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 ℝⁿ (n ≥ 2) of class C^0,1. The existence of L^∞ 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^∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W^1,p estimates for different ranges of the exponent p 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 H¹ 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 L¹ 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 L^∞ estimates.

It is known that the information that “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 coefficients 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 [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 p > 2, as long as arbitrary elliptic L^∞ coefficients are admissible [19]. 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 by some techniques from [32–34]).

2. Statement of the Problem

Let Ω be a domain (i.e., connected open set) in ℝⁿ (n ≥ 2) of class C^0,1 and bounded. Its boundary ∂Ω is constituted by two disjoint open (n − 1)-dimensional sets, Γ_D and Γ, such that . The Dirichlet situation Γ_D = ∂Ω (or equivalently Γ = ∅) and the Neumann situation Γ = ∂Ω (or equivalently Γ_D = ∅) are available.

Let us consider the following boundary value problem, in the sense of distributions:

()

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 (6) into account, since any bounded Lipschitz domain has the cone property. Here |·| stands for the (n − 1)-Lebesgue measure. Also |A| stands for the Lebesgue measure of a set A of ℝⁿ. The significance of |·| depends on the kind of the set.

Defining the W^1,q norm by

()

with C_* being any of the Poincaré constants,

()

where Σ ⊂ ∂Ω and ⨍_A means the integral average over the set A of positive measure, the Sobolev and trace inequalities read

()

Hence further we call (7) the Sobolev inequality and for the general situation the W^1,q-Sobolev inequality. Analogously, the trace inequality may be stated. For 1 ≤ q < n, q^* = qn/(n − q) and q_* = q(n − 1)/(n − q) are the critical Sobolev and trace exponents such that they correspond, respectively, to

and

. For 1 < q < n, the best constants of the Sobolev and trace inequalities are, respectively, (for smooth functions that decay at infinity, see [35, 36])

()

We observe that q^* > 1 is arbitrary if q = n. Here Γ stands for the gamma function. Set by ω_n the volume of the unit ball B₁(0) of ℝⁿ; that is, ω_n = π^n/2/Γ(n/2 + 1) and Γ(n/2 + 1) = (n/2)! if n is even, and Γ(n/2 + 1) = π^1/22^−(n+1)/2n(n − 2)(n − 4) ⋯ 1 if n is odd. Moreover, the relationship σ_n−1 = nω_n holds true, where σ_n−1 = 2π^n/2/Γ(n/2) denotes the area of the unit sphere ∂B₁(0).

For n > 1, from the fundamental theorem of calculus applied to each of the n variables separately, it follows that

()

We emphasize that the above explicit constant is not sharp, since there exists the limit constant S₁ = π^−1/2n⁻¹[Γ(1 + n/2)] ^1/n [35].

Definition 1. One says that u is weak solution to (1)–(3), if it verifies u = g a.e. on Γ_D, and

()

where g ∈ L²(Γ_D), f ∈ L²(Ω), and f ∈ L^t(Ω), with

; that is, t = 2n/(n + 2) if n > 2 and any t > 1 if n = 2, h ∈ L^s(Γ), with s = 2(n − 1)/n if n > 2 and any s > 1 if n = 2, and a ∈ L^∞(Ω) satisfies 0 < a_# ≤ a ≤ a^# a.e. in Ω.

Since Ω is bounded, we have that Ω ⊂ B_∂(Ω)(x), where δ(Ω): = diam⁡(Ω), for every x ∈ Ω. 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 a∇u ∈ L²(Ω) and ∇·(a∇u) ∈ L²(Ω).

3. Some W^1,q Constants (q ≤ 2)

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

3.1. H¹ Solvability

We recall the existence result in the Hilbert space H¹ 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 |Γ_D | > 0, then there exists u ∈ H¹(Ω) being a weak solution to (1)–(3). If g = 0, then u is unique. Letting as an extension of g ∈ L²(Γ_D) (i.e., it is such that a.e. on Γ_D), the following estimate holds:

()

where C_n(A, B) = S₂A + K₂B if n > 2,

if t < 2, and

if t ≥ 2. In particular,

is unique.

Proof. For g ∈ L²(Γ_D) there exists an extension such that a.e. on Γ_D. 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 g = 0, and then u ≡ w.

Taking as a test function in (13), applying the Hölder inequality, and using the lower and upper bounds of a, we obtain

()

For n > 2, this inequality reads

()

implying (12).

Consider the case of dimension n = 2. For t, s > 1, using the Hölder inequality in (10) if t^′ ≤ 2, in (7) if t^′ > 2, and in (8) for any s > 1, we have

()

This concludes the proof of Proposition 2.

Proposition 3 (Neumann). If |Γ_D | = 0, then there exists a unique u ∈ V₂ being a weak solution to (1)–(3). Moreover, the following estimate holds:

()

where C_n(A, B) is given as in Proposition 2.

Proof. The existence and uniqueness of a weak solution u ∈ V₂ 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 u ∈ V₂ in Proposition 3 should be understood as u ∈ V₂(∂Ω) solving (11) for all v ∈ V₂(∂Ω) or u ∈ V₂(Ω) solving (11) for all v ∈ V₂(Ω).

3.2. W^1,q Solvability (q ≤ n/(n − 1))

The existence of a solution is recalled in the following proposition in accordance with L¹ theory, that is, via solutions obtained by limit approximation (SOLA) (cf. [1–4, 37]), in order to determine the explicit constants.

