# Explicit Estimates for Solutions of Mixed Elliptic Problems

Authors: Luisa Consiglieri
Journal: International Journal of Partial Differential Equations
 (2014)

---

## Abstract

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains inℝn (n≥2) of class C0,1. The existence of L∞ and W1,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 W1,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.


---

## Body

## 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 (Section3.1) 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 W1,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 aLp 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 W1,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

LetΩ be a domain (i.e., connected open set) in ℝn(n≥2) of class C0,1 and bounded. Its boundary ∂Ω is constituted by two disjoint open (n-1)-dimensional sets, ΓD and Γ, such that ∂Ω=Γ-D∪Γ-. The Dirichlet situation ΓD=∂Ω (or equivalently Γ=∅) and the Neumann situation Γ=∂Ω (or equivalently ΓD=∅) are available.Let us consider the following boundary value problem, in the sense of distributions:(1)-∇·(a∇u)=f-∇·f,inΩ;(2)(a∇u-f)·n=h,on⁡Γ;(3)u=g,on⁡ΓD,
where n is the unit outward normal to the boundary ∂Ω.Set for anyq≥1(4)Vq={WΓD1,q(Ω)={v∈W1,q(Ω):v=0on⁡ΓD},hhhhhhhhhhhhhhhhhhhhhhhif|ΓD|>0,Vq(∂Ω)={v∈W1,q(Ω):∫∂Ωvds=0},Vq(Ω)={v∈W1,q(Ω):∫Ωvdx=0},hhhhhhhhhhhhhhhhhhhhotherwise,
the Banach space endowed with the seminorm of W1,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 theW1,q norm by
(5)∥v∥1,q,Ω∶=1C*+1(∥v∥q,Ω+∥∇v∥q,Ω),
with C* being any of the Poincaré constants,
(6)∥v-⨍Σvds∥q,Ω≤C*∥∇v∥q,Ω,∀v∈W1,q(Ω);∥v-⨍Ωvdx∥q,Ω≤C*∥∇v∥q,Ω,∀v∈W1,q(Ω),
where Σ⊂∂Ω and ⨍A means the integral average over the set A of positive measure, the Sobolev and trace inequalities read
(7)∥v∥q*,Ω≤Sq∥∇v∥q,Ω;(8)∥v∥q*,∂Ω≤Kq∥∇v∥q,Ω,∀v∈Vq.
Hence further we call (7) the Sobolev inequality and for the general situation the W1,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 W1,q(Ω)↪Lq*(Ω) and W1,q(Ω)↪Lq*(∂Ω). 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])
(9)Sq=π-1/2n-1/q(q-1n-q)1-1/q×[Γ(1+n/2)Γ(n)Γ(n/q)Γ(1+n-n/q)]1/n;Kq=π(1-q)/2(q-1n-q)q-1×[Γ(q(n-1)×(2(q-1))-1)Γ((n-1)×(2(q-1))-1)](q-1)/(n-1).
We observe that q*>1 is arbitrary if q=n. Here Γ stands for the gamma function. Set by ωn the volume of the unit ball B1(0) of ℝn; that is, ωn=πn/2/Γ(n/2+1) and Γ(n/2+1)=(n/2)! if n is even, and Γ(n/2+1)=π1/22-(n+1)/2n(n-2)(n-4)⋯1 if n is odd. Moreover, the relationship σn-1=nωn holds true, where σn-1=2πn/2/Γ(n/2) denotes the area of the unit sphere ∂B1(0).Forn>1, from the fundamental theorem of calculus applied to each of the n variables separately, it follows that
(10)∥u∥n/(n-1),Ω≤n-1/2∥∇u∥1,Ω.
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 thatu is weak solution to (1)–(3), if it verifies u=g a.e. on ΓD, and
(11)∫Ω‍a∇u·∇vdx=∫Ω‍f·∇vdx+∫Ω‍fvdx+∫Γ‍hvds,∀v∈V2,
where g∈L2(ΓD), f∈L2(Ω), and f∈Lt(Ω), with t=(2*)′; that is, t=2n/(n+2) if n>2 and any t>1 if n=2, h∈Ls(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.SinceΩ 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. SomeW1,q Constants (q≤2)

