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Direct methods in the calculus of variations for differential forms
Journal of Inequalities and Applications volume 2013, Article number: 407 (2013)
The purpose of this paper is to establish the general theory of the direct methods to functionals I defined on the Grassmann algebra employing the classical approaches. In this paper, various notions of convexity conditions for weak lower semicontinuity of I are discussed, and existence theorems for minimizers of I are obtained. Lastly, we present some examples to illustrate our main results.
Differential forms as invaluable tools have been available and applicable to various fields of study, see [1–6], while the systematic investigation of variational theory for differential forms has been less studied. Direct methods is a classical and fundamental method, used in variational problems, which has been widely applied for solving differential equations that possess variational structure, see [7–10]. More precisely, direct methods work for the minimization problem for certain energy functionals I, whose critical points are usually related to solutions of certain differential equations. Various conditions for the existence of minimizers of I have been greatly studied, see [9–21].
This paper is intended to generalize and apply the classical direct methods in variational problems for differential forms with an aim to enlarge the range of applications of the variational principle. Our work is motivated by Iwaniec and Lutoborski , who first used differential forms to express variational problems.
Let Ω be a nonempty, bounded, open subset of . Suppose that integers and is a continuous function satisfying the growth condition
for all , where . Write . Given , we consider the integral
defined for vectorial differential forms
where is imposed by the growth condition (1.1). Let
We consider the existence of the minimizer of I in ℳ, that is, we want to prove that
In particular, associated with the variational kernel , (1.2) is reduced to the p-Dirichlet integral
and the minimizers of the p-Dirichlet integral are exactly the weak solutions to the p-harmonic equation
2 Notation and preliminary results
We denote by the space of l-covectors in , and the direct sum
is a graded algebra with respect to the wedge product ∧. We shall make use of the exterior derivative
and its formal adjoint operator
known as the Hodge codifferential, where the symbol ∗ denotes the Hodge star duality operator. Note that each of the operators d and applied twice gives a zero.
Let be the class of infinitely differentiable l-forms on . Since Ω is a smooth domain, near each boundary point one can introduce a local coordinate system such that on ∂ Ω, and such that the -curve is orthogonal to ∂ Ω. Near this boundary point, every differential form can be decomposed as , where
are called the tangential and the normal part of ω, respectively. Now, the duality between d and is expressed by the integration by parts formula
for all and , provided or . The symbol denotes the inner product, i.e., let and , then .
Due to (2.1), extended definitions for d and can be introduced as the introduction of weak derivatives.
Definition 2.1 
Suppose that and . If
for every test form , we say that ω has generalized exterior derivative v and write .
The notion of the generalized exterior coderivative can be defined analogously.
Definition 2.2 
Suppose that and . If
for every test form , we say that ω has generalized exterior coderivative v and write .
The generalized exterior derivative has many properties similar to those of the weak derivative. For example, (i1) if the generalized exterior derivative exists, it is unique; (i2) if ω is differentiable in the conventional sense, then its generalized exterior derivative is identical to its classical exterior differential dω. Analogous results hold for generalized exterior coderivative.
If the generalized exterior derivative of ω, , exists, then also has its generalized exterior derivative . Moreover, .
d and have analogous expressions, i.e., for , we have
where ∂ denotes the ordinary derivative and the weak derivative. So next, we use d to represent the action instead of , similar for and .
Next, we present briefly some spaces of differential forms
which are used throughout this paper.
Finally, we introduce the definition of weak convergence for sequences in spaces of differential forms as needed. Throughout the paper, let be the Hölder conjugate pair, .
Definition 2.3 
We say that weakly converges to φ in if
whenever and write in .
We also need some properties of spaces of differential forms.
Proposition 2.4 
For, is reflexive.
Proposition 2.5 
inif and only ifinandin, where, and the partial differentiation is applied to the coefficients ofφ.
3 Weak lower semicontinuity
In many variational problems, weak lower semicontinuity is an essential condition for the existence of minimizers, using the minimization method. This will motivate the study of various notions of convexity conditions. In this section, we study the conditions for weak lower semicontinuity of the integral functional in the Sobolev space .
Definition 3.1I is called weak lower semicontinuous on if
whenever in .
3.1 The convex case
Theorem 3.2For a given, letbe continuous and convex. ThenIis weakly lower semicontinuous on.
Proof Suppose that in , we then need to show
We may assume that is finite and convergent.
Since in , then, according to Mazur lemma , for any , there exists a sequence of convex linear combinations
such that () in and pointwise almost everywhere.
Since W is continuous, we have
as . On the other hand, we have by the convexity of W that
Therefore, we obtain from Fatou lemma that
Let in the inequality above, we obtain
that is, . The theorem follows. □
3.2 The non-convex case
This section considers the case when W is not convex. Fruitful results have been obtained for the weak lower semicontinuity of integral functionals I when W fails to be convex, see [9–21]. We consider the counterpart in the case of differential forms.
Definition 3.3 Let be continuous. Then W is said to be quasiconvex if
holds for every bounded open set with for every fixed and for all .
Remark (i) This definition is actually the so-called -quasiconvex, see . For the general notion of (relatively is -quasiconvex) and other results, readers can refer to [9, 10, 13, 16] and references therein.