Proposition 5. Let g = 0 on Γ_D (possibly empty) and let f ∈ L²(Ω), f ∈ L¹(Ω), h ∈ L¹(Γ), and a ∈ L^∞(Ω) satisfy 0 < a_# ≤ a ≤ a^# a.e. in Ω. For any 1 ≤ q < n/(n − 1) there exists u ∈ V_q solving (11) for every . Moreover, one has the following estimate:

()

with ϰ = 2 if |Γ_D | > 0, ϰ = 4 if |Γ_D | = 0, and

()

where ℓ∈]2, +∞[ is explicitly given in (30).

Proof. For each m ∈ ℕ, take

()

Applying Propositions 2 and 3, there exists a unique solution u_m ∈ V₂ to the following variational problem:

()

In particular, (21) holds for all

(q^′ > n).

In order to pass to limit (21) on m (m → ∞) let us establish the estimate (18) for ∇u_m. The method for estimating is due to Boccardo and Gallouët (see, e.g., [1, 37]).

Case (|Γ_D | > 0). Let us choose

()

as a test function in (21). Hence it follows that

()

and consequently

()

By the Hölder inequality with exponents 2/q and 2/(2 − q) > 1, we have

()

Set

()

Let us choose s > 0 such that (s + 1)q/(2 − q) = q^* = nq/(n − q) which is possible since 1 ≤ q < n/(n − 1); that is, s = (n + q − nq)/(n − q). Then, gathering the above two inequalities and inserting (7) for u_m ∈ V₂↪V_q with (q ≤ 2), we deduce

()

using the Young inequality AB ≤ ϵA^a/a + B^b/(bϵ^b/a), for A, B ≥ 0, ϵ > 0, and a, b > 1 such that 1/a + 1/b = 1, with ϵ = 1, and a = 2(n − q)/[n(2 − q)] if n > 2.

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 ϵ = [2S_qM(s)] ^{−2q/(2−q)}. Then, we obtain

()

where ℓ is given by

()

as q → 2⁻. Hence, we find (18) with ϰ = 2.

Case (|Γ_D | = 0). We choose, for s > 0,

()

as a test function in (21). Since |v | ≤ 2 a.e. in Ω, it follows that

()

Then, we argue as in the above case, concluding (18) 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 (11) for all .

Remark 6. In terms of Proposition 5, the terms on the right-hand side of (11) have sense, since for q^′ > n, that is, q < n/(n − 1).

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 V₂, if the data belong to the convenient L² 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 g = 0 on Γ_D (possibly empty), let a ∈ L^∞(Ω) satisfy 0 < a_# ≤ a ≤ a^# a.e. in Ω, and, for each x ∈ Ω, let δ_x ∈ ℳ(Ω) be the Dirac delta function. For any 1 ≤ q < n/(n − 1) there exists u ∈ V_q solving

()

for every i = 1, …, n. Moreover, one has the following estimate:

()

where the constants C₁(Ω, n, q), C₂(n, q, A), and ϰ are determined in Proposition 5.

Proof. Since the Dirac delta function δ_x ∈ ℳ(Ω) can be approximated by a sequence {f_m} _m∈ℕ ⊂ L^∞(Ω) such that

()

identity (21) holds, with f being replaced by f_m, f = 0 in Ω, and h = 0 on Γ, for all v ∈ V₂ and in particular for all v ∈ C₀(Ω)∩V₁. Then, we may proceed by using the argument already used in the proof of Proposition 5, with ∥f∥_1,Ω = 1 and ∥f∥_2,Ω = ∥h∥_1,Γ = 0, to conclude (34).

4. L^∞ 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 |Γ_D| > 0.

Proposition 9. Let p > n ≥ 2, |Γ_D | > 0, and u ∈ H¹(Ω) be any weak solution to (1)–(3) in accordance with Definition 1. If g ∈ L^∞(Γ_D), f ∈ L^p(Ω), f ∈ L^np/(p+n)(Ω), and h ∈ L^(n−1)p/n(Γ), then one has

()

where C_n = 2^{n(p−2)/[2(p−n)]}S₂ if n > 2 and C₂ = 2^{(3p−2)/[2(p−2)]}.

Proof. Let k ≥ k₀ = ess sup⁡{|g(x)| : x ∈ Γ_D}. Choosing v = sign⁡(u)(|u | − k) ⁺ = as a test function in (11), then ∇v = ∇u ∈ L²(A(k)), and we deduce

()

where A(k) = {x ∈ Ω:|u(x)| > k}. Using the Hölder inequality, it follows that

()

Making use of (7)-(8) and with q = p^′ < n and the Hölder inequality, we get

()

if provided by p^′ < 2 ≤ n. Inserting (38)-(39) into (37) we obtain

()

where the positive constant C_n,p is

()

Taking into account that A(h) ⊂ A(k) when h > k > k₀, we find

()

Case (n > 2). Take α = 2^* = 2n/(n − 2) in (42). Making use of (7) and with q = 2 and inserting (40), we deduce

()

Therefore, we conclude

()

where β > 1 if and only if p > n. By appealing to [13, Lemma 4.1] we obtain

()

This means that the essential supremum does not exceed the well-determined constant M∶ = k₀ + S₂C_n,p | Ω|^(β−1)/α2^β/(β−1).

Case (n = 2). Choose α = 2 in (42). Using (10) for followed by the Hölder inequality and inserting (40), we obtain

()

Therefore, we find

()

where β > 1 if and only if p > 2. Then, (36) holds by appealing to [13, Lemma 4.1] as in the anterior case (n > 2).

This completes the proof of Proposition 9.