The presented results in this section are valid whethera 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=aij∂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 spaceH1 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|ΓD|>0, then there exists u∈H1(Ω) being a weak solution to (1)–(3). If g=0, then u is unique. Letting g~∈H1(Ω) as an extension of g∈L2(ΓD) (i.e., it is such that g~=g a.e. on ΓD), the following estimate holds:
(12)∥∇u∥2,Ω≤(a#a#+1)∥∇g~∥2,Ω+1a#(∥f∥2,Ω+Cn(∥f∥t,Ω,∥h∥s,Γ)),
where Cn(A,B)=S2A+K2B if n>2, C2(A,B)=|Ω|1/t′S2t/(3t-2)A+|Ω|1/(2s′)K2s/(2s-1)B if t<2, and C2(A,B)=|Ω|1/t′A/2+|Ω|1/(2s′)K2s/(2s-1)B if t≥2. In particular, u-g~∈HΓD1(Ω) is unique.Proof.
Forg∈L2(ΓD) there exists an extension g~∈H1(Ω) such that g~=g a.e. on ΓD. The existence and uniqueness of a weak solution w∈HΓD1(Ω) are well known via the Lax-Milgram lemma, to the variational problem:
(13)∫Ω‍a∇w·∇vdx=∫Ω‍(f-a∇g~)·∇vdx+∫Ω‍fvdx+∫Γ‍hvds,
for all v∈HΓD1(Ω). Therefore, the required solution is given by u=w+g~.
Ifg=0, g~=0 and then u≡w.
Takingv=w∈HΓD1(Ω) as a test function in (13), applying the Hölder inequality, and using the lower and upper bounds of a, we obtain
(14)a#∥∇w∥2,Ω2≤(∥f∥2,Ω+a#∥∇g~∥2,Ω)∥∇w∥2,Ω+∥f∥t,Ω∥w∥t′,Ω+∥h∥s,Γ∥w∥s′,Γ.
Forn>2, this inequality reads
(15)a#∥∇w∥2,Ω≤∥f∥2,Ω+a#∥∇g~∥2,Ω+S2∥f∥2n/(2+n),Ω+K2∥h∥2(n-1)/n,Γ,
implying (12).
Consider the case of dimensionn=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
(16)∥w∥t′,Ω≤|Ω|1/t′-1/2∥w∥2,Ω≤12|Ω|1/t′-1/2∥∇w∥1,Ω≤12|Ω|1/t′∥∇w∥2,Ω,(t≥2);∥w∥t′,Ω≤S2t/(3t-2)∥∇w∥2t/(3t-2),Ω≤S2t/(3t-2)|Ω|1/t′∥∇w∥2,Ω,(t<2);∥w∥s′,Γ≤K2s/(2s-1)∥∇w∥2s/(2s-1),Ω≤K2s/(2s-1)|Ω|1/(2s′)∥∇w∥2,Ω.
This concludes the proof of Proposition 2.Proposition 3 (Neumann).
If|ΓD|=0, then there exists a unique u∈V2 being a weak solution to (1)–(3). Moreover, the following estimate holds:
(17)∥∇u∥2,Ω≤1a#(∥f∥2,Ω+Cn(∥f∥t,Ω,∥h∥s,Γ)),
where Cn(A,B) is given as in Proposition 2.Proof.
The existence and uniqueness of a weak solutionu∈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 solutionu∈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.W1,q Solvability (q≤n/(n-1))

