Variational Approaches to Characterize Weak Solutions for Some Problems of Mathematical Physics Equations
This paper is aimed at providing three versions to solve and characterize weak solutions for Dirichlet problems involving the -Laplacian and the -pseudo-Laplacian. In this way generalized versions for some results which use Ekeland variational principle, critical points for nondifferentiable functionals, and Ghoussoub-Maurey linear principle have been proposed. Three sequences of generalized statements have been developed starting from the most abstract assertions until their applications in characterizing weak solutions for some mathematical physics problems involving the abovementioned operators.
Obtaining and/or characterization of weak solutions for problems of mathematical physics equations involving -Laplacian and -pseudo-Laplacian is a subject matter previously discussed by the author through several approach methods in [1–6]. New similar results can be found by other authors, for instance, Amiri and Zivari-Rezapour in , El Khalil and Ouanan in , Rhouma and Sayeb in , Yoshida in . The importance of these operators also devolves from their involvement in actual modelings of natural phenomena as thermal transfer by Lanchon-Ducauquis, Tulita, and Meuris in  or glacier sliding or flow by Partridge in  and Pélissier in . In this paper three methods are proposed following three sequences of results, starting from abstract statements and finishing with their application to find weak solutions of Dirichlet problems for -Laplacian and for -pseudo-Laplacian as well. In the first succession of assertions, two results of Ghoussoub from  have been generalized replacing the frame of Hilbert space by reflexive strictly convex Banach space and the condition imposed to the goal function to be of -Fréchet class by the weaker condition to have a lower semicontinuous and Gâteaux differentiable function. These two theoretical statements were used, together with other results concerning Dirichlet problems for both cited generalized operators, in finding weak solutions for these problems. The second proposition sequence involves critical points for nondifferentiable functional. In this context, two results of Chang from  have been generalized changing the space in and introducing and in some problems formulated for by Costa and Gonçalves in . The last series of assertions starts from Ghoussoub-Maurey linear principle which is used here to characterize weak solutions for some mathematical physics equations. Moreover, the problems were discussed and solved using important findings for these operators obtained by the author in [5, 6] in connection with specific properties of the Sobolev spaces involved.
2. Critical Points and Weak Solutions for Elliptic Type Equations
2.1. Theoretical Support
In order to introduce the first result, a theoretical support will be given starting with the following.
Ekeland Principle (see [1, 17, 18]). Let be a complete metric space and bounded from below, lower semicontinuous, and proper. For any and of withand for any , there exists in such that
Definition 1. Let be a real normed space, be a bornology on , and let . Let be in and a nonempty subset of . verifies the Palais-Smale condition on the level around (or relative to ), , with respect to , when a sequence of points in for whichthis sequence has a convergent subsequence.
To clarify the above notation, let be a bornology on and let locally finite in the point . By definition is -differentiable in , if there exists in the dual such that for every in we have (uniform limit on for ). is the -derivative of in , and it is denoted by .
Through the minimization on of a functional (minimization with constraints) global critical points of this may be obtained.
As a preliminary, we generalize some results from  introducing Banach space instead of Hilbert space and the Gâteaux differentiability instead of -class Fréchet.
Proposition 2. Let be a real reflexive strictly convex Banach space, let be lower semicontinuous and Gâteaux differentiable and let be a closed subset of such that for every from with the metric gradient , for sufficiently small ,Then, if is lower bounded on , for every a minimizing sequence for on , there exists a sequence in such thatwhere and .
Remark 3. This result is reported in  as Lemma 9 in the frame of the Hilbert spaces having the function of -class Fréchet, but condition (5) is more complicated due to another condition imposed to the set .
Explanations. Let us introduce the definition of the metric gradient in order to provide other observations relative to this central notion for this statement. In a real normed space , consider the Gâteaux differentiable functional . The metric gradient of is the multiple-valued mapping , , where is the duality mapping on corresponding to the identity and the canonical injection of into , . Consequently, for any = . If is reflexive, for any , is nonempty. being strictly convex, is single-valued. So, if is reflexive and strictly convex, then , , and the following equalities hold:
Go now to the proof of Proposition 2.
Proof. Denote and let be from . For , hence , we have . Apply the enunciated Ekeland principle with , in with known properties. Thus we obtain the sequence satisfying (6), (7) , andVerify (5). It is sufficient to work under the assumption . Thus apply the hypothesis made in the statement with respect to with and denoting, for , replace in (9) and find,multiply this inequality by , , and take the limit for in order to keep the sense of the inequality. Remark that ; . Consider that the existence of the limit for implies the existence of the limit for together with their equality, = = , taking into account the definition of the Gâteaux derivative and the above considerations on the metric gradient, and (5) is also fulfilled.
Remark 4. The Gâteaux derivative from the above statement can be replaced by any -derivative and the result remains the same. In the case of the Fréchet derivative, it must remove the condition “ lower semicontinuous” from the statement.