Remark 10. The Dirichlet problem studied by Stampacchia in [13] coincides with (1)–(3), with Γ = ∅, f = g = 0, and n > 2.

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 p > 2(n − 1) if n > 2,

()

with 1/b = 1/p + (n − 2)/[2n(n − 1)]. For n = 2, p > 2α/(α − 1), and α > 1, then any weak solution

to (1)–(3) satisfies

()

Proof. Let k ≥ k₀ = ess sup⁡{|g(x)| : x ∈ Γ_D}. For each b > 2, f ∈ L^b(Ω), f ∈ L^nb/(b+n)(Ω), and h ∈ L^(n−1)b/n(Γ), (40) reads

()

where

. With this definition, the integral I from the proof of Proposition 9 reads

()

and, for h > k > k₀, we have

()

Case (n > 2). Take α = 2(n − 1)/(n − 2) < 2^* = 2n/(n − 2). Making use of (7)-(8) and with q = 2, we deduce

()

Since there exist different exponents and our objective is to find one β > 1, we apply (50) twice (1/b = 1/p + 1/α − 1/2^* < 1/n and b = p > 2(n − 1) > n > 2), obtaining

()

Therefore, we conclude

()

where β > 1 if and only if p > 2(n − 1). Notice that

()

Case (n = 2). Using (7) with q = 2α/(α + 2) < 2, (8) with q = 2α/(α + 1) < 2, and the Hölder inequality, we have

()

Thus, we deduce

()

Applying (50) twice (b = 2αp/(p + 2α) > 2 and b = p > 2α/(α − 1) > 2), we conclude

()

where β = α(1/(2α) + 1/2 − 1/p) > 1 if and only if p > 2α/(α − 1).

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 B_R(x)⊂⊂Ω and denotes a cut-off function, and it corresponds to [13, Lemma 5.2] if the lower bound of a is related with its upper bound by a_# = 1/a^#.

Proposition 12. Let n ≥ 2, |Γ_D | > 0, f = 0 in Ω, f, g, h = 0, respectively, in Ω, on Γ_D, and on Γ, and let u be the unique weak solution u to (1)–(3) in accordance with Proposition 2. Then one has the following.

(1)
The Caccioppoli inequality is shown as
()
for any η ∈ W^1,∞(ℝⁿ).
(2)
For arbitrary x ∈ Ω, R > 0, and k₀ ≥ 0,
()
where and Ω(x, r) = Ω∩B_r(x) for any r > 0.

Proof. (1) Let us choose as a test function in (11). Thus, applying the Hölder inequality we deduce

()

Then, using the upper and lower bounds of a, we conclude (60).

(2) Let x ∈ Ω be fixed but arbitrary. Arguing as in Proposition 9, let k ≥ k₀, and with the definition of the set A(k, r) = {z ∈ Ω(x, r):|u(z)| > k}, property (42) is still valid. In particular, we have, for h > k > k₀,

()

Fix 0 < r < R ≤ R₀, and let us take as a test function in (11), where η ∈ W^1,∞(ℝⁿ) is the cut-off function defined by η ≡ 1 in B_r(x), η ≡ 0 in ℝⁿ∖B_R(x), and η(y) = (R−|y − x|)/(R − r) for all y ∈ B_R(x)∖B_r(x). Thus, we have that 0 ≤ η ≤ 1 in ℝⁿ and |∇η | ≤ 1/(R − r) a.e. in B_R(x) and that (60) reads

()

Making use of (7) and with exponent q = 2n/(n + 2) < 2 ≤ n and the Hölder inequality, we have

()

Applying the properties of η, 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 γ = 1, α = 2/(3n), and β = 1 + 2/(3n) > 1, we use (66) and inequality (63) with r replaced by R, obtaining

()

Then, taking R = R₀ and σ = 1/2, (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 C¹(B_R(x)).

Let us prove the corresponding Neumann version of Proposition 12.

Proposition 14. Let n ≥ 2, |Γ_D | = 0, f = 0 in Ω, f, h = 0, respectively, in Ω and on Γ, and let u be the unique weak solution u to (1)–(3) in accordance with Proposition 3. For arbitrary x ∈ Ω, R > 0, and k₀ ∈ ℝ, then (61) holds with .

Proof. Fix k₀ ∈ ℝ, x ∈ Ω, and 0 < r < R ≤ R₀ as arbitrary. Arguing as in Proposition 12, (64) is true by taking v = η²sign⁡(u)(|u | − k) ⁺ − ⨍_∂Ωη²sign⁡(u)(|u | − k) ⁺ds ∈ V₂(∂Ω) or v = η²sign⁡(u)(|u | − k) ⁺ − ⨍_Ωη²sign⁡(u)(|u | − k) ⁺dx ∈ V₂(Ω) as a test function in (11) and observing that ∇v = η²∇u + 2η∇ηsign⁡(u)(|u | − k) ⁺ ∈ L²(A(k, R)).

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

()

Considering 1 + 1/(R − r)<(R₀ + 1)/(R − r) and denoting the new constant by the same symbol c, we may proceed as in the proof of Proposition 12. Thus, the proof of Proposition 14 is complete, taking R = R₀ into account.

