International Journal of Partial Differential EquationsVolume 2014 (2014), Article ID 845760, 16 pageshttp://dx.doi.org/10.1155/2014/845760

Research Article

Explicit Estimates for Solutions of Mixed Elliptic Problems

Luisa Consiglieri

Núcleo de Investigadores Científicos, Universidad Central del Ecuador, Ciudadela Universitaria Avenida América, 290-4799 Quito, Ecuador

Received 25 October 2013; Accepted 13 January 2014; Published 31 March 2014

Academic Editor: Zhenhua Guo

Copyright © 2014 Luisa Consiglieri. 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.

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in () of class . The existence of and estimates is assured for and any (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive estimates for different ranges of the exponent depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

1. Introduction

The knowledge of the data makes all the difference in the real-world applications of boundary value problems. Quantitative estimates are of extreme importance in any other area of science such as engineering, biology, geology, and even physics, to mention a few. In the existence theory to the nonlinear elliptic equations, fixed point arguments play a crucial role. The solution may exist such that it is estimated in an appropriate functional space, where the boundedness constant is frequently given in an abstract way. Their derivation is so complicated that it is difficult to express them, or they include unknown ones that are achieved by a contradiction proof, as, for instance, the Poincaré constant for nonconvex domains. The majority of works consider the same symbol for any constant that varies from line to line along the whole paper (also known as universal constant). In conclusion, the final constant of the boundedness appears completely unknown from the physical point of view. In presence of this, our first concern is to exhibit the dependence on the data of the boundedness constant. To this end, first (Section 3.1) we solve in the Dirichlet, mixed, and Neumann problems to an elliptic second-order equation in divergence form with discontinuous coefficient, and simultaneously we establish the quantitative estimates with explicit constants. Besides in Section 3.2 we derive estimative constants involving and measure data, via the technique of solutions obtained by limit approximation (SOLA) (cf. [1–4]).

Dirichlet, Neumann, and mixed problems with respect to uniformly elliptic equation in divergence form are widely investigated in the literature (see [5–13] and the references therein) when the leading coefficients are functions on the spatial variable, and the boundary values are given by assigned Lebesgue functions. Meanwhile, many results on the regularity for elliptic PDE are appearing [14–28] (see Section 6 for details). Notwithstanding their estimates seem to be inadequate for physical and technological applications. For this reason, the explicit description of the estimative constants needs to be carried out. Since the smoothness of the solution is invalidated by the nonsmoothness of the coefficients and the domain, Section 4 is devoted to the direct derivation of global and local estimates.

It is known that the information that “The gradient of a quantity belongs to a space with larger than the space dimension” is extremely useful for the analysis of boundary value problems to nonlinear elliptic equations in divergence form with leading coefficients , where is a known function, usually the temperature function, such as the electrical conductivity in the thermoelectric [29, 30] and thermoelectrochemical [31] problems. It is also known that one cannot expect in general that the integrability exponent for the gradient of the solution of an elliptic equation exceeds a prescribed number , as long as arbitrary elliptic coefficients are admissible [19]. Having this in mind, in Section 6 we derive estimates of weak solutions, which verify the representation formula, of the Dirichlet, Neumann, and mixed problems to an elliptic second-order equation in divergence form. The proof is based on the existence of Green kernels, which are described in Section 5, whenever the coefficients are whether continuous or only measurable and bounded (inspired by some techniques from [32–34]).

2. Statement of the Problem

Let be a domain (i.e., connected open set) in of class and bounded. Its boundary is constituted by two disjoint open -dimensional sets, and , such that . The Dirichlet situation (or equivalently ) and the Neumann situation (or equivalently ) are available.

Let us consider the following boundary value problem, in the sense of distributions: where is the unit outward normal to the boundary .

Set for any the Banach space endowed with the seminorm of , taking the Poincaré inequalities (6) into account, since any bounded Lipschitz domain has the cone property. Here stands for the -Lebesgue measure. Also stands for the Lebesgue measure of a set of . The significance of depends on the kind of the set.

