds for Cp ( Ω ) =
3 21 + 2ν− p p p 1
2
2
2
1 p 2p p − p ρ+ σ , − 1 ··· −ν+2
2
2
where 2 2 − 1 · · · 2 − ν + 2 = 1 if ν = 1. (b) Let n ≥ 3 and p ∈ [2, 2n/(n − 2)]. With s := n(p−1 − 2−1 + n−1 ) ∈ [0, 1], (1) holds for
p p
p
Cp ( Ω ) =
√
n−1
n (n − 2)
1−s
s sρ + σ
2s
.
References [1] M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems, Japan J. Appl. Math. 5 (2) (1988) 313–332. [2] M. Plum, Computer-assisted existence proofs for two-point boundary value problems, Computing 46 (1) (1991) 19–34. [3] M.T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim. 22 (3–4) (2001) 321–356. [4] M.T. Nakao, Y. Watanabe, Numerical verification methods for solutions of semilinear elliptic boundary value problems, Nonlinear Theory Appl. IEICE 2 (1) (2011) 2–31. [5] M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl. 324 (1) (2001) 147–187. [6] M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresber. Deutsch. Math. Verein. 110 (1) (2008) 19–54. [7] A. Takayasu, X. Liu, S. Oishi, Remarks on computable a priori error estimates for finite element solutions of elliptic problems, Nonlinear Theory Appl. IEICE 5 (1) (2014) 53–63. [8] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (4) (1976) 573–598. [9] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1) (1976) 353–372. [10] T. Carroll, J. Ratzkin, Interpolating between torsional rigidity and principal frequency, J. Math. Anal. Appl. 379 (2) (2011) 818–826. [11] S. Juhnke, J. Ratzkin, A numerical investigation of level sets of extremal sobolev functions, Involve 8 (5) (2015) 787–799. [12] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer Science & Business Media, 2001. [13] C.-S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2 , Manuscripta Math. 84 (1) (1994) 13–19. [14] M.T. Nakao, K. Hashimoto, Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing 75 (1) (2005) 1–14. [15] K. Tanaka, A. Takayasu, X. Liu, S. Oishi, Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation, Japan J. Indust. Appl. Math. 31 (3) (2014) 665–679. [16] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 69, SIAM, 2011. [17] X. Liu, S. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal. 51 (3) (2013) 1634–1654. [18] M. Grossi, et al., A uniqueness result for a semilinear elliptic equation in symmetric domains, Adv. Differential Equations 5 (1–3) (2000) 193–212. [19] F. Pacella, Uniqueness of positive solutions of semilinear elliptic equations and related eigenvalue problems, Milan J. Math. 73 (1) (2005) 221–236. [20] S. Rump, INTLAB - INTerval LABoratory, in: T. Csendes (Ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77–104. http://www.ti3.tuhh.de/rump/. [21] M. Kashiwagi, KV library, 2015. http://verifiedby.me/kv/. [22] S. Kimura, N. Yamamoto, On explicit bounds in the error for the H01 -projection into piecewise polynomial spaces, Bull. Inform. Cybernet. 31 (2) (1999) 109–115. [23] T. Kinoshita, Y. Watanabe, M.T. Nakao, Recurrence relations of orthogonal polynomials in H01 and H02 , Nonlinear Theory Appl. IEICE 6 (3) (2015) 404–409. [24] M. Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan J. Indust. Appl. Math. 26 (2–3) (2009) 419–442. [25] X. Liu, T. Okayama, S. Oishi, High-precision eigenvalue bound for the Laplacian with singularities, in: Computer Mathematics, Springer, 2014, pp. 311–323.