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 coefficient. Our concern is to estimate the solutions with explicit constants, for domains in $\mathbb{R}^n$ ($n\geq 2$) of class $C^{0,1}$. The existence of $L^\infty$ and $W^{1,q}$-estimates is assured for $q=2$ and any $q<n/(n-1)$ (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative $L^\infty$-estimates is based on the DeGiorgi technique developed by Stampacchia. By using the potential theory, we derive $W^{1,p}$-estimates for different ranges of the exponent $p$ 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.


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 H 1 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 W 1,q (q < n/(n − 1)) estimative constants involving L 1 and measure data, via the technique of solutions obtained by limit approximation (SOLA) (cf. [4,10,13,35]).
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, 15-17, 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 L ∞ -estimates.
It is known that the information 'The gradient of a quantity belongs to a L p space with p 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 a(x, T ) = a(x, T (x)) ∈ L ∞ (Ω), where T is a known function, usually the temperature function, such as the electrical conductivity in the thermoelectric [8,9] 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 p > 2, as long as arbitrary elliptic L ∞ -coefficients are admissable [17]. Having this in mind, in Section 6 we derive W 1,p -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 [25,27,31]).
Let us consider the following boundary value problem, in the sense of distributions, − ∇ · (a∇u) = f − ∇ · f in Ω; (1) (a∇u − f) · n = h on Γ; where n is the unit outward normal to the boundary ∂Ω.
Set for any q ≥ 1 the Banach space endowed with the seminorm of W 1,q (Ω), taking the Poincaré inequalities (4)-(5) into account, since any bounded Lipschitz domain has the cone property.
The presented results in this Section are valid whether a is a matrix or a function such that obeys the measurable and boundedness properties. We emphasize that in the matrix situation a∇u · ∇v = a ij ∂ i u∂ j v, under the Einstein summation convention.
Here we restrict to the function situation for the sake of simplicity.
3.1. H 1 -solvability. We recall the existence result in the Hilbert space H 1 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 |Γ D | > 0, then there exists u ∈ H 1 (Ω) being a weak solution to (1)-(3). If g = 0, then u is unique. Letting g ∈ H 1 (Ω) as an extension of g ∈ L 2 (Γ D ), i.e. it is such that g = g a.e. on Γ D , the following estimate holds ∇u 2,Ω ≤ (a # /a # + 1) ∇ g 2,Ω + (10) For g ∈ L 2 (Γ D ) there exists an extension g ∈ H 1 (Ω) such that g = g a.e. on Γ D . The existence and uniqueness of a weak solution w ∈ H 1 Γ D (Ω) is well-known via the Lax-Milgram Lemma, to the variational problem . Therefore, the required solution is given by u = w + g. If g = 0, g = 0 and then u ≡ w.
Consider the case of dimension n = 2. For t, s > 1, using the Hölder inequality in (8) if t ′ ≤ 2, in (6) if t ′ > 2, and in (7) for any s > 1, we have This concludes the proof of Proposition 3.1.
Proposition 3.2 (Neumann). If |Γ D | = 0, then there exists a unique u ∈ V 2 being a weak solution to (1)-(3). Moreover, the following estimate holds where C n (A, B) is given as in Proposition 3.1.
Proof. The existence and uniqueness of a weak solution u ∈ V 2 is consequence of the Lax-Milgram Lemma (see Remark 3.1). The estimate (12) follows the same argument used to prove (10).
Proof. For each m ∈ N, take Applying Propositions 3.1 and 3.2, there exists a unique solution u m ∈ V 2 to the following variational problem In particular, (14) holds for all v ∈ V q ′ (q ′ > n).
In order to pass to the limit (14) on m (m → ∞) let us establish the estimate (13) for ∇u m .
For n = 2, s > 0 is chosen such that (s + 1)q/(2 − q) < q * = 2q/(2 − q) which is possible since 1 ≤ q < 2, that is s < 1. Using the above Young inequality with a = 2/(s + 1), we find Let us choose, for instance, s = 2 − q < 1, and ǫ = [2S q M(s)] −2q/(2−q) . Then, we obtain where ℓ is given by , as a test function in (14). Since |v| ≤ 2 a.e. in Ω, it follows that Then, we argue as in the above case, concluding (13) with κ = 4. For both cases, we can extract a subsequence of u m , still denoted by u m , such that it weakly converges to u in W 1,q (Ω), where u ∈ V q solves the limit problem (9) for all v ∈ V q ′ .