Defining the norm by with being any of the Poincaré constants, where and means the integral average over the set of positive measure, the Sobolev and trace inequalities read Hence further we call (7) the Sobolev inequality and for the general situation the -Sobolev inequality. Analogously, the trace inequality may be stated. For , and are the critical Sobolev and trace exponents such that they correspond, respectively, to and . For , the best constants of the Sobolev and trace inequalities are, respectively, (for smooth functions that decay at infinity, see [35, 36]) We observe that is arbitrary if . Here stands for the gamma function. Set by the volume of the unit ball of ; that is, and if is even, and if is odd. Moreover, the relationship holds true, where denotes the area of the unit sphere .

For , from the fundamental theorem of calculus applied to each of the variables separately, it follows that We emphasize that the above explicit constant is not sharp, since there exists the limit constant [35].

Definition 1. One says that is weak solution to (1)–(3), if it verifies a.e. on , and where , , and , with ; that is, if and any if , , with if and any if , and satisfies a.e. in .

Since is bounded, we have that , where , for every . We emphasize that the existence of equivalence between the strong (1)–(3) and weak (11) formulations is only available under sufficient data. For instance, the Green formula may be applied if and .

3. Some Constants ()

The presented results in this section are valid whether is a matrix or a function such that it obeys the measurable and boundedness properties. We emphasize that in the matrix situation , under the Einstein summation convention. Here we restrict ourselves to the function situation for the sake of simplicity.

3.1. Solvability

We recall the existence result in the Hilbert space in order to express its explicit constants in the following propositions, namely, Propositions 2 and 3 corresponding to the mixed and the Neumann problems, respectively.

Proposition 2. If , then there exists being a weak solution to (1)–(3). If , then is unique. Letting as an extension of (i.e., it is such that a.e. on ), the following estimate holds: where if , if , and if . In particular, is unique.

Proof. For there exists an extension such that a.e. on . The existence and uniqueness of a weak solution are well known via the Lax-Milgram lemma, to the variational problem: for all . Therefore, the required solution is given by .If , and then .Taking as a test function in (13), applying the Hölder inequality, and using the lower and upper bounds of , we obtain For , this inequality reads implying (12).Consider the case of dimension . For , using the Hölder inequality in (10) if , in (7) if , and in (8) for any , we have This concludes the proof of Proposition 2.

Proposition 3 (Neumann). If , then there exists a unique being a weak solution to (1)–(3). Moreover, the following estimate holds: where is given as in Proposition 2.

Proof. The existence and uniqueness of a weak solution are consequence of the Lax-Milgram lemma (see Remark 4). Estimate (17) follows the same argument used to prove (12).

Remark 4. The meaning of the Neumann solution in Proposition 3 should be understood as solving (11) for all or solving (11) for all .

3.2. Solvability

The existence of a solution is recalled in the following proposition in accordance with theory, that is, via solutions obtained by limit approximation (SOLA) (cf. [1–4, 37]), in order to determine the explicit constants.

Proposition 5. Let on (possibly empty) and let , , , and satisfy a.e. in . For any there exists solving (11) for every . Moreover, one has the following estimate: with if , if , and where is explicitly given in (30).

Proof. For each , take Applying Propositions 2 and 3, there exists a unique solution to the following variational problem: In particular, (21) holds for all ().In order to pass to limit (21) on    let us establish the estimate (18) for . The method for estimating is due to Boccardo and Gallouët (see, e.g., [1, 37]).Case . Let us choose as a test function in (21). Hence it follows that and consequently By the Hölder inequality with exponents and , we have Set Let us choose such that which is possible since ; that is, . Then, gathering the above two inequalities and inserting (7) for with , we deduce using the Young inequality , for , , and such that , with , and if .For , is chosen such that which is possible since ; that is, . Using the above Young inequality with , we find Let us choose, for instance, , and . Then, we obtain where is given by as . Hence, we find (18) with .Case . We choose, for , as a test function in (21). Since a.e. in , it follows that Then, we argue as in the above case, concluding (18) with .For both cases, we can extract a subsequence of , still denoted by , such that it weakly converges to in , where solves the limit problem (11) for all .

Remark 6. In terms of Proposition 5, the terms on the right-hand side of (11) have sense, since for , that is, .

Remark 7. The existence of a solution, which is given at Proposition 5, is in fact unique for the class of SOLA solutions (cf. [1–3]). By the uniqueness of solution in the Hilbert space, this unique SOLA solution is the weak solution of , if the data belong to the convenient Hilbert spaces.

