The best constant of Sobolev inequality corresponding to anti-periodic boundary value problem

In this paper we establish the best constant of Lp Sobolev inequality for a function with anti-periodic boundary conditions. The best constant is expressed by Lq norm of (M − 1)-th order Euler polynomial. Lyapunov-type inequality for certain higher order differential equation including 1-dim p-Laplacian is obtained by the usage of this constant.


Introduction
It is well-known that sharp Sobolev inequalities are very important in the study of partial and ordinary differential equations, especially in the study of problems arising from geometry and physics.They are relevant for the study of boundary value problems.In this paper, we treat the anti-periodic case.Throughout the paper, we assume p > 1, a < b and 1  p + 1 q = 1, i.e. q = p p−1 > 1 is the conjugate exponent of p.For M = 1, 2, 3, . . .let us consider a sequence of Sobolev spaces and the following one-dimensional Sobolev inequality: where u ∈ W M and • ∞ and • p are the usual L ∞ and L p norms.When p = 2, the engineering meaning of this inequality is that the square of the maximum bending of a string (M = 1) or a beam (M = 2) is estimated from above by the constant multiple of the potential energy due to internal forces.Notice that anti-periodic boundary value problems appear in physics also in other situations, see, for example [4,8].The purpose Email: jozef.kiselak@upjs.sk 2 J. Kisel'ák of this paper is to derive a Sobolev inequality corresponding to an anti-periodic boundary value problem, and to obtain the best constant by using the property as the reproducing kernel of Green function.As an application, we give Lyapunov-type inequalities for certain half-linear higher order differential equations with anti-periodic boundary conditions.

Polynomials
To state the conclusion, we need to introduce the Appell polynomials (or sequences).The sequence P n (x) is Appel for g(t) if and only if for all x in the field C of field characteristic 0 and 1 g(t) is analytic function, see [9], where the author summarizes properties of more general Sheffer sequences and gives a number of specific examples.
The Bernoulli polynomials have been studied since the 18th century.There are many applications in mathematics and physics.Many functions are used to obtain the generating function of them, but also Euler and Genocchi polynomials.We first define Bernoulli polynomials B n (x) using the generating function (1.2) with g(t) = e t −1 t , i.e.

e tx e
Although it does not immediately yield their explicit form, the manipulation of (1.3), (1.4), (1.5), along with the uniqueness theorem for power series expansions, leads to many properties of these polynomials.For example symmetry is easily obtained in this way.For k ∈ N 0 and all x ∈ R we have where P k can be replaced by B k or E k .Since all these polynomial sequences are Appell sequences, also property concerning derivation must hold (it is sometimes used as an equivalent definition).We summarize this: The best Sobolev constant to anti-periodic BVP where P k can be replaced by B k , E k or G k .Notice that another convention followed by some authors (see [1, p. 169]) defines this concept in a different way, conflicting with Appell's original definition, by using the identity instead.This is reflected in the fact that B k (E k , G k ) is 1 k! multiple of the "original" one.Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been studied.We need only few of them, which can be summarized in the following lemma, see e.g.[2].
Lemma 1.1.We have G n (x) = n E n−1 (x), ∀n ∈ N and ∀x ∈ R. (1.13) In the Table 1.1 we give explicit forms of the first four Bernoulli, Euler and Genocchi polynomials.

Boundary value problem
In this section, we present the main theorems of this paper.For the case p = 2, the problem of finding the best constants of (1.1) is solved completely, whereas the method of maximizing the diagonal value of reproducing kernels was used, see references in Table 2.1.For the general case the difficulty of obtaining the best constants increases and cases of clamped and Dirichlet boundary conditions remain unsolved, again see Table 2.1.
We consider the boundary value problem In [14] the authors obtained a Green function for even M and a = 0, b = 1.As it is pointed out in [13], using the method of reflection and some algebra one can show that for the problem (2.1) the expression of the Green function has the form Once the Green's function is obtained, one can write down the solution of the problem (2.1) very easily using an integral.
Proof.Differentiating u(x), the properties listed in Lemma 2.2 and the following fact yields the existence and uniqueness of the solution.It is not difficult to show that for general M the result from [13] should be modified to where Due to the symmetry of Bernoulli polynomials it is true that B n Proof.Obviously (a) and (b) holds.Differentiating (2.2) k times with respect to x, we obtain . From (2.4) and symmetry property (1.6), we have The best Sobolev constant to anti-periodic BVP

5
So we have (c).Again from (2.4), it is true that Item (d) follows from symmetry property (1.9) and (e) is equivalent to (d).This completes the proof.