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. [4,10,13]). By the uniqueness of solution in the Hilbert space, this unique SOLA solution is the weak solution of V 2 , if the data belong to the convenient L 2 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 M(Ω) = (C 0 (Ω)) ′ .
in Ω, and for each x ∈ Ω, δ x ∈ M(Ω) be the Dirac delta function. For for every i = 1, · · · , n. Moreover, we have the following estimate where the constants C 1 (Ω, n, q), C 2 (n, q, A), and κ are determined in Proposition 3.3.
Proof. Since the Dirac delta function δ x ∈ M(Ω) can be approximated by a sequence the identity (14) holds, with f replaced by f m , f = 0 in Ω, and h = 0 on Γ, for all v ∈ V 2 and in particular for all v ∈ C 0 (Ω) ∩ V 1 . Then, we may proceed by using the argument already used in the proof of Proposition 3.3, with f 1,Ω = 1, and f 1,Ω = h 1,Γ = 0, to conclude (17).

L ∞ -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 |Γ D | > 0.
Then, using the upper and lower bounds of a, we conclude (26).
Applying the properties of η, the W 1,q -Sobolev inequality for η(|u| − k) + ∈ W 1,q (Ω) with exponent q = 2n/(n + 2) < 2 ≤ n, and the Hölder inequality, we have  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 4.5. Let n ≥ 2, a ∈ L ∞ (Ω) satisfies 0 < a # ≤ a ≤ a # a.e. in Ω, x ∈ Ω, and R > 0 be such that |Ω ∩ ∂B R (x)| > 0. If u ∈ H 1 (Ω(x, R)) solves the local variational formulation Proof. First we argue as in Proposition 4.3, with k 0 = 0. The validity of the properties (28) and (29) remain. The application of the W 1,2n/(n+2) -Sobolev inequality is available for η(|u| − k) + ∈ W 1,2n/(n+2) (A(x, R)). Thus, we conclude the proof of Proposition 4.5 as in the proof of Proposition 4.4.

Green kernels
In this Section, we reformulate some properties of the Green kernels. where δ x is the Dirac delta function at the point x, in the following sense: there is q > 1 such that E verifies the variational formulation If |Γ D | > 0, 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 E = G and E = N, respectively.
In order to prove the nonnegativeness assertion, first calculate Then, G ρ = |G ρ |, and by passing to the limit as ρ tends to 0, the nonnegativeness claim holds.
Since our concern is on weak solutions to (1)-(3) in accordance with Definition 2.1, we reformulate for n ≥ 2 the existence result due to Kenig and Pipher on solutions to the Neumann problem in bounded Lipschitz domains if n > 2, with no information of its boundary behavior.
Hereafter, ∂ x i denotes the partial derivative ∂/∂x i . Proposition 5.3. Let n ≥ 2, 1 ≤ q < n/(n − 1), E be the symmetric function that is either the Green function G or the Neumann function N in accordance with Propositions 5.1 and 5.2, respectively. If a ∈ L ∞ (Ω) verifies 0 < a # ≤ a ≤ a # a.e.
Remark 5.3. Notice that q ′ > n implies that E is not an admissible test function in for each x ∈ Ω, and for every i = 1, · · · , n, which comes from Definition 5.1, i.e. due to differentiate (34) under the integral sign in x i . We emphasize that for each x ∈ Ω and any r > 0 such that r < dist(x, ∂Ω), the symmetric function E(x, ·) ∈ V q ∩ H 1 (Ω \ B r (x)) verifies, by construction, the limit system of identities Next, we prove additional estimates for the derivative of the weak solution to (1) with f = 0 and f = 0, if we strengthen the hypotheses on the regularity of the coefficient a. Indeed we proceed as in [25] where the coefficient is assumed Dini-continuous to be enable to derive some more pointwise estimates for the derivative of the Green kernels.