Finally, we state the following version of Proposition 5, which will be required in Section 5, with datum belonging to the space of all signed measures with finite total variation .

Proposition 8. Let on (possibly empty), let satisfy a.e. in , and, for each , let be the Dirac delta function. For any there exists solving for every . Moreover, one has the following estimate: where the constants , , and are determined in Proposition 5.

Proof. Since the Dirac delta function can be approximated by a sequence such that identity (21) holds, with being replaced by , in , and on , for all and in particular for all . Then, we may proceed by using the argument already used in the proof of Proposition 5, with and , to conclude (34).

4. Constants

In this section, we establish some maximum principles, by recourse to the Stampacchia technique [13], via the analysis of the decay of the level sets of the solution. We begin by deriving the explicit estimates in the mixed case .

Proposition 9. Let , , and be any weak solution to (1)–(3) in accordance with Definition 1. If , , , and , then one has where if and .

Proof. Let . Choosing      as a test function in (11), then , and we deduce where . Using the Hölder inequality, it follows that Making use of (7)-(8) and with and the Hölder inequality, we get if provided by . Inserting (38)-(39) into (37) we obtain where the positive constant is Taking into account that when , we find Case . Take in (42). Making use of (7) and with and inserting (40), we deduce Therefore, we conclude where if and only if . By appealing to [13, Lemma 4.1] we obtain This means that the essential supremum does not exceed the well-determined constant .Case . Choose in (42). Using (10) for followed by the Hölder inequality and inserting (40), we obtain Therefore, we find where if and only if . Then, (36) holds by appealing to [13, Lemma 4.1] as in the anterior case ().This completes the proof of Proposition 9.

Remark 10. The Dirichlet problem studied by Stampacchia in [13] coincides with (1)–(3), with , , and .

Let us extend Proposition 9 up to the boundary.

Proposition 11. Under the conditions of Proposition 9, any weak solution to (1)–(3) satisfies, for if , with . For , , and , then any weak solution to (1)–(3) satisfies

Proof. Let . For each , , , and , (40) reads where . With this definition, the integral from the proof of Proposition 9 reads and, for , we have Case . Take . Making use of (7)-(8) and with , we deduce Since there exist different exponents and our objective is to find one , we apply (50) twice ( and ), obtaining Therefore, we conclude where if and only if . Notice that Case . Using (7) with , (8) with , and the Hölder inequality, we have Thus, we deduce Applying (50) twice ( and ), we conclude where if and only if .Finally, we find (48)-(49) by appealing to [13, Lemma 4.1] similarly as to obtain (36).

Next, let us state the explicit local estimates. The Caccioppoli inequality (60) coincides with the interior Caccioppoli inequality whenever and denotes a cut-off function, and it corresponds to [13, Lemma 5.2] if the lower bound of is related with its upper bound by .

Proposition 12. Let , , in , , respectively, in , on , and on , and let be the unique weak solution to (1)–(3) in accordance with Proposition 2. Then one has the following. (1) The Caccioppoli inequality is shown as  for any .(2) For arbitrary , , and ,  where and for any .

Proof. Let us choose as a test function in (11). Thus, applying the Hölder inequality we deduce Then, using the upper and lower bounds of , we conclude (60). Let be fixed but arbitrary. Arguing as in Proposition 9, let , and with the definition of the set , property (42) is still valid. In particular, we have, for , Fix , and let us take as a test function in (11), where is the cut-off function defined by in , in , and for all . Thus, we have that in and a.e. in and that (60) reads Making use of (7) and with exponent and the Hölder inequality, we have Applying the properties of , inserting (64) into (65), and gathering the second inequality from (63), we get In order to apply [13, Lemma 5.1] that leads to with , , and , we use (66) and inequality (63) with replaced by , obtaining Then, taking and , (61) holds.Therefore, the proof of Proposition 12 is finished.

Remark 13. The cut-off function explicitly given in Proposition 12 does not belong to .Let us prove the corresponding Neumann version of Proposition 12.

Proposition 14. Let , , in , , respectively, in and on , and let be the unique weak solution to (1)–(3) in accordance with Proposition 3. For arbitrary , , and , then (61) holds with .

