Journal of Calculus of Variations

Volume 2013 (2013), Article ID 936915, 11 pages

http://dx.doi.org/10.1155/2013/936915

Research Article

Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term

Koji Kikuchi

Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan

Received 30 April 2013; Accepted 11 July 2013

Academic Editor: Adam Kowalewski

Copyright © 2013 Koji Kikuchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. Usually, the term is added, and it is well known that this equation is well posed in the space of functions. In this paper, the viscosity term is changed to , and it is proved that if initial data is slightly smooth (but belonging to is sufficient), then a weak solution exists uniquely in the space of BV functions.

1. Introduction

Let be a bounded domain in with the Lipschitz continuous boundary . In [1] and in the author’s previous works [2–4], the following:

is investigated, which is in these works referred to as the equation of motion of vibrating membrane. Up to now, neither existence nor uniqueness of a solution to (1) is obtained. In [1–3], we only have that a sequence of approximate solutions to (1) converges to a function in an appropriate function space, and that if satisfies the energy conservation law, it is a weak solution to (1). In [1], approximate solutions are constructed by the Ritz-Galerkin method and in [2, 3] by Rothe’s method. In [2], the boundary condition is not essentially discussed, and the observation is added in [3]. In these works, the limit should satisfy the energy conservation law, and existence theorem of a global weak solution has not been established yet. Instead, in [4], linear approximation for (1) is established. On the other hand, the equation with the strong viscosity term is investigated by several authors. For example, in [5], it is investigated in the context of control theory, and it is asserted that if and , there exists a unique solution for each . Namely, the equation with strong viscosity term is well posed in , and since is a smaller class than the space of BV functions, this suggests that the influence of the term is too strong.

In this paper, replacing the strong viscosity term with , we investigate it in the space of BV functions. Namely, our problem of this paper is as follows:

with initial and boundary conditions

We should note that the term “viscosity” probably means implying regularity. However, in this paper, we only investigate existence and uniqueness of (2)–(4), regularity is not investigated. This is the reason that in the title there is a quotation mark.

A function is said to be a function of bounded variation or a BV function in if the distributional derivative is an valued finite Radon measure in . The vector space of all functions of bounded variation in is denoted by . It is a Banach space equipped with the norm (see, e.g., [6–8]). We should note that, for , the operator is multivalued. It is usually defined by the use of the subdifferential of the area functional. Namely, for a function , we regard as

where

Here, readers should note that is not . We are imposing (4), and in the analysis in the space of BV functions, the most appropriate weak formulation of (4) is to replace with (cf. [3], see, also [4, Appendix C]).

Now, we present our definition of a weak solution to (2)–(4).

Definition 1. A function is a weak solution to (2)–(4) in if satisfies that

(i), , (ii)s- in , (iii) there exist and such that for -a.e. ( denotes the one-dimensional Lebesgue measure), and, for any ,

If a function is a weak solution to (2)–(4) in for each , then we say that is a weak solution to (2)–(4) in .

Our main theorem is as follows.

Theorem 2. Suppose that and . We further suppose that . Then, there exists a unique weak solution to (2)–(4) in .

Remark 3. If , then the element is unique. Indeed, for each , is differentiable at and for each . Since and is arbitrary, is uniquely determined.

2. Reduction of the Problem

In order to solve (2)–(4), we give a formal observation. Let us put

then (2) becomes , which can be regarded as an ordinary differential equation to . By the variation-of-constants formula, we obtain that . Noting that , we have

where . Hence, formally, (2) is reduced to

Definition of a weak solution to this equation is as follows.

Definition 4. Let ,  , and . A function is a weak solution to (10) with (3) and (4) in if satisfies that

(i), , (ii)s- in , (iii)for any and for -a.e. ,

Similar to the case of (2), we say that is a weak solution to (10) with (3) and (4) in if a function is a weak solution to (10) with (3) and (4) in for each .

The previous observation is just formal. In the following proposition, we show it rigorously.

Proposition 5. Definitions 1 and 4 are equivalent.

Proof. It is sufficient to show that, for each , a function is a weak solution to (2)–(4) in if and only if it is a weak solution to (10) with (3) and (4) in .