Proof. By density, since u ∈ W 1,1 (Ω) there exists a sequence {u m } m∈N ⊂ C 1 (Ω) such that u m → u in W 1,1 (Ω). In particular, u m → u in L 1 (Ω) and ∇u m → ∇u a.e. in Ω. Thus, it is sufficient to prove the estimate (44), under the assumption u ∈ C 1 (Ω). Fix x ∈ Ω, and R > 0. For an arbitrary y ∈ Ω(x, R) we can choose 0 < r < min{R, 1, δ(y)} and M > 0 such that In order to determine the final constant in (44), let η ∈ C 1 0 (B d (y)) ∩ W 2,∞ (Ω) be the cut-off function explicitly given by Thus, η satisfies 0 ≤ η ≤ 1, For w ∈ B := B d (y), we multiply (43) by G L (w, ·)η/a(y) where G L is the fundamental solution of Laplace equation, and we integrate over B to get taking into account the use of integration by parts. Differentiating the above identity with respect to w and setting w = y it results in ∇u(y) = I 1 + I 2 , where Using the lower bound of a, the definition of G L , and the properties of η, we have c 2 ν |y − z| n−1 |x − y| 2 + 2(n + 1) |y − z| n |x − y| dz.
By appealing to (46), we obtain Considering that, for all x, y ∈ Ω and z ∈ B d (y), Let us analyze the first integral of RHS in (50). From the definition of the radius d, we consider two different cases: |x − y| = νr and otherwise. In the first case, from z ∈ B d (y) we have |y − z| < |x − y|/ν. Hence, we find (ν − 1)|y − z| < |x − z| and consequently If d = |x − y|/2 and z ∈ B d (y), clearly (51) holds denoting ν = 2.
Proposition 5.5. Let n ≥ 2, 1 ≤ q < n/(n − 1), E be the symmetric function that is either the Green function G or the Neumann function N in accordance with Propositions 5.1 and 5.2, respectively. If a ∈ L ∞ (Ω) satisfies 0 < a # ≤ a ≤ a # a.e.
Considering that, for all x, y ∈ Ω and z ∈ B d (y) with d ≤ |x − y|/2, |x − y| −n/q |y − z| n−1 dz = 2 n/q dnω n |x − y| −n/q , where the Riesz potential is calculated by the spherical transformation as in the above proof. Next, from d ≤ |x − y|/2 we find (54).
6. W 1,p -constants (p > n) Let p > n, g = 0 on Γ D (possibly empty), and u ∈ V p solve (9) for all v ∈ V p ′ . 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 [22, pp. 107-109] into account. This result can be proved if the convexity of Ω is replaced by weaker assumptions, for instance when Ω is a plane bounded domain with Lipschitz and piecewise C 2 boundary whose angles are all convex [22, p. 151], or when Ω is a plane bounded domain with curvilinear polygonal C 1,1 boundary whose angles are all strictly convex [22, p. 174]. For general bounded domains with Lipschitz boundary, the higher integrability of the exponents for the gradients of the solutions may be assured [2,39], under particular restrictions on the coefficients. In [17,26], the authors figure out configurations of (discontinuous) coefficient functions and geometries of the domain, such that the required result does hold. In [29], the authors derive global W 1,∞ and piecewise C 1,α 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 n-dimensional elliptic equations of divergence structure [32]. Adopting this rather weak hypothesis, the works [23,24,34] extend to mixed boundary value problem the result due to Meyers.
For a domain of class C 1,1 , W 1,p -regularity of the solution is found for 1 < p < ∞ in [15,36] under the hypotheses on the coefficients of the principal part are to belong to the Sarason class [37] of vanishing mean oscillation functions (VMO). In [19], the author extends the W 1,p -solvability to the Neumann problem for a range of integrability exponent p ∈]2n/(n + 1) − ε, 2n/(n − 1) + ε[, where ε > 0 depends on n, the ellipticity constant, and the Lipschitz character of Ω. Notwithstanding, the results concerning VMO-coefficients are irrevelant for real world applications. The reason is that the VMO-property forbids jumps across a hypersurface, what is the generic case of discontinuity.
For Lipschitz domains with small Lipschitz constant, the Neumann problem is solved in [16], where the leading coefficient is assumed to be measurable in one direction, to have small BMO semi-norm in the other directions, and to have small BMO semi-norm in a neighborhood of the boundary of the domain. We refer to [6] for the optimal W 1,p 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 V such that C(Ω) ⊂ VMO ⊂ V ⊂ BMO which is defined as the BMO space with their BMO semi-norms sufficiently small. In [7] the authors obtain the global W 1,p regularity theory 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.
Having the results established in Section 5 in mind, we find a W 1,p -estimate for weak solutions where the regularity (42) of the leading coefficient is not a necessary condition.
Finally, inserting the above inequality into (58), the proof of Proposition 6.2 is finished.