The existence of a solution is recalled in the following proposition in accordance withL1 theory, that is, via solutions obtained by limit approximation (SOLA) (cf. [1–4, 37]), in order to determine the explicit constants.Proposition 5.
Letg=0 on ΓD (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∈Vq solving (11) for every v∈Vq′. Moreover, one has the following estimate:
(18)∥∇u∥q,Ω≤C1(Ω,n,q)(∥f∥2,Ωa#+ϰ(∥f∥1,Ω+∥h∥1,Γ)a#)+C2(n,q,∥f∥2,Ωa#+ϰ(∥f∥1,Ω+∥h∥1,Γ)a#),
with ϰ=2 if |ΓD|>0, ϰ=4 if |ΓD|=0, and
(19)C1(Ω,n,q)=|Ω|1/q-1/2{(n-q)3/2q(n-2)(n+q-nq)1/2×21/q+(3n-q(n+1))/[2(n-q)],ifn>2,(2-q)-1/22(6-q)/2-1/q,ifn=2,C2(n,q,A)={A2(n-q)/[q(n-2)](n-qn+q-nq)(n-q)/[q(n-2)]×2(2-q)(nq-n+q)/[q2(n-2)]Sqn(2-q)/[q(n-2)],ifn>2A2/(q-1)|Ω|1/q-1/2×[(q-1)(3-q)(3-q)/(q-1)](2-q)/(2q)(2-q)1/(q-1)×2ℓSq(3-q)/(q-1),ifn=2,
where ℓ∈]2,+∞[ is explicitly given in (30).Proof.
For eachm∈ℕ, take
(20)fm=mfm+|f|∈L∞(Ω),hm=mhm+|h|∈L∞(Γ).
Applying Propositions 2 and 3, there exists a unique solution um∈V2 to the following variational problem:
(21)∫Ω‍a∇um·∇vdx=∫Ω‍f·∇vdx+∫Ω‍fmvdx+∫Γ‍hmvds,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhihhhh∀v∈V2.
In particular, (21) holds for all v∈Vq′ (q′>n).
In order to pass to limit (21) on m  (m→∞) let us establish the estimate (18) for ∇um. The method for estimating is due to Boccardo and Gallouët (see, e.g., [1, 37]).
Case
(
|
Γ
D
|
>
0
)
. Let us choose
(22)v=sign⁡(um)[1-1(1+|um|)s]∈WΓD1,2(Ω)∩L∞(Ω),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhifors>0,
as a test function in (21). Hence it follows that
(23)a#∫Ω‍s|∇um|2(1+|um|)s+1dx≤s∥f∥2,Ω∥∇um(1+|um|)(s+1)/2∥2,Ω+∥f∥1,Ω+∥h∥1,Γ,
and consequently
(24)∫Ω‍|∇um|2(1+|um|)s+1dx≤1(a#)2∥f∥2,Ω2+2sa#(∥f∥1,Ω+∥h∥1,Γ).
By the Hölder inequality with exponents2/q and 2/(2-q)>1, we have
(25)∫Ω‍|∇um|qdx≤(∫Ω‍|∇um|2(1+|um|)s+1dx)q/2×(∫Ω‍(1+|um|)(s+1)q/(2-q)dx)(2-q)/2.
Set
(26)M(s)∶=∥f∥2,Ωa#+(2(∥f∥1,Ω+∥h∥1,Γ)a#s)1/2.
Let us chooses>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 um∈V2↪Vq with (q≤2), we deduce
(27)∥∇um∥q,Ω≤M(s)2(2-q)(nq/(n-q)-1)/(2q)×(|Ω|1/q-1/2+(Sq∥∇um∥q,Ω)n(2-q)/[2(n-q)])≤M(n+q-nqn-q)2(2-q)(nq/(n-q)-1)/(2q)|Ω|1/q-1/2+n(2-q)2(n-q)∥∇um∥q,Ω+q(n-2)2(n-q)×[M(n+q-nqn-q)]2(n-q)/[q(n-2)]×2(2-q)(nq-n+q)/[q2(n-2)]Sqn(2-q)/[q(n-2)],
using the Young inequality AB≤ϵAa/a+Bb/(bϵb/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.
Forn=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