Suppose that is a weak solution to (10) with (3) and (4) in . Conditions (i) and (ii) of Definition 1 are the same as those of Definition 4. Thus, we only have to show (iii) of Definition 1. Let

Then, by (iii) of Definition 4, we have that for -a.e. . Thus, by a direct calculation, we have that satisfies (iii) of Definition 1.

Next, we suppose that is a weak solution to (2)–(4) in . For each and each , we put

Then , and since , we have the following by (iii) of Definition 1:

By integration by parts, we have

Furthermore, we have the following by Fubini’s theorem:

Finally, noting that , we have the following by (14), (15), and (16):

Since and are arbitrary, we have, for -a.e. ,

which means that satisfies Definition 4, (iii).

Now, Theorem 2 is reduced to the following.

Theorem 6. Suppose that and . We further suppose that and let . Then, there exists a unique weak solution to (10), (3), and (4) in .

Our strategy of proving Theorem 6 is the contracting mapping theorem. For this purpose, given that , we solve

and show that the map is a contraction. A weak solution to (19) with (3) and (4) is defined as follows.

Definition 7. Let . A function is a weak solution to (19) with (3) and (4) in if satisfies that

(i), , (ii)s- in , (iii)for any and for -a.e. ,

The proof of Theorem 6 consists of two parts. The first part is solving (19), and the second part is to show that the map is a contraction.

3. Existence and Uniqueness of a Solution to (19)

Let and be as in Theorem 6. In this section, we show that there exists a unique solution to (19) with (3) and (4) in for each .

Uniqueness is easy. Suppose that and are solutions to (19) with (3) and (4) in , and inserting to (iii) of Definition 7, integrating it from to , obtaining another inequality by replacing and , and adding these two inequalities, we have

Since , we have the uniqueness of a solution to (19).

It is sufficient to show the existence in for -a.e. . Approximate solutions are constructed by Rothe’s time semidiscretization method. In Rothe’s method, we should solve elliptic equations with respect to space variables. Here, we solve them by a direct variational method (namely, this is the method of discrete Morse semiflow, cf. [9] and references cited therein).

Suppose that with and , and let . For a positive number , we construct a sequence in the following way. For , we let be as in (3), and for , it is defined as a minimizer of the following functional:

in the class , where

Since

is bounded from below, and hence, the existence of a minimizer of follows.

Lemma 8 (energy inequality). consider the following:

Proof. Since is a minimizer of , we have

Hence, for each ,

Thus, by induction on , we have the conclusion.

Next, we define approximate solutions and for as follows: for ,

Then Lemma 8 shows for each

Now, we estimate the second term of the left hand side of (29). Then,

where , and thus, it is easy to see that

By (29), we have, for each and for each ,

where

Proposition 9. It holds that

(1)  is uniformly bounded with respect to  ;(2)for any  ,    is uniformly bounded with respect to  ;(3)for any  ,    is uniformly bounded with respect to  ; Then there exist a sequence    with    as    and a function    such  that(4)  converges to    as    weakly in  ;(5)for any  ,    converges to    as    weakly star in  ;(6)for any  ,    converges to    as    strongly in    for each  ;(7)for any  ,    converges to    as    strongly in    for each  ;(8);(9)for  -a.e.  ,    converges to    as    in the sense of distributions;(10)s-   in  .

Proof. Assertion (1) immediately follows from (32). Since we have

for each , , Assertion (1) implies that, for each , is uniformly bounded with respect to . Given that , we let be an integer such that . Then,

By (32),

Thus, we have

for each , where . Hence,

Now, we have that is uniformly bounded with respect to since

Since is increasing with respect to , Assertion follows from (32). Since is convex, we have

and Assertion ) also holds.

Assertion is a direct consequence of Assertion . Assertion follows from Assertion . Furthermore, (34) and Assertion imply that the function is equicontinuous with respect to . By Sobolev’s theorem compactly for each . This means that, for any , is contained in a sequentially compact subset of which is independent of and . Thus, by the Ascoli-Arzela theorem, we obtain Assertion .