Notation 1. -differentiable;
Proposition 5. Let be a real reflexive strictly convex Banach space and lower semicontinuous and Gâteaux differentiable and is a nonempty convex closed subset such that , the identity map. If is lower bounded on , then for every a minimizing sequence for on , there is a sequence in such thatMoreover, if satisfies , where , then
Proof. Applying Proposition 2, (4) is satisfied indeed: if and , then, being convex,Let be the sequence given by the statement. , hence . , hence , clearly , and consequently has a convergent subsequence , . This implies , and is a global critical point of contained in .
2.2. Weak Solutions
Open set of class in . Use the notations (the norm is that Euclidean from ): , , , . Let be an open nonempty set in , , and its boundary. By definition, is of class if from a neighbourhood of in and bijective such that , , , and . Weak solution. Let be an open bounded nonempty set in , , , and . Consider the problemsActually on means , where is the trace operator, a continuous linear mapping from in , . We have and . from is by definition a weak solution for and , respectively, if on and
Remark 6. Here is the weak gradient, and it is equal to the weak derivatives; . is endowed in the first case with the norm ; that is, , which is equivalent to the norm . For the second case , equip the same vector space with the norm , which is equivalent to . Nemytskii Operator. Let be , , the Lebesgue measure in , open nonempty Lebesgue measurable and . By definition is a Carathéodory function if is Lebesgue measurable and is continuous , . In this case, for every from one may consider the function , , Nemytskii operator. Suppose Then . Assume that satisfies the growth condition: with , where , , and , .
Then ; is continuous and bounded on . If is bounded and , then with continuous; moreover, with continuous (see, e.g., ), where , and , is of Fréchet class and , so it is also Gâteaux differentiable.
Theorem 7. Let be an open bounded nonempty set in and a Carathéodory function with the growth condition:where , when , and when , and where ,
Then the energy functional,where is Gâteaux differentiable on and
Proof. One may consider , in both cases, as the sum of two other functions. The second of these functions being Gâteaux differentiable (see the above consideration), it is sufficient to remark that also the maps and are Gâteaux differentiable on [2, 19] and then is Gâteaux differentiable on.
Corollary 8. Letand f be as in Theorem 7. Then the weak solutions of and , respectively, are precisely the critical points of the functional:
Proof. Indeed, if is a weak solution of and , respectively, then ((15) and (16), resp., Theorem 7); hence . The inverse assertion is obvious.
Weak Subsolutions and Weak Supersolutions of and . Let be an open bounded set of class in , , let be a Carathéodory function, and let . is a weak subsolution and a weak supersolution, respectively, of or if
Proposition 9. Letbe an open bounded of class set in , and a Carathéodory function and , from bounded weak subsolution and weak supersolution of , respectively, with a.e. on. Suppose that verifies (17) and there is such that the function is strictly increasing in s on . Then there is a weak solution of in with the property
Proof. Take the equivalent norm on :Consider the functional : is Gâteaux differentiable and its critical points are the weak solutions of (see Corollary 8). is also lower bounded, the norm on actually being of Fréchet class (see, e.g.,  or ). Use Proposition 5, being a reflexive strictly convex Banach space (see, e.g., ). Let be is closed convex. We also getHere denotes the metric gradient of . Since is reflexive and strictly convex (see, e.g., ), thus is univaluated and it has the above described properties. Indeed, let be in and. We should prove that . and . Since the definition relation of the subsolution for actually means in with almost everywhere (a.e.) on and that of supersolution for is in verifying a.e. on, we will prove that a.e. on and a.e. on using the Gâteaux derivatives of in and , respectively. (take into account that , is a linear map and some properties of the metric gradient). Also . is lower bounded on , being actually continuous (for this see, for instance, ). Until now, applying Proposition 5, for every a minimizing sequence for on , there is a sequence in such that , , . So since , we have already, also the last property from condition is verified. Finish the proof applying once again Proposition 5.
Example 10. Consider the problem (open bounded of class in , )where ; is continuous with on . Then is a weak subsolution, , sufficiently big, is a weak supersolution, (condition (17)), and is increasing in on ; consequently, according to Proposition 9, (32) has a weak solutionwith a.e. on.
Proposition 11. Letbe an open bounded of class set in , and a Carathéodory function and , from bounded weak subsolution and weak supersolution, respectively, of with a.e. on. Suppose that f verifies (17) and there is such that the function is strictly increasing in s on . Then there is a weak solution of in with the property
Proof. Follow step by step the above proof for Proposition 9 considering the real reflexive strictly convex Banach space endowed with the norm or the equivalent norm which both are also equivalent to the other two norms used at Proposition 9. The function is from (19) having the weak derivative given in Theorem 7. Using similar calculus, obtain similar conclusion.