(28)(1+|um|)(s+1)q/(2-q)≤2(s+1)q/(2-q)-1(1+|um|(s+1)q/(2-q))≤2((s+2)q-2)/(2-q)+1-s2(s+12ϵ)(s+1)/(1-s)×22[(s+2)q-2]/[(2-q)(1-s)]+ϵ|um|q*.
Let us choose, for instance, s=2-q<1, and ϵ=[2SqM(s)]-2q/(2-q). Then, we obtain
(29)∥∇um∥q,Ωhhhh≤M(s)(22[(s+2)q-2]/[(2-q)(1-s)](s+12ϵ)(s+1)/(1-s)](2-q)/(2q)ϵ(2-q)/(2q)Sq∥∇um∥q,Ω+|Ω|1/q-1/2hhhhhhhhhhhhh×[2((s+2)q-2)/(2-q)+1-s2(s+12ϵ)(s+1)/(1-s)hhhhhhhhhhhhhhhhh×22[(s+2)q-2]/[(2-q)(1-s)](s+12ϵ)(s+1)/(1-s)](2-q)/(2q))hhhh≤12∥∇um∥q,Ω+M(2-q)|Ω|1/q-1/2hhhhhh×[122((4-q)q-2)/(2q)hhhhhhhhh+12[(q-1)(3-q)(3-q)/(q-1)](2-q)/(2q)hhhhhhhhh×2ℓ[M(2-q)](3-q)/(q-1)Sq(3-q)/(q-1)12],
where ℓ is given by
(30)ℓ=2[(s+2)q-2]-(2-3q)(s+1)2q(1-s)=-5q2+19q-102q(q-1)⟶2+,
as q→2-. Hence, we find (18) with ϰ=2.
Case
(
|
Γ
D
|
=
0
)
. We choose, for s>0,
(31)v=-sign⁡(um)(1+|um|)s+⨍∂Ωsign⁡(um)(1+|um|)sds∈V2(∂Ω);v=-sign⁡(um)(1+|um|)s+⨍Ωsign⁡(um)(1+|um|)sdx∈V2(Ω),
as a test function in (21). Since |v|≤2 a.e. in Ω, it follows that
(32)∫Ω‍|∇um|2(1+|um|)s+1dx≤1(a#)2∥f∥2,Ω2+4sa#(∥f∥1,Ω+∥h∥1,Γ).
Then, we argue as in the above case, concluding (18) with ϰ=4.
For both cases, we can extract a subsequence ofum, still denoted by um, such that it weakly converges to u in W1,q(Ω), where u∈Vq solves the limit problem (11) for all v∈Vq′.Remark 6.
In terms of Proposition5, the terms on the right-hand side of (11) have sense, since v∈W1,q′(Ω)↪C(Ω-) for q′>n, that is, q<n/(n-1).Remark 7.
The existence of a solution, which is given at Proposition5, 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 Proposition5, which will be required in Section 5, with datum belonging to the space of all signed measures with finite total variation ℳ(Ω)=(C0(Ω))′.Proposition 8.
Letg=0 on ΓD (possibly empty), let a∈L∞(Ω) satisfy 0<a#≤a≤a# a.e. in Ω, and, for each x∈Ω, let δx∈ℳ(Ω) be the Dirac delta function. For any 1≤q<n/(n-1) there exists u∈Vq solving
(33)∫Ω‍a∇u·∇vdx=〈δx,v〉ℳ(Ω)×C0(Ω),∀v∈C0(Ω)∩V1,
for every i=1,…,n. Moreover, one has the following estimate:
(34)∥∇u∥q,Ω≤C1(Ω,n,q)ϰa#+C2(n,q,ϰa#),
where the constants C1(Ω,n,q), C2(n,q,A), and ϰ are determined in Proposition 5.Proof.
Since the Dirac delta functionδx∈ℳ(Ω) can be approximated by a sequence {fm}m∈ℕ⊂L∞(Ω) such that
(35)∥fm∥1,Ω=1,lim⁡m→∞∫Ω‍fmφdx=〈δx,φ〉ℳ(Ω)×C0(Ω),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi∀φ∈C0(Ω),
identity (21) holds, with f being replaced by fm, 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).