Proof. Fix , , and as arbitrary. Arguing as in Proposition 12, (64) is true by taking   −   or as a test function in (11) and observing that .Applying the properties of , the -Sobolev inequality for with exponent , and the Hölder inequality, we have Considering and denoting the new constant by the same symbol , we may proceed as in the proof of Proposition 12. Thus, the proof of Proposition 14 is complete, taking into account.

Remark 15. The set is open and bounded but may be neither convex nor connected (see Figure 1).

Figure 1: 2D schematic representations of a Lipschitz domain (with ) representing an electrolytic cell, being the union of the recipient and air contact boundaries, representing the surface of two electrodes submerged in the electrolyte, and denoting the subset centered at three different points.

Finally, we state the following local version that will be required in Section 5. Here the boundary conditions do not play any role, since one can localize the problem around any point by multiplying with a suitable cut-off function and paying for this by a modified variational formulation.

Proposition 16. Let , satisfy a.e. in , , and let be such that . If solves the local variational formulation then one has

Proof. First we argue as in Proposition 12, with . The validity of properties (63) and (64) remains. The application of the -Sobolev inequality is available for . Thus, we conclude the proof of Proposition 16 as in the proof of Proposition 14.

5. Green Kernels

In this section, we reformulate some properties of the Green kernels.

Definition 17. For each , one says that is a Green kernel associated with (1)–(3), if it solves where is the Dirac delta function at the point , in the following sense: there is such that verifies the variational formulation: If , we call it the Green function; otherwise we call it simply the Neumann function (also called Green function for the Neumann problem or Green function of the second kind), and we write and , respectively.

The existence of the Green function verifying is standard if (see, e.g., [32, 34]), with being the unique solution to for all , for any , and for such that . Moreover, satisfies, for some positive constant and [32, Theorem 1.1], In order to provide explicit estimates and to simultaneously extend to and a mixed boundary value problem, let us build the Green kernels for .

Proposition 18. Let and , and let be a measurable (and bounded) function defined in satisfying . Then, for each and any such that , there exists a unique Green function according to Definition 17 and enjoying the following estimates: with and the constants and being explicitly given in Proposition 5. Moreover, a.e. , and for a.e. such that , where

Proof. For any and such that , the existence and uniqueness of solving (75), for all , are due to Proposition 2 with a.e. in , a.e. on, respectively, and , and belonging to if and to if . Moreover, (12) reads Therefore, for any such that , there exists such that In order for to correspond to the well defined one in (74), the estimate (77) is true for due to (18) with , by applying Proposition 5 with , , and . Then, we can extract a subsequence of , still denoted by , weakly converging to in as tends to , with solving (73) for all . A well-known property of passage to the weak limit implies (77). Estimate (78) is consequence of the Sobolev embedding with continuity constant given in (7).In order to prove the nonnegativeness assertion, first calculate Then, , and, by passing to the limit as tends to , the nonnegativeness claim holds.For each such that , we may take such that verifies in . Applying (71), followed by the Hölder inequality since means , we obtain with .Hence, using (78) we conclude (79), which completes the proof of Proposition 18.

Remark 19. Since as and as , the integrability exponent of in (78) obeys . In conclusion, Proposition 18 ensures that for any .

For each , the Neumann function is defined as being the solution of the regularity problem [33, Definition 2.5], where is the Green function solving (75) and , with mean value zero over , is the unique solution to the variational formulation [33, Lemma 2.3]: Here, is the -harmonic measure [38]; that is, it is unique probability measure on such that due to the Riesz representation theorem applied to the continuous linear functional , where is the solution to the Dirichlet problem (1) with and and (3) with . The question of solvability of the regularity problem is assigned by the gradient of the solution having nontangential limits at almost every point of the boundary [33, 39].

Remark 20. For each , admits an extension across (cf. [33, Lemmas 2.9 and 2.11]) to the domain which is such that where is the homothety function that reduces into its half, that is, the homothetic boundary with measure . That is, each is the reflection of across in the following sense: where is such that , for some .

Since our concern is on weak solutions to (1)–(3) in accordance with Definition 1, we reformulate for the existence result due to Kenig and Pipher on solutions to the Neumann problem in bounded Lipschitz domains if , with no information about its boundary behavior.

