Explicit estimates in mixed elliptic problems

Explicit estimates for solutions of mixed elliptic problems

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 coefficient. 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 DeGiorgi technique developed by Stampacchia. By using the potential theory, we derive -estimates for different ranges of the exponent depending on 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.

Key words and phrases:
elliptic equation; -theory; potential theory; regularity
2010 Mathematics Subject Classification:
35J25, 35D30, 35B50, 35C15, 35J08, 35B65

1. Introduction

The knowledge of the data makes all the difference on the real world applications of boundary value problems. Quantitative estimates are of extremely importance in any other area of science such as engineering, biology, geology, 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 belongs to a bounded set of a 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 explicit 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. [4, 13, 10, 35]).

Dirichlet, Neumann, and mixed problems with respect to uniformly elliptic equation in divergence form is widely investigated in the literature (see [1, 3, 14, 20, 21, 22, 28, 33, 38] and the references therein) when the leading coefficient is a function 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 [2, 6, 7, 15, 16, 17, 19, 23, 24, 26, 29, 32, 34, 36, 39] (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 carry out. Since the smoothness of the solution is invalidated by the nonsmoothness of the coefficient and the domain, Section 4 is devoted to the direct derivation of global and local -estimates.

It is known that the information ’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 coefficient , where is a known function, usually the temperature function, such as the electrical conductivity in the thermoelectric [9, 8] and thermoelectrochemical [11] 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 admissable [17]. 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 in some techniques from [25, 31, 27]).

2. Statement of the problem

Let be a domain (that is, 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,

(1) in
(2) on
(3) on

where is the unit outward normal to the boundary .

Set for any

the Banach space endowed with the seminorm of , taking the Poincaré inequalities (4)-(5) 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 anyone of the Poincaré constants

(4)
(5)

where , and means the integral average over the set of positive measure, the Sobolev and trace inequalities read

(6)
(7)

Hence further we call (6) 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 correspond, respectively, to and . For , the best constants of the Sobolev and trace inequalities are, respectively, (for smooth functions that decay at infinity, see [40] and [5])

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

(8)

We emphasize that the above explicit constant is not sharp, since there exists the limit constant [40].

Definition 2.1.

We say that is weak solution to (1)-(3), if it verifies a.e. on , and

(9)

where , , , with , i.e. 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 (9) formulations is only available under sufficiently 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 obeys the measurable and boundedness properties. We emphasize that in the matrix situation , under the Einstein summation convention. Here we restrict 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 3.1 and 3.2 corresponding to the mixed and the Neumann problems, respectively.

Proposition 3.1.

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

(10)

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 is well-known via the Lax-Milgram Lemma, to the variational problem

(11)

for all . Therefore, the required solution is given by .

If , and then .

Taking as a test function in (11), applying the Hölder inequality, and using the lower and upper bounds of , we obtain

For , this inequality reads

implying (10).

Consider the case of dimension . For , using the Hölder inequality in (8) if , in (6) if , and in (7) for any , we have

This concludes the proof of Proposition 3.1. ∎

Proposition 3.2 (Neumann).

If , then there exists a unique being a weak solution to (1)-(3). Moreover, the following estimate holds

(12)

where is given as in Proposition 3.1.

Proof.

The existence and uniqueness of a weak solution is consequence of the Lax-Milgram Lemma (see Remark 3.1). The estimate (12) follows the same argument used to prove (10). ∎

Remark 3.1.

The meaning of the Neumann solution in Proposition 3.2 should be understood as solving (9) for all , or solving (9) for all .

3.2. -solvability ()

The existence of a solution is recalled in the following proposition in accordance to -theory, that is via solutions obtained by limit approximation (SOLA) (cf. [4, 13, 10, 35]), in order to determine the explicit constants.

Proposition 3.3.

Let on (possibly empty), , , , and satisfy a.e. in . For any there exists solving (9) for every . Moreover, we have the following estimate

(13)

with if , if , and

where is explicitly given in (16).

Proof.

For each , take

Applying Propositions 3.1 and 3.2, there exists a unique solution to the following variational problem

(14)

In particular, (14) holds for all ().

In order to pass to the limit (14) on let us establish the estimate (13) for .

Case . From -data theory (see, for instance, [35]), let us choose

as a test function in (14). Hence it follows that

and consequently

By the Hölder inequality with exponents and , we have

(15)

Set

Let us choose such that which is possible since , that is . Then, gathering the above two inequalities, and inserting (6) 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

(16)

as . Hence, we find (13) with .

Case . We choose, for ,

as a test function in (14). Since a.e. in , it follows that

Then, we argue as in the above case, concluding (13) 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 (9) for all . ∎

Remark 3.2.

In terms of Proposition 3.3, the terms on the right hand side of (9) have sense, since for that is, .

Remark 3.3.

The existence of a solution, which is given at Proposition 3.3, is in fact unique for the class of SOLA solutions (cf. [4, 13, 10]). 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 3.3, which will be required in Section 5, with datum belonging to the space of all signed measures with finite total variation .

Proposition 3.4.

Let on (possibly empty), satisfy a.e. in , and for each , be the Dirac delta function. For any there exists solving

for every . Moreover, we have the following estimate

(17)

where the constants , , and are determined in Proposition 3.3.

Proof.

Since the Dirac delta function can be approximated by a sequence such that

the identity (14) holds, with 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 3.3, with , and , to conclude (17). ∎

4. -constants

In this Section, we establish some maximum principles, by recourse to the De Giorgi technique [38], 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 4.1.

Let , , and be any weak solution to (1)-(3) in accordance with Definition 2.1. If , , , and , then we have

(18)

where if , and .

Proof.

Let . Choosing as a test function in (9), then , and we deduce

(19)

where . Using the Hölder inequality, it follows that

Making use of (6)-(7) and with , and the Hölder inequality, we get

if provided by . Inserting last three inequalities into (19) we obtain

(20)

where the positive constant is