## 3.1.H1 Solvability

We recall the existence result in the Hilbert spaceH1 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|ΓD|>0, then there exists u∈H1(Ω) being a weak solution to (1)–(3). If g=0, then u is unique. Letting g~∈H1(Ω) as an extension of g∈L2(ΓD) (i.e., it is such that g~=g a.e. on ΓD), the following estimate holds:
(12)∥∇u∥2,Ω≤(a#a#+1)∥∇g~∥2,Ω+1a#(∥f∥2,Ω+Cn(∥f∥t,Ω,∥h∥s,Γ)),
where Cn(A,B)=S2A+K2B if n>2, C2(A,B)=|Ω|1/t′S2t/(3t-2)A+|Ω|1/(2s′)K2s/(2s-1)B if t<2, and C2(A,B)=|Ω|1/t′A/2+|Ω|1/(2s′)K2s/(2s-1)B if t≥2. In particular, u-g~∈HΓD1(Ω) is unique.Proof.
Forg∈L2(ΓD) there exists an extension g~∈H1(Ω) such that g~=g a.e. on ΓD. The existence and uniqueness of a weak solution w∈HΓD1(Ω) are well known via the Lax-Milgram lemma, to the variational problem:
(13)∫Ω‍a∇w·∇vdx=∫Ω‍(f-a∇g~)·∇vdx+∫Ω‍fvdx+∫Γ‍hvds,
for all v∈HΓD1(Ω). Therefore, the required solution is given by u=w+g~.
Ifg=0, g~=0 and then u≡w.
Takingv=w∈HΓD1(Ω) as a test function in (13), applying the Hölder inequality, and using the lower and upper bounds of a, we obtain
(14)a#∥∇w∥2,Ω2≤(∥f∥2,Ω+a#∥∇g~∥2,Ω)∥∇w∥2,Ω+∥f∥t,Ω∥w∥t′,Ω+∥h∥s,Γ∥w∥s′,Γ.
Forn>2, this inequality reads
(15)a#∥∇w∥2,Ω≤∥f∥2,Ω+a#∥∇g~∥2,Ω+S2∥f∥2n/(2+n),Ω+K2∥h∥2(n-1)/n,Γ,
implying (12).
Consider the case of dimensionn=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
(16)∥w∥t′,Ω≤|Ω|1/t′-1/2∥w∥2,Ω≤12|Ω|1/t′-1/2∥∇w∥1,Ω≤12|Ω|1/t′∥∇w∥2,Ω,(t≥2);∥w∥t′,Ω≤S2t/(3t-2)∥∇w∥2t/(3t-2),Ω≤S2t/(3t-2)|Ω|1/t′∥∇w∥2,Ω,(t<2);∥w∥s′,Γ≤K2s/(2s-1)∥∇w∥2s/(2s-1),Ω≤K2s/(2s-1)|Ω|1/(2s′)∥∇w∥2,Ω.
This concludes the proof of Proposition 2.Proposition 3 (Neumann).
If|ΓD|=0, then there exists a unique u∈V2 being a weak solution to (1)–(3). Moreover, the following estimate holds:
(17)∥∇u∥2,Ω≤1a#(∥f∥2,Ω+Cn(∥f∥t,Ω,∥h∥s,Γ)),
where Cn(A,B) is given as in Proposition 2.Proof.
The existence and uniqueness of a weak solutionu∈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 solutionu∈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.W1,q Solvability (q≤n/(n-1))