Proposition 21. Let and , and let be a measurable (and bounded) function defined in satisfying . Then, for each , there exists a Neumann function solving (72) that satisfies (77)-(78) and (79), with .

Proof. For each and such that , the existence of a unique Neumann function solving (75), for all , is consequence of Proposition 3 with , , and for if and any if . Arguing as in the proof of Proposition 18, belongs to , uniformly for , according to (18) with . Therefore, we may pass to the limit as , finding , solving (72). The remaining estimates (78)-(79), under , are obtained exactly as in the proof of Proposition 14.

Hereafter, denotes the partial derivative .

Proposition 22. Let and , and let be the symmetric function that is either the Green function or the Neumann function in accordance with Propositions 18 and 21, respectively. If verifies a.e. in , then, for every ,   is uniformly bounded for . In particular, it satisfies (77)-(78) and (79), where , with if and if .

Proof. For each , we may approximate by , where solves (75) for every such that . Since is a Dirac delta function, Proposition 8 ensures that verifies (77) and also (78) by the Sobolev inequality (7), with , where if and if . Consequently, (77)-(78) hold, by passage to the weak limit.To prove estimate (79) for , let us take such that . Thus, verifies in , for every . Therefore, we proceed by using the argument already used in the proof of Proposition 18, with being replaced by .

Remark 23. Notice that implies that is not an admissible test function in for each and for every , which comes from Definition 17, that is, due to differentiation of (73) under the integral sign in . We emphasize that, for each and any such that , the symmetric function verifies, by construction, the limit system of identities for any such that , and for all     .

Next, we prove additional estimates for the derivative of the weak solution to (1) with and , if we strengthen the hypotheses on the regularity of the coefficient . Indeed we proceed as in [32] where the coefficient is assumed Dini-continuous to allow the derivation of a few more pointwise estimates for the derivative of the Green kernels.

Proposition 24. Let satisfy a.e. in . If there exists a function such that, a.e. , then for each and , any function solving in the sense of distributions, enjoys, a.e. , where for some and with .

Proof. By density, since there exists a sequence such that in . In particular, in and a.e. in . Thus, it is sufficient to prove estimate (93), under the assumption .Fix and . For an arbitrary we can choose and such that for some constant . Since is bounded, we can take and define as in (94). Notice that implies .In order to determine the final constant in (93), let be the cut-off function explicitly given by Thus, satisfies , For , we multiply (92) by where is the fundamental solution of Laplace equation: with and we integrate over to get taking into account the use of integration by parts. Differentiating the above identity with respect to and setting it results in where Using the lower bound of , the definition of , and the properties of , we have By appealing to (95), we obtain Considering that, for all and , we obtain Let us analyze the first integral of the right-hand side in (106). From the definition of the radius , we consider two different cases: and otherwise. In the first case, from we have . Hence, we find and consequently If and , clearly (107) holds denoting .Returning to (106), substituting the value of from (99) with , and dividing by , we write it as In an -dimensional Euclidean space, the spherical coordinate system consists of a radial coordinate and angular coordinates and , and the Cartesian coordinates are , , and . Since the Jacobian of this transformation is and applying (91), we deduce Inserting this last inequality into (108), we find (93).

Remark 25. Observing (110), assumption (91) can be replaced by belonging to the VMO space of vanishing mean oscillation functions which is constituted by the functions belonging to the BMO space that verify where ranges in the class of the balls with radius contained in . We recall that the John-Nirenberg space BMO of the functions of bounded mean oscillation is defined as where ranges in the class of the balls contained in .

Remark 26. The upper bound in (93) is not optimal; it depends on the choice of the cut-off function through the constants and (cf. (97) and (108)).

Proposition 27. Let and , and let be the symmetric function that is either the Green function or the Neumann function in accordance with Propositions 18 and 21, respectively. If satisfies a.e. in and (91), then a.e. , with where if , if , and the constants and are explicitly given in Proposition 5.

Proof. Let be arbitrary. Using property (93) and applying (79), we get with Considering that, for all and with , we compute where the Riesz potential is calculated by the spherical transformation as in the above proof. Next, from we find (113).

6. Constants ()

Let , on (possibly empty), and solve (11) for all . Its existence depends on several factors.