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

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

2D schematic representations of a Lipschitz domain Ω (with |Ω | = 1) representing an electrolytic cell, Γ being the union of the recipient and air contact boundaries, Γ_D representing the surface of two electrodes submerged in the electrolyte, and Ω(x, r) = Ω∩B_r(x) 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 n ≥ 2, a ∈ L^∞(Ω) satisfy 0 < a_# ≤ a ≤ a^# a.e. in Ω, x ∈ Ω, and let R > 0 be such that |Ω∩∂B_R(x)| > 0. If u ∈ H¹(Ω(x, R)) solves the local variational formulation

()

then one has

()

Proof. First we argue as in Proposition 12, with k₀ = 0. The validity of properties (63) and (64) remains. 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 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 x ∈ Ω, one says that E is a Green kernel associated with (1)–(3), if it solves

()

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.

The existence of the Green function G verifying

()

is standard if n > 2 (see, e.g., [32, 34]), with

being the unique solution to

()

for all

, for any x ∈ Ω, and for ρ > 0 such that B_ρ(x) ⊂ Ω. Moreover, G satisfies, for some positive constant C(n) and n > 2 [32, Theorem 1.1],

()

In order to provide explicit estimates and to simultaneously extend to n = 2 and a mixed boundary value problem, let us build the Green kernels for n ≥ 2.

Proposition 18. Let n ≥ 2 and 1 ≤ q < n/(n − 1), and let a be a measurable (and bounded) function defined in Ω satisfying 0 < a_# ≤ a ≤ a^#. Then, for each x ∈ Ω and any r > 0 such that r < dist⁡(x, ∂Ω), there exists a unique Green function according to Definition 17 and enjoying the following estimates:

()

with

and the constants C₁(Ω, n, q) and C₂(n, q, A) being explicitly given in Proposition 5. Moreover, G(x, y) ≥ 0 a.e. x, y ∈ Ω, and

()

for a.e. x, y ∈ Ω such that x ≠ y, where

()

Proof. For any x ∈ Ω and ρ > 0 such that B_ρ(x)⊂⊂Ω, the existence and uniqueness of solving (75), for all , are due to Proposition 2 with f = 0 a.e. in Ω, g, h = 0 a.e. on, respectively, Γ_D and Γ, and belonging to L^2n/(n+2)(Ω) if n > 2 and to L²(Ω) if n = 2. Moreover, (12) reads

()

Therefore, for any r > 0 such that B_r(x)⊂⊂Ω, there exists G ∈ H¹(Ω∖B_r(x)) such that

()

In order for G to correspond to the well defined one in (74), the W^1,q estimate (77) is true for G^ρ due to (18) with ϰ = 2, by applying Proposition 5 with f = 0, g, h = 0, and . Then, we can extract a subsequence of G^ρ, still denoted by G^ρ, weakly converging to G in W^1,q(Ω) as ρ tends to 0, with G ∈ V_q 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, G^ρ = |G^ρ|, and, by passing to the limit as ρ tends to 0, the nonnegativeness claim holds.

For each x, y ∈ Ω such that x ≠ y, we may take r < R = |x − y | /2 such that G(x, ·) ∈ H¹(Ω(y, R)) verifies ∇·(a∇G) = 0 in Ω(y, R). Applying (71), followed by the Hölder inequality since qn/(n − q) ≥ 2 means q ≥ 2n/(n + 2), we obtain

()

with

Hence, using (78) we conclude (79), which completes the proof of Proposition 18.

Remark 19. Since qn/(n − q) → n/(n − 1) as q → 1⁺ and qn/(n − q) → n/(n − 2) as q → [n/(n − 1)] ⁻, the integrability exponent of G in (78) obeys n/(n − 1) < qn/(n − q) < n/(n − 2). In conclusion, Proposition 18 ensures that G ∈ L^p(Ω) for any p ∈ [1, n/(n − 2)[.

For each x ∈ Ω, the Neumann function is defined as N = G + w being the solution of the regularity problem [33, Definition 2.5], where

is the Green function solving (75) and w ∈ H¹(Ω), with mean value zero over ∂Ω, is the unique solution to the variational formulation [33, Lemma 2.3]:

()

Here, ω^x is the L-harmonic measure [38]; that is, it is unique probability measure on ∂Ω such that

()

due to the Riesz representation theorem applied to the continuous linear functional

, where L_g(x) is the solution to the Dirichlet problem (1) with f = 0 and f = 0 and (3) with Γ_D = ∂Ω. 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 x ∈ Ω, N(x, ·) admits an extension across ∂Ω (cf. [33, Lemmas 2.9 and 2.11]) to the domain which is such that

()

where 𝒯 is the homothety function that reduces ∂Ω into its half, that is, the homothetic boundary with measure |∂Ω | /2. That is, each

is the reflection of 𝒯(y^δ) across ∂Ω in the following sense:

()

where y^δ ∈ Ω is such that y^δ = y^* − δ(y^* − 𝒯(y^*))/|y^* − 𝒯(y^*)|, for some 0 < δ < δ(Ω).

Since our concern is on weak solutions to (1)–(3) in accordance with Definition 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 about its boundary behavior.

Proposition 21. Let n ≥ 2 and 1 ≤ q < n/(n − 1), and let a be a measurable (and bounded) function defined in Ω satisfying 0 < a_# ≤ a ≤ a^#. Then, for each x ∈ Ω, there exists a Neumann function N = N(x, ·) ∈ V_q solving (72) that satisfies (77)-(78) and (79), with .

Proof. For each x ∈ Ω and ρ > 0 such that B_ρ(x) ⊂ Ω, the existence of a unique Neumann function N^ρ(x, ·) ∈ V₂ solving (75), for all v ∈ V₂, is consequence of Proposition 3 with f = 0, g, h = 0, and for t = 2n/(n + 2) if n > 2 and any t < 2 if n = 2. Arguing as in the proof of Proposition 18, N^ρ belongs to W^1,q(Ω), uniformly for x ∈ Ω, according to (18) with ϰ = 4. Therefore, we may pass to the limit as ρ → 0, finding N ∈ V_q, solving (72). The remaining estimates (78)-(79), under , are obtained exactly as in the proof of Proposition 14.