Now, we have, for ,

the right hand side of which converges to as by (38) and Assertion (6). Now, we have Assertion (7). Assertions (2) and (7) imply Assertions (8) and (9).

Letting in (34), we have

Thus, by Assertion (1) the left hand side is uniformly bounded with respect to and, hence, passing to a subsequence if necessary, converges weakly in , and by Assertion (6), the weak limit is . Then, by the lower semicontinuity of norm, we have

which implies Assertion (10).

Now, our purpose is to show that is a weak solution to (19). Proposition 9 implies that satisfies (i) and (ii) of Definition 7.

Since is a minimizer of , we have

Let us write, for , . Then, for each and for -a.e. ,

namely, for each ,

By Proposition 9 (7) we have that, for -a.e. ,

strongly in as . Let be a number such that (46) and (47) hold. We insert an arbitrary function in (46). Integrating it from to , we have the following by Proposition 9 (4), (7), Fatou’s lemma, and the lower semicontinuity:

For a while, we write for simplicity. First, we note the following identity:

Let be the integer such that . By (49), we have

By (42) and Proposition 9 (1), is uniformly bounded with respect to and . On the other hand, since ,

Hence, we have by (37)

and thus by (37) again

as . By Proposition 9 (1),

as . Since , , (47) implies that in . In particular, we have . Then, since , we have

Summing up, we have

It is not difficult to show that strongly in . Hence, by (48), we finally have

for each . By the convexity of , for each function with , we have . Thus, (57) implies that

It is easy to extend this inequality to all nonnegative functions . Hence, (iii) of Definition 7 holds for -a.e. .

4. Proof That Is a Contraction

Let , be functions in . In this section, we write

Then,

Let , be a solution to (19) with (3) and (4) for , , respectively. By (iii) of Definition 7,

Summing these, we have

Integrating from to , we have the following by (60) and by the fact that :

We further integrate this from to and write it . Then,

Here,

Thus,

These two terms are estimated as follows:

Summing up, we have

Hence, when , putting

we have

As , converges to . Thus, if is sufficiently small, . This means that the map from to is a contraction in . Hence, there is a fixed point and it is a solution to (10) with (3) and (4) in .

End of the proof of Theorem 6.

Uniqueness of a Local Solution. Let , be solutions to (10) with (3) and (4). Then, in the same calculus as before, we obtain

This implies the uniqueness.

Existence of a Time Global Solution. Suppose that is a solution to (10) with (3) and (4) in . First, we remark that

and the right hand side belongs to . Hence, we are able to solve (10) with (3) and (4) from . By the change of variable , (10) becomes

where

Hence, solving (10) with (4) and initial condition , , we obtain a function in which solves (10). For , put . Then, is a solution to (10) with (3) and (4) in . Repeating this process, we obtain a time global solution. Uniqueness of the local solution implies uniqueness of the global solution.

5. Uniform Estimates

Let be the small number presented in the previous section, and let be a solution to (10) with (3) and (4) in . As we see in the previous section, it is obtained as a fixed point of the map in , where is a solution to (19) with (3) and (4). The fixed point is obtained as in the following way. Let be an arbitrary element of and put . There is a constant which is determined by such that and . Thus, we have

and hence,

namely, is a Cauchy sequence in and it converges to a function . Since , we have that by letting . Thus, is the fixed point of .

Letting in (78), we have

By the lower semicontinuity, an energy inequality for a solution to (19) with (3) and (4) is obtained by letting in (32):

Since is increasing with respect to , we have . Hence,

Recall that is presented as in (33). By the proof of Proposition 9 (5), we have

Letting in (79), we finally have

Now, a solution to (10) is a solution to (19) for , the fixed point of . Hence, by (81), we have

By (83) and (84), there exists a constant such that

Now, by (83) and (85), we obtain uniform estimates for , , for the small .

Solving (10) from (), we have the following by (83):

where is as in (76). By (81) and Chebyshev’s inequality, for sufficiently large , . Let be an arbitrary small positive number and put . Then, there exists an such that . Hereby, we have by (83) and (85) that, for such an , there exists a constant such that

Repeating this process, we have uniform estimates for , , for each . However, their upper bounds depend on .

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