Explicit Estimates for Solutions of Mixed Elliptic Problems
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.
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  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 . 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 .
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.
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.
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 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.
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).
In this section, we establish some maximum principles, by recourse to the Stampacchia technique , via the analysis of the decay of the level sets of the solution. We begin by deriving the explicit estimates in the mixed case .
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.
Let us extend Proposition 9 up to the boundary.
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.
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).
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 ,