Hereafter, denotes the partial derivative ∂/∂x_i.

Proposition 22. Let n ≥ 2 and 1 ≤ q < n/(n − 1), and let E be the symmetric function that is either the Green function G or the Neumann function N in accordance with Propositions 18 and 21, respectively. If a ∈ L^∞(Ω) verifies 0 < a_# ≤ a ≤ a^# a.e. in Ω, then, for every i = 1, …, n, is uniformly bounded for x ∈ Ω. In particular, it satisfies (77)-(78) and (79), where , with ϰ = 2 if |Γ_D | > 0 and ϰ = 4 if |Γ_D | = 0.

Proof. For each x ∈ Ω, we may approximate by , where E^ρ = E^ρ(x, ·) ∈ V₂ solves (75) for every ρ > 0 such that B_ρ(x) ⊂ Ω. Since is a Dirac delta function, Proposition 8 ensures that verifies (77) and also (78) by the Sobolev inequality (7), with , where ϰ = 2 if |Γ_D | > 0 and ϰ = 4 if |Γ_D | = 0. Consequently, (77)-(78) hold, by passage to the weak limit.

To prove estimate (79) for , let us take y ∈ Ω such that R = |x − y | /2 > 0. Thus, E(x, ·) ∈ H¹(Ω(y, R))∩V_q verifies in Ω(y, R), for every i = 1, …, n. Therefore, we proceed by using the argument already used in the proof of Proposition 18, with G being replaced by .

Remark 23. 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 17, that is, due to differentiation of (73) 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∩H¹(Ω∖B_r(x)) verifies, by construction, the limit system of identities

()

for any φ ∈ W^1,∞(ℝⁿ) such that

, and for all v ∈ V₁ ∩ H¹(Ω∩supp⁡(φ)).

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 [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 a ∈ L^∞(Ω) satisfy 0 < a_# ≤ a ≤ a^# a.e. in Ω. If there exists a function ω : [0, ∞[→[0, ∞[ such that, a.e. x, y ∈ Ω,

()

then for each x ∈ Ω and R > 0, any function u ∈ W^1,1(Ω) solving

()

in the sense of distributions, enjoys, a.e. y ∈ Ω,

()

where

()

for some 2 ≤ ν < δ(Ω)/r and 0 < r < min⁡{1, δ(y)} with δ(y): = dist⁡(y, ∂Ω).

Proof. By density, since u ∈ W^1,1(Ω) there exists a sequence such that u_m → u in W^1,1(Ω). In particular, u_m → u in L¹(Ω) and ∇u_m → ∇u a.e. in Ω. Thus, it is sufficient to prove estimate (93), under the assumption .

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

()

for some constant b ∈ ]0,1/2[. Since Ω is bounded, we can take 2 ≤ ν < δ(Ω)/r and define d as in (94). Notice that d ≤ r implies B_d(y)⊂⊂Ω.

In order to determine the final constant in (93), let be the cut-off function explicitly given by

()

Thus, η satisfies 0 ≤ η ≤ 1,

()

For w ∈ B : = B_d(y), we multiply (92) by G_L(w, ·)η/a(y) where G_L is the fundamental solution of Laplace equation:

()

with

()

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

()

where

()

Using the lower bound of a, the definition of G_L, and the properties of η, we have

()

By appealing to (95), we obtain

()

Considering that, for all x, y ∈ Ω and z ∈ B_d(y),

()

we obtain

()

Let us analyze the first integral of the right-hand side in (106). 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 (107) holds denoting ν = 2.

Returning to (106), substituting the value of C_L from (99) with r ≤ 1, and dividing by b > 0, we write it as

()

In an n-dimensional Euclidean space, the spherical coordinate system consists of a radial coordinate t and n − 1 angular coordinates ϕ₁, …, ϕ_n−2 ∈ [0, π] and ϕ_n−1 ∈ [0,2π[, and the Cartesian coordinates are z₁ = y₁ + tcos⁡(ϕ₁), z₂ = y₂ + tsin(ϕ₁)cos⁡(ϕ₂), … , z_n−1 = y_n−1 + tsin(ϕ₁) ⋯ sin(ϕ_n−2)cos⁡(ϕ_n−1), and z_n = y_n + tsin(ϕ₁) ⋯ sin(ϕ_n−1). Since the Jacobian of this transformation is tⁿ⁻¹sinⁿ⁻²(ϕ₁) ⋯ sin(ϕ_n−2) and

()

applying (91), we deduce

()

Inserting this last inequality into (108), we find (93).

Remark 25. Observing (110), assumption (91) can be replaced by a belonging to the VMO space of vanishing mean oscillation functions which is constituted by the functions f belonging to the BMO space that verify

()

where B_ρ ranges in the class of the balls with radius ρ contained in Ω. We recall that the John-Nirenberg space BMO of the functions of bounded mean oscillation is defined as

()

where B ranges in the class of the balls contained in Ω.

Remark 26. The upper bound in (93) is not optimal; it depends on the choice of the cut-off function through the constants c₁ and c₂ (cf. (97) and (108)).