Reproducing kernel
Thanks to the solution of the boundary value problem (2.1) we have found reproducing kernel for some specific Hilbert space, which helps us to solve the problem of finding the best constant for Sobolev inequality (1.1).We denote by W * M the dual space of W M , i.e.
We show that the Green function G(2M; x, y) is a reproducing kernel for function spaces For any u ∈ W M , we have the following reproducing relation Proof.We first prove the first part.For any two smooth functions u and v, we have Now, we put v(x) = G(2M; x, y), y ∈ (a, b) and integrate this identity with respect to x over intervals a < x < y and y < x < b, we obtain where It suffices now to use symmetry property (1.6).The second part results directly from the fact that u ∈ W M .
, M is even Anti-periodic [14] this paper and see [6] Neumann Note that most of the authors solved specific problem on the interval [0, 1], but it can be simply extended to [a, b].We recall that α 0 is the unique solution to the equation is used.
We now present the main result of the paper.The technique used in the proof of the following theorem is much like that employed in [3].
Theorem 2.4.The best constant of the Sobolev inequality or the supremum of the Sobolev functional where the supremum is attained for where f (x) = (−1) The best Sobolev constant to anti-periodic BVP 7 Proof.Applying Hölder's inequality to the identity (2.5) in Lemma 2.3, we have , previous inequality can be rewritten as follows This shows that the best constant is not greater than right-hand side of (2.7).Now, we prove second part of the theorem.Let f be defined as above.Then u is the solution to the boundary value problem (2.1).Note that u ∈ W M .Interchanging x and y, we obtain Moreover, we have .
Since u(a) ≤ sup a≤y≤b |u(y)|, this together with (2.8) shows that we have constructed u in which the supremum of Sobolev inequality is attained.

Application
The well-known Lyapunov inequality states that if r : [a, b] → R is a continuous function, then a necessary condition for the Dirichlet boundary value problem
Now we establish a Lyapunov-type inequality for the half-linear equation of higher order with anti-periodic boundary value conditions.Notice that for m = 1 problems (3.5) and (3.

J. Kisel'ák Theorem 2 . 1 .
For any f ∈ BC(a, b) boundary value problem (2.1) has one and only one classical solution u(x) given by u(x) = b a G(M; x, y) f (y) dy, a ≤ x ≤ b, where the Green function has the form G(M; x, y) = (−1)

9 Applying.
Multiplying the first equation in (3.5) by u (m−1) (t) and integrating over [a, b], we have b a |u (m) (t)| p dt = b a r(t)|u(t)| p−2 u(t)u (m−1) (t) dt.The best Sobolev constant to anti-periodic BVP Theorem 2.4 to u and u (m−1) respectively, yields b a |u (m) (t)| p dt ≤ Since u is nonzero solution, by dividing both sides by b a |u (m) (t)| p dt, we obtain inequality (3.6).Moreover, this inequality is strict, since u(t) is not a constant.

Table 1 . 1 :
Explicit forms of the first four polynomials.

Table 2 .
1: Various boundary conditions and best constants.

Table 2 .
[10]pecial cases of best constants and best functions.4cannotbe replaced by a larger number.Such result has found many practical uses in problems as oscillation theory or eigenvalue problems (spectral properties of differential equations).Several proofs and generalizations or improvements for various boundary conditions have appeared in the literature.Recently, the author in[10]obtained a new Lyapunov-type inequality, a generalization of (3.2), for a certain anti-periodic problem.