From Jensen’s inequality, we see that every convex function is quasiconvex. In fact, we have by Jensen’s inequality that
Since , the Stokes theorem  yields
Therefore, we have
Quasiconvexity implies that the function W is convex with respect to each , , which together with (1.1) yields a Lipschitz-type condition(3.2)
Lemma 3.4LetandWbe quasiconvex satisfying the growth condition (1.1). Suppose that
Proof Let , then as . We choose cut-off functions such that , and . Let
We take as test functions in the definition of quasiconvexity to obtain
Since W is continuous, thus, W is bounded on bounded sets, together with that as , we then have the last integral in the right hand side of (3.4) converges to zero as . Therefore, we have
which is the desired result. □
Theorem 3.5Letbe quasiconvex and satisfy (1.1). ThenIis weakly lower semicontinuous in.
Proof Suppose that in , we then need to show that
We may assume that is finite and convergent, i.e.,
Let be disjoint cubes parallel to the axes, and edge-length is . Denote , and we then have that
as . Let
with . For , set
Then is constant on and write . Then we have
as . Therefore, for arbitrary , we can choose k large enough such that
Note that in , we have is uniformly bounded. We then consider
For : Since , W is continuous and Ω is bounded, we have
For : From (3.2) and the Hölder inequality, we see that
For : Applying Lemma 3.4 for , yields
which implies that
For : It follows from (3.2) and the Hölder inequality that
Substituting (3.6)-(3.9) into (3.5), we have by letting that
Let in the above inequality, we obtain
The theorem follows. □
Polyconvexity is also an important condition for the weak lower semicontinuity of I, which is quasiconvex, but not necessarily convex. We give below the definition of polyconvex, and we want to establish the weakly lower semicontinuity of I on , provided W is polyconvex.
Definition 3.6 Let be the number of elements of the set
for . A function is called polyconvex if there exists be convex in each of its variables, such that
Proceeding with the proof in much the same way as , the equivalent definition for polyconvexity is given by Iwaniec and Lutoborski.
Definition 3.7 
A continuous function is called polyconvex if
where and are variable points from and , , are functions of ζ only.
Remark (i) Note that (3.10) implies convexity of W in each variable, and we have from the definitions that
The non-zero terms in the right side of (3.10) imply that
For , and assume that
where or . Observe that the coefficients of (3.11) are subdeterminant of the Jacobian of , , where
We apply the following lemma.
Lemma 3.8 [, Corollary 5.32]
Letbe anysubdeterminant, and letbe a number. Letbe any sequence weakly convergent toinas. Thenweakly in.
By adding the condition
we have that for ,
in as when in . Therefore, we obtain the following theorem for the weakly lower semicontinuity of I.
Theorem 3.9Letbe polyconvex, and let. ThenIis weakly lower semicontinuous, provided functionsin (3.10) belong to.
Proof Suppose that in . We may assume that is finite and convergent as before. We first obtain from (3.10) that
Next, observe that for ,
and since and , , we then have by the Hölder inequality that
Therefore, integrating both sides of (3.12) in Ω, and then letting , we obtain
that is, we have
which is the desired result. □
4 Existence of minimizers
To prove the existence of minimizers of I, another important point is requiring its coercivity condition. The counterpart in the variational problem of differential forms is introduced by Iwaniec and Lutoborski  as follows.
Definition 4.1 
Suppose that W satisfies (1.1), we say that W is coercive in the mean if
for all test forms .
Theorem 4.2In addition to the hypotheses, which ensureIto be weakly lower semicontinuous, assume thatWsatisfies the growth condition (1.1) and mean-coercivity condition (4.1). Then the minimizer ofIis achieved at some.
It follows from (1.1) and (4.1) that m is finite. By the definition of infimum, there exists a minimizing sequence such that
as . Then the mean coerciveness (4.1) yields is bounded in , which implies that is bounded in , . From Proposition 2.4, we see that has a weakly convergent subsequence, denoted by again, in . Then we may assume that
as . Write , then we get
and . Since I is weakly lower semicontinuous, it follows that
i.e., as desired. □
In this section, we present two examples to illustrate our main results. The first example is to explain the existence result Theorem 4.2 for the stored-energy function W, which is polyconvex.
Example 1 We consider a special case of , , and . Let with
and assume that is defined by
where ∗ is the Hodge star duality operator.
Step 1. Direct computation implies
then we see that W satisfies the growth condition (1.1) with . Thus, we have , and .
Step 2. Note that and
Hence we have .
Step 3. Applying Proposition 9.1 in , yields W satisfies (4.1).
Step 4. Suppose that and
Then , and , . Thus, we have
where . Write
Then by applying
we get from (5.1) that
Similarly, we have
By substituting (5.3) and (5.4) into (5.2), we obtain that
Then we have from (5.5) that the corresponding terms in (3.10) are
Since and Ω is bounded, then we get that . Therefore, it follows from Theorem 4.2 that the minimizer of I can be achieved.
Finally, we give an example for a stored-energy function W, which is quasiconvex.
Example 2 We consider the case , and . Let and
for . Write , , where denotes the element of i th-row and j th-column in A. Fix , then we have by simple calculation that
for all ordered l-tuples , . Let , then it follows from (5.6) that
denotes the principal minors of A. Let denote the sum of the principal minors of A, then
Suppose that A is positive semidefinite, then , and we have from  that W is quasiconvex.
aInequality between two volume forms should be understood as inequality between their coefficients with respect to the standard basis, that is to say, we say that an n-form α on is nonnegative if for some nonnegative function λ.
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This work was supported by the National Natural Science Foundation of China (Grant No. 11071048).
The authors declare that they have no competing interests.
All authors contributed equally to this paper. They read and approved the final manuscript.
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Wang, T., Bao, G. & Li, G. Direct methods in the calculus of variations for differential forms. J Inequal Appl 2013, 407 (2013). https://doi.org/10.1186/1029-242X-2013-407
- weak lower semicontinuity
- differential forms
- variational problem
- direct methods
- existence theorem