Proposition 27. Let n ≥ 2 and 1 ≤ q < n/(n − 1), and let E be the symmetric function that is either the Green function G or the Neumann function N in accordance with Propositions 18 and 21, respectively. If a ∈ L^∞(Ω) satisfies 0 < a_# ≤ a ≤ a^# a.e. in Ω and (91), then a.e. x, y ∈ Ω,

()

with

()

where ϰ = 2 if |Γ_D | > 0, ϰ = 4 if |Γ_D | = 0, and the constants C₁(Ω, n, q) and

are explicitly given in Proposition 5.

Proof. Let x ∈ Ω be arbitrary. Using property (93) and applying (79), we get

()

with

()

Considering that, for all x, y ∈ Ω and z ∈ B_d(y) with d≤|x − y | /2,

()

we compute

()

where the Riesz potential is calculated by the spherical transformation as in the above proof. Next, from d≤|x − y | /2 we find (113).

6. W^1,p Constants (p > n)

Let p > n, g = 0 on Γ_D (possibly empty), and u ∈ V_p 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 Ω is replaced by weaker assumptions, for instance, when Ω is a plane bounded domain with Lipschitz and piecewise C² boundary whose angles are all convex [10, page 151] or when Ω is a plane bounded domain with curvilinear polygonal C^1,1 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 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 [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 C^1,1, W^1,p regularity of the solution is found for 1 < p < ∞ 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 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 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 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 seminorms being sufficiently small. In [16] the authors obtain the global W^1,p 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 u ∈ V_p (p > n) and the Green kernel E associated with (1)–(3); that is,

is either the Green or the Neumann functions, E = G and E = N, in accordance with Propositions 18 and 21, respectively. To this end, we take

and v = u ∈ V_p as test functions in (11) and (73), respectively, obtaining the Green representation formula:

()

where 𝒯, 𝒮, and 𝒦 are the layer potential operators defined by

()

For every 0 < λ < n, u ∈ L^s(ℝⁿ), and v ∈ L^t(ℝⁿ), with s, t > 1 and λ/n + 1/s + 1/t = 2, the Hardy-Littlewood-Sobolev inequality in its general form states the following:

()

where the constant is sharp [41], if s = t = 2n/(2n − λ), defined by

()

In the presence of the Hardy-Littlewood-Sobolev inequality, we prove the following W^1,p estimate.

Proposition 28. Let p > 1, f ∈ L^t(Ω) with t ∈ ]pn/(p + n), p[, f = 0 in Ω, g = 0 on Γ_D (possibly empty), h = 0 on Γ, and let a ∈ L^∞(Ω) satisfy 0 < a_# ≤ a ≤ a^# a.e. in Ω and (91). If u ∈ V_p solves (11), for all , then u satisfies

()

with 1/q = 1 + 1/p − 1/t, C(n, p^′, n/q) relative to (121), and C(Ω, n, q, a) determined in Proposition 27. In particular, for 2 < p < 2n/(n − 1) we have

()

Proof. Since ∇u ∈ L^p(Ω), (119) holds. Differentiating it, for i = 1, …, n, we deduce

()

Let

be arbitrary such that

. Using (113) for any 1 < q < n/(n − 1) and applying the Fubini-Tonelli theorem, we find

()

Next, using (121) with λ = n/q, s = p^′, and 1/t = 1 + 1/p − 1/q, we conclude (123).

For the particular situation, we choose 1 < q = p/2 < n/(n − 1) and we use (121) with s = t = p^′.

Having the results established in Section 5 in mind, we find a W^1,p estimate for weak solutions where regularity (91) of the leading coefficient is not a necessary condition.

Proposition 29. Let p > n, f ∈ L^p(Ω), f ∈ L^p(Ω), g = 0 on Γ_D (possibly empty), h ∈ L^p(Γ), a ∈ L^∞(Ω) satisfy 0 < a_# ≤ a ≤ a^# a.e. in Ω, and let u ∈ V_p solve (11), for all . Then u satisfies

()

with the constants C₁(Ω, n, p^′) and C₂(n, p^′, 1/a_#) being explicitly given in Proposition 5.

Proof. Differentiating (119), for i = 1, …, n, 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 |Ω|^1/p is due to the embedding

. Considering that

uniformly for x ∈ Ω (cf. Proposition 22) and consequently also

uniformly in x ∈ Ω, 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.