The regularity theory for solutions of the class of divergence form elliptic equations in convex domains guarantees the existence of a unique strong solution if the coefficient is uniformly continuous, taking the Korn perturbation method [10, pp. 107–109] into account. This result can be proved if the convexity of is replaced by weaker assumptions, for instance, when is a plane bounded domain with Lipschitz and piecewise boundary whose angles are all convex [10, page 151] or when is a plane bounded domain with curvilinear polygonal boundary whose angles are all strictly convex [10, page 174]. For general bounded domains with Lipschitz boundary, the higher integrability of the exponents for the gradients of the solutions may be assured [14, 28], under particular restrictions on the coefficients. In [19, 23], the authors figure out configurations of (discontinuous) coefficient functions and geometries of the domain, such that the required result does hold. In [24], the authors derive global and piecewise estimates with piecewise Hölder continuous coefficients, which depend on the shape and on the size of the surfaces of discontinuity of the coefficients, but they are independent of the distance between these surfaces. When the coefficient of the principal part of the divergence form elliptic equation is only supposed to be bounded and measurable, Meyers extends Boyarskii result to -dimensional elliptic equations of divergence structure [25]. Adopting this rather weak hypothesis, the works [21, 22, 26] extend the regularity result for the Dirichlet problem due to Meyers to a similar one for the mixed boundary value problem.

For a domain of class , regularity of the solution is found for in [17, 27] under the hypotheses that the coefficients of the principal part are to belong to the Sarason class [40] of vanishing mean oscillation functions (VMO). In [20], the author extends the solvability to the Neumann problem for a range of integrability exponent , where depends on , the ellipticity constant, and the Lipschitz character of . Notwithstanding, the results concerning VMO coefficients are irrelevant for real-world applications. The reason is that the VMO property forbids jumps across a hypersurface, which is the generic case of discontinuity.

For Lipschitz domains with small Lipschitz constant, the Neumann problem is solved in [18], where the leading coefficient is assumed to be measurable in one direction, to have small BMO seminorm in the other directions, and to have small BMO seminorm in a neighborhood of the boundary of the domain. We refer to [15] for the optimal regularity theory regarding Dirichlet problem on bounded domains whose boundary is so rough that the unit normal vector is not well defined but is well approximated by hyperplanes at every point and at every scale (Reifenberg flat domain); and the coefficient belongs to the space such that which is defined as the BMO space with their BMO seminorms being sufficiently small. In [16] the authors obtain the global regularity theory to a linear elliptic equation in divergence form with the conormal boundary condition via perturbation theory in harmonic analysis and geometric measure theory, in particular on maximal function approach.

Let us begin by establishing the relation between any weak solution and the Green kernel associated with (1)–(3); that is, is either the Green or the Neumann functions, and , in accordance with Propositions 18 and 21, respectively. To this end, we take and as test functions in (11) and (73), respectively, obtaining the Green representation formula: where , , and are the layer potential operators defined by

For every , , and , with and , the Hardy-Littlewood-Sobolev inequality in its general form states the following: where the constant is sharp [41], if , defined by

In the presence of the Hardy-Littlewood-Sobolev inequality, we prove the following estimate.

Proposition 28. Let , with , in , on (possibly empty), on , and let satisfy a.e. in and (91). If solves (11), for all , then satisfies with , relative to (121), and determined in Proposition 27. In particular, for we have

Proof. Since , (119) holds. Differentiating it, for , we deduce Let be arbitrary such that . Using (113) for any and applying the Fubini-Tonelli theorem, we find Next, using (121) with ,  , and , we conclude (123).For the particular situation, we choose and we use (121) with .

Having the results established in Section 5 in mind, we find a estimate for weak solutions where regularity (91) of the leading coefficient is not a necessary condition.

Proposition 29. Let , , , on (possibly empty), , satisfy a.e. in , and let solve (11), for all . Then satisfies with the constants and being explicitly given in Proposition 5.

Proof. Differentiating (119), for , we deduce Let be arbitrary such that ; applying the Fubini-Tonelli theorem and next the Hölder inequality, it follows Let us estimate the last integral on the right-hand side in (129), since the two other integrals are similarly bounded: where is due to the embedding . Considering that uniformly for (cf. Proposition 22) and consequently also uniformly in , then we obtain Finally, inserting the above inequality into (129), the proof of Proposition 29 is finished.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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