Example 12. Consider the problem (open bounded of class in , ):where and is continuous with on. Then is a weak subsolution, , sufficiently big, is a weak supersolution, (condition (17)), and is increasing in on ; consequently, according to Proposition 11, (34) has a weak solutionwith a.e. on .
3. Critical Points for Nondifferentiable Functionals
The sense of the title actually is “not compulsory differentiable.” Start this section with the following.
Definition 13. is a critical point (in the sense of Clarke subderivative) for the real function if . In this case is a critical value (in the sense of Clarke subderivative) for .
To clarify this notion, the Clarke subderivative should be introduced. Let be a real normed space, , , , and . We set . is by definition Clarke derivative (or the generalized directional derivative) of the function at in the direction . The functional from is by definition Clarke subderivative (or generalized gradient) of in if . The set of these generalized gradients is designated by .
Here it is a generalization at -Laplacian and -pseudo-Laplacian of an application from  of this concept.
Let be a bounded domain of with the smooth boundary (topological boundary). Consider the nonlinear boundary value problems and where is a measurable function with subcritical growth; that is,where , for and for or .
Set as in 
SupposeWe emphasize that is verified in the following two cases:(i)is independent of .(ii) is Baire measurable and is decreasing , in which case we have
Definition 14. from is solution of and , respectively, if on in the sense of trace andDefine with , regular differential manifold, e.g., , open of class with bounded. In this situation, there exists a unique linear continuous operator , the trace, such that is surjective and . This gives a sense for , for any in . Moreover, .
Let be , but in the first case , the norm endowing is , that is, , which is equivalent to the norm . For the second case , equip the same set by the norm , which is equivalent to .
Associate to the locally Lipschitz functional :and associate to where . Set, a map defined on , taking values in , is locally Lipschitz (use ). The functional , , is also locally Lipschitz (again ). Using Sobolev embedding , we obtain that is locally Lipschitz on , which implies locally Lipschitz on , and consequently, according to a local extremum result for Lipschitz functions (if is a point of local extremum for , then ), the critical points of for Clarke subderivative can be taken into account. One may state the following.
Proposition 15. Suppose and are satisfied. Then is locally Lipschitz on (Ω) and (i) in a.e,(ii)if , where endowed with the norm for the problem and for the problem , respectively, then
Proof. The proof for (i) can be found in , Theorem , which remained here the same, while the problem was solved for the Laplacian with only. In order to prove (ii), use Theorem from  observing for both cases ( endowed with each one of those two norms) that is reflexive and dense in as can be seen, for instance, summarized in .
Proposition 16. If and are verified, every critical point of is solution for and , respectively.
Proof. Problem . Let be a critical point for . We have since , and apply some rules of subdifferential calculus concerning finite sums. , where .
Using (44) and a specific property of a function Lipschitz around , the Clarke derivative of ), we findBut (a property of the Clarke derivative, see ) and thusthat is, and, using Proposition 15, . Since , it results and (37):Problem . Let be a critical point for . We havesince , and apply some rules of subdifferential calculus concerning finite sums. , where .
Using (49) and a mentioned property of a function Lipschitz around , we findBut (a property of the Clarke derivative, see ) and thusthat is, and, using Proposition 15, . Since , it results and (38):
4. An Application of Ghoussoub-Maurey Linear Principle to -Laplacian and to -Pseudo-Laplacian
Start with the statement of this generalized perturbed variational principle.
Theorem 17 (Ghoussoub-Maurey linear principle). Let be reflexive separable space and lower semicontinuous and proper:(I)If φ is bounded from below on the closed bounded nonempty subset , the set is of type and everywhere dense.(II)If, for any from , is bounded from below, the set is of type and everywhere dense.
To clarify the involved notions, let be a real normed space, , nonempty subset of , and . strongly exposes from below in , when and , . “ strongly exposes from above in ” has a similar definition. Remark that, taking in the given definition, we fall on the definition of strongly minimum point. And also, a set of type means a set which is a countable intersection of open sets. A set of type means a set which is a countable union of closed sets.
We imply this theorem in two generalizations of a minimization problem of the form where is open set of class in , , , , , the critical exponent for the Sobolev embedding (for the necessary explanations, here and in the following, see [1, I, § , last section]).
Let be an open bounded set of class in , . Consider the problems () and (), where is a Carathéodory function with the growth conditionThe functionals ,with , are of -class Fréchet and their critical points are the weak solutions of the problems () and (), respectively.
Problem (). Let be the first eigenvalue of in with homogeneous boundary condition. We have (see [2, ])
And now give an answer for (56). Use the norm on (see above). Denote the dual of by , where is the conjugate of (i.e., ).
Proposition 18. Under the above assumptions and in addition the growth conditionwith , for some and , from , the following assertions hold: (i)The set of functions from , having the property that the functional , has in only one point an attained minimum includes a set everywhere dense.(ii)The set of functions from , having the property