References

1 Boccardo L. and Gallouët T., Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis. (1989) 87, no. 1, 149–169, 2-s2.0-0000930026.
10.1016/0022-1236(89)90005-0
Web of Science® Google Scholar
2 Consiglieri L., Mathematical Analysis of Selected Problems from Fluid Thermo-Mechanics. The (p–q) Coupled Fluid-Energy Systems, 2011, Lambert Academic Publishing, Saarbrücken, Germany.
Google Scholar
3 Dall′Aglio A., Approximated solutions of equations with L¹ data. Application to the H-convergence of quasi-linear parabolic equations, Annali di Matematica Pura ed Applicata. (1996) 170, 207–240, https://doi.org/10.1007/BF01758989, MR1441620.
10.1007/BF01758989
Web of Science® Google Scholar
4 Prignet A., Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure, Annales de la Faculté des Sciences de Toulouse. Mathématiques. (1997) 6, no. 2, 297–318, MR1611840, https://doi.org/10.5802/afst.867, ZBL0895.35103.
10.5802/afst.867
Google Scholar
5 Agmon S., Douglis A., and Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Communications on Pure and Applied Mathematics. (1959) 12, 623–727.
10.1002/cpa.3160120405
Web of Science® Google Scholar
6 Beirao da Veiga H., Sur la régularité des solutions de l′équation div A(x, u, ∇u) = B(x, u, ∇u) avec des conditions aux limites unilatérales et mêlées, Annali di Matematica Pura ed Applicata. (1972) 93, 173–230, MR0326164, https://doi.org/10.1007/BF02412021.
10.1007/BF02412021
Google Scholar
7 Dauge M., Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, 1988, 1341, Springer, Berlin, Germany, Lecture Notes in Mathematics, MR961439.
10.1007/BFb0086682
Web of Science® Google Scholar
8 Giaquinta M., Introduction to Regularity Theory for Nonlinear Elliptic Systems, 1993, Birkhäuser, Berlin, Germany, Lectures in Mathematics ETH Zürich, MR1239172.
Google Scholar
9 Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second Order, 1983, Springer, New York, NY, USA, MR737190.
10.1007/978-3-642-61798-0
Google Scholar
10 Grisvard P., Elliptic Problems in Nonsmooth Domains, 1985, 24, Pitman, London, UK, Monographs and Studies in Mathematics, MR775683.
Web of Science® Google Scholar
11 Ladyzenskaya O. A. and Ural′ceva N. N., Linear and Quasilinear Elliptic Equations, 1968, 46, Academic Press, London, UK, Mathematics in Science and Engineering.
Google Scholar
12 Nirenberg L., On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa. (1959) 13, 115–162, MR0109940, ZBL0088.07601.
Google Scholar
13 Stampacchia G., Équations Elliptiques du Second Ordre à Coefficients Discontinus, 1964, Séminaire Jean Leray.
Google Scholar
14 Agranovich M. S., Regularity of variational solutions to linear boundary value problems in Lipschitz domains, Functional Analysis and Its Applications. (2006) 40, no. 4, 313–329, 2-s2.0-33846145381, https://doi.org/10.1007/s10688-006-0048-6.
10.1007/s10688-006-0048-6
Web of Science® Google Scholar
15 Byun S.-S. and Wang L., Elliptic equations with BMO coefficients in Reifenberg domains, Communications on Pure and Applied Mathematics. (2004) 57, no. 10, 1283–1310, https://doi.org/10.1002/cpa.20037, MR2069724, ZBL1112.35053.
10.1002/cpa.20037
Web of Science® Google Scholar
16 Byun S.-S. and Wang L., The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proceedings of the London Mathematical Society. (2005) 90, no. 1, 245–272, https://doi.org/10.1112/S0024611504014960, MR2107043, ZBL1087.35028.
10.1112/S0024611504014960
Web of Science® Google Scholar
17 Di Fazio G., L^p estimates for divergence form elliptic equations with discontinuous coefficients, Bollettino della Unione Matematica Italiana. (1996) 10, no. 2, 409–420, MR1405255, ZBL0865.35048.
Google Scholar
18 Dong H. and Kim D., Elliptic equations in divergence form with partially BMO coefficients, Archive for Rational Mechanics and Analysis. (2010) 196, no. 1, 25–70, https://doi.org/10.1007/s00205-009-0228-7, MR2601069, ZBL1206.35249.
10.1007/s00205-009-0228-7
Web of Science® Google Scholar
19 Elschner J., Kaiser H.-C., Rehberg J., and Schmidt G., W^1,q regularity results for elliptic transmission problems on heterogeneous polyhedra, Mathematical Models and Methods in Applied Sciences. (2007) 17, no. 4, 593–615, https://doi.org/10.1142/S0218202507002030, MR2316300, ZBL1206.35059.
10.1142/S0218202507002030
Web of Science® Google Scholar
20 Geng J., W^1,p estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains, Advances in Mathematics. (2012) 229, no. 4, 2427–2448, https://doi.org/10.1016/j.aim.2012.01.004, MR2880228, ZBL1234.35048.
10.1016/j.aim.2012.01.004
Web of Science® Google Scholar
21 Gröger K., A W^1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Mathematische Annalen. (1989) 283, no. 4, 679–687, https://doi.org/10.1007/BF01442860, MR990595.
10.1007/BF01442860
Web of Science® Google Scholar
22 Gröger K. and Rehberg J., Resolvent estimates in W^−1,p for second order elliptic differential operators in case of mixed boundary conditions, Mathematische Annalen. (1989) 285, no. 1, 105–113, https://doi.org/10.1007/BF01442675, MR1010194, ZBL0659.35032.
10.1007/BF01442675
Web of Science® Google Scholar
23 Haller-Dintelmann R., Kaiser H.-C., and Rehberg J., Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de Mathématiques Pures et Appliquées. (2008) 89, no. 1, 25–48, https://doi.org/10.1016/j.matpur.2007.09.001, MR2378088, ZBL1132.35022.
10.1016/j.matpur.2007.09.001
Web of Science® Google Scholar
24 Li Y. Y. and Vogelius M., Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Archive for Rational Mechanics and Analysis. (2000) 153, no. 2, 91–151, https://doi.org/10.1007/s002050000082, MR1770682, ZBL0958.35060.
10.1007/s002050000082
Web of Science® Google Scholar
25 Meyers N. G., An L^p-estimate for the gradient of solutions of second order elliptic divergence equations, Annali della Scuola Normale Superiore di Pisa. (1963) 17, 189–206, MR0159110.
Google Scholar
26 Ott K. A. and Brown R. M., The mixed problem for the Laplacian in Lipschitz domains, Potential Analysis. (2013) 38, no. 4, 1333–1364, https://doi.org/10.1007/s11118-012-9317-6, MR3042705, ZBL1272.35098.
10.1007/s11118-012-9317-6
Web of Science® Google Scholar
27 Ragusa M. A., Regularity of solutions of divergence form elliptic equations, Proceedings of the American Mathematical Society. (2000) 128, no. 2, 533–540, https://doi.org/10.1090/S0002-9939-99-05165-5, MR1641085, ZBL0949.35037.
10.1090/S0002-9939-99-05165-5
Web of Science® Google Scholar
28 Savaré G., Regularity results for elliptic equations in Lipschitz domains, Journal of Functional Analysis. (1998) 152, no. 1, 176–201, https://doi.org/10.1006/jfan.1997.3158, MR1600081, ZBL0889.35018.
10.1006/jfan.1997.3158
Web of Science® Google Scholar
29 Consiglieri L., The Joule-Thomson effect on the thermoelectric conductors, Zeitschrift für Angewandte Mathematik und Mechanik.(2009) 89, no. 3, 218–236, https://doi.org/10.1002/zamm.200800108, MR2510635, ZBL1162.35358.
10.1002/zamm.200800108
Web of Science® Google Scholar
30 Consiglieri L., A limit model for thermoelectric equations, Annali dell′Università di Ferrara. Matematiche. (2011) 57, no. 2, 229–244, https://doi.org/10.1007/s11565-011-0129-1, MR2855359, ZBL1255.35207.
10.1007/s11565-011-0129-1
Google Scholar
31 Consiglieri L., On the posedness of thermoelectrochemical coupled systems, The European Physical Journal Plus. (2013) 128, no. 5, article 47.
10.1140/epjp/i2013-13047-7
Web of Science® Google Scholar
32 Grüter M. and Widman K.-O., The Green function for uniformly elliptic equations, Manuscripta Mathematica. (1982) 37, no. 3, 303–342, https://doi.org/10.1007/BF01166225, MR657523.
10.1007/BF01166225
Web of Science® Google Scholar
33 Kenig C. E. and Pipher J., The Neumann problem for elliptic equations with nonsmooth coefficients, Inventiones Mathematicae. (1993) 113, no. 3, 447–509, https://doi.org/10.1007/BF01244315, MR1231834.
10.1007/BF01244315
Web of Science® Google Scholar
34 Littman W., Stampacchia G., and Weinberger H. F., Regular points for elliptic equations with discontinuous coefficients, Annali della Scuola Normale Superiore di Pisa. (1963) 17, no. 1-2, 43–77, MR0161019, ZBL0116.30302.
Google Scholar
35 Talenti G., Best constant in Sobolev inequality, Annali di Matematica Pura ed Applicata. (1976) 110, 353–372, MR0463908, https://doi.org/10.1007/BF02418013, ZBL0353.46018.
10.1007/BF02418013
Google Scholar
36 Fernández Bonder J. and Saintier N., Estimates for the Sobolev trace constant with critical exponent and applications, Annali di Matematica Pura ed Applicata IV. (2008) 187, no. 4, 683–704, https://doi.org/10.1007/s10231-007-0062-1, MR2413374, ZBL1223.35131.
10.1007/s10231-007-0062-1
Web of Science® Google Scholar
37 Boccardo L. and Gallouët T., Nonlinear elliptic equations with right-hand side measures, Communications in Partial Differential Equations. (1992) 17, no. 3-4, 641–655, https://doi.org/10.1080/03605309208820857, MR1163440, ZBL0812.35043.
10.1080/03605309208820857
Web of Science® Google Scholar
38 Dahlberg B. E. J., Real analysis and potential theory, Proceedings of the International Congress of Mathematicians, 1983, Warsaw, Poland, 953–959, MR804749.
Google Scholar
39 Escauriaza L., Fabes E. B., and Verchota G., On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proceedings of the American Mathematical Society. (1992) 115, no. 4, 1069–1076, https://doi.org/10.2307/2159357, MR1092919, ZBL0761.35013.
10.1090/S0002-9939-1992-1092919-1
Web of Science® Google Scholar
40 Sarason D., Functions of vanishing mean oscillation, Transactions of the American Mathematical Society. (1975) 207, 391–405, MR0377518, https://doi.org/10.1090/S0002-9947-1975-0377518-3, ZBL0319.42006.
10.1090/S0002-9947-1975-0377518-3
Web of Science® Google Scholar
41 Lieb E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Mathematics. (1983) 118, no. 2, 349–374, https://doi.org/10.2307/2007032, MR717827, ZBL0527.42011.
10.2307/2007032
Web of Science® Google Scholar

Citing Literature

All articles

Explicit Estimates for Solutions of Mixed Elliptic Problems

Abstract

1. Introduction

2. Statement of the Problem

3. Some W^1,q Constants (q ≤ 2)

3.1. H¹ Solvability

3.2. W^1,q Solvability (q ≤ n/(n − 1))

4. L^∞ Constants

5. Green Kernels

6. W^1,p Constants (p > n)

Conflict of Interests

References

Citing Literature

Figures

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Explicit Estimates for Solutions of Mixed Elliptic Problems

Abstract

1. Introduction

2. Statement of the Problem

3. Some W1,q Constants (q ≤ 2)

3.1. H1 Solvability

3.2. W1,q Solvability (q ≤ n/(n − 1))

4. L∞ Constants

5. Green Kernels

6. W1,p Constants (p > n)

Conflict of Interests

References

Citing Literature

Figures

References

Related

Information

3. Some W^1,q Constants (q ≤ 2)

3.1. H¹ Solvability

3.2. W^1,q Solvability (q ≤ n/(n − 1))

4. L^∞ Constants

6. W^1,p Constants (p > n)