The existence of a solution is recalled in the following proposition in accordance withL1 theory, that is, via solutions obtained by limit approximation (SOLA) (cf. [1–4, 37]), in order to determine the explicit constants.Proposition 5.
Letg=0 on ΓD (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∈Vq solving (11) for every v∈Vq′. Moreover, one has the following estimate:
(18)∥∇u∥q,Ω≤C1(Ω,n,q)(∥f∥2,Ωa#+ϰ(∥f∥1,Ω+∥h∥1,Γ)a#)+C2(n,q,∥f∥2,Ωa#+ϰ(∥f∥1,Ω+∥h∥1,Γ)a#),
with ϰ=2 if |ΓD|>0, ϰ=4 if |ΓD|=0, and
(19)C1(Ω,n,q)=|Ω|1/q-1/2{(n-q)3/2q(n-2)(n+q-nq)1/2×21/q+(3n-q(n+1))/[2(n-q)],ifn>2,(2-q)-1/22(6-q)/2-1/q,ifn=2,C2(n,q,A)={A2(n-q)/[q(n-2)](n-qn+q-nq)(n-q)/[q(n-2)]×2(2-q)(nq-n+q)/[q2(n-2)]Sqn(2-q)/[q(n-2)],ifn>2A2/(q-1)|Ω|1/q-1/2×[(q-1)(3-q)(3-q)/(q-1)](2-q)/(2q)(2-q)1/(q-1)×2ℓSq(3-q)/(q-1),ifn=2,
where ℓ∈]2,+∞[ is explicitly given in (30).Proof.
For eachm∈ℕ, take
(20)fm=mfm+|f|∈L∞(Ω),hm=mhm+|h|∈L∞(Γ).
Applying Propositions 2 and 3, there exists a unique solution um∈V2 to the following variational problem:
(21)∫Ω‍a∇um·∇vdx=∫Ω‍f·∇vdx+∫Ω‍fmvdx+∫Γ‍hmvds,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhihhhh∀v∈V2.
In particular, (21) holds for all v∈Vq′ (q′>n).
In order to pass to limit (21) on m  (m→∞) let us establish the estimate (18) for ∇um. The method for estimating is due to Boccardo and Gallouët (see, e.g., [1, 37]).
Case
(
|
Γ
D
|
>
0
)
. Let us choose
(22)v=sign⁡(um)[1-1(1+|um|)s]∈WΓD1,2(Ω)∩L∞(Ω),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhifors>0,
as a test function in (21). Hence it follows that
(23)a#∫Ω‍s|∇um|2(1+|um|)s+1dx≤s∥f∥2,Ω∥∇um(1+|um|)(s+1)/2∥2,Ω+∥f∥1,Ω+∥h∥1,Γ,
and consequently
(24)∫Ω‍|∇um|2(1+|um|)s+1dx≤1(a#)2∥f∥2,Ω2+2sa#(∥f∥1,Ω+∥h∥1,Γ).
By the Hölder inequality with exponents2/q and 2/(2-q)>1, we have
(25)∫Ω‍|∇um|qdx≤(∫Ω‍|∇um|2(1+|um|)s+1dx)q/2×(∫Ω‍(1+|um|)(s+1)q/(2-q)dx)(2-q)/2.
Set
(26)M(s)∶=∥f∥2,Ωa#+(2(∥f∥1,Ω+∥h∥1,Γ)a#s)1/2.
Let us chooses>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 um∈V2↪Vq with (q≤2), we deduce
(27)∥∇um∥q,Ω≤M(s)2(2-q)(nq/(n-q)-1)/(2q)×(|Ω|1/q-1/2+(Sq∥∇um∥q,Ω)n(2-q)/[2(n-q)])≤M(n+q-nqn-q)2(2-q)(nq/(n-q)-1)/(2q)|Ω|1/q-1/2+n(2-q)2(n-q)∥∇um∥q,Ω+q(n-2)2(n-q)×[M(n+q-nqn-q)]2(n-q)/[q(n-2)]×2(2-q)(nq-n+q)/[q2(n-2)]Sqn(2-q)/[q(n-2)],
using the Young inequality AB≤ϵAa/a+Bb/(bϵb/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.
Forn=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
(28)(1+|um|)(s+1)q/(2-q)≤2(s+1)q/(2-q)-1(1+|um|(s+1)q/(2-q))≤2((s+2)q-2)/(2-q)+1-s2(s+12ϵ)(s+1)/(1-s)×22[(s+2)q-2]/[(2-q)(1-s)]+ϵ|um|q*.
Let us choose, for instance, s=2-q<1, and ϵ=[2SqM(s)]-2q/(2-q). Then, we obtain
(29)∥∇um∥q,Ωhhhh≤M(s)(22[(s+2)q-2]/[(2-q)(1-s)](s+12ϵ)(s+1)/(1-s)](2-q)/(2q)ϵ(2-q)/(2q)Sq∥∇um∥q,Ω+|Ω|1/q-1/2hhhhhhhhhhhhh×[2((s+2)q-2)/(2-q)+1-s2(s+12ϵ)(s+1)/(1-s)hhhhhhhhhhhhhhhhh×22[(s+2)q-2]/[(2-q)(1-s)](s+12ϵ)(s+1)/(1-s)](2-q)/(2q))hhhh≤12∥∇um∥q,Ω+M(2-q)|Ω|1/q-1/2hhhhhh×[122((4-q)q-2)/(2q)hhhhhhhhh+12[(q-1)(3-q)(3-q)/(q-1)](2-q)/(2q)hhhhhhhhh×2ℓ[M(2-q)](3-q)/(q-1)Sq(3-q)/(q-1)12],
where ℓ is given by
(30)ℓ=2[(s+2)q-2]-(2-3q)(s+1)2q(1-s)=-5q2+19q-102q(q-1)⟶2+,
as q→2-. Hence, we find (18) with ϰ=2.
Case
(
|
Γ
D
|
=
0
)
. We choose, for s>0,
(31)v=-sign⁡(um)(1+|um|)s+⨍∂Ωsign⁡(um)(1+|um|)sds∈V2(∂Ω);v=-sign⁡(um)(1+|um|)s+⨍Ωsign⁡(um)(1+|um|)sdx∈V2(Ω),
as a test function in (21). Since |v|≤2 a.e. in Ω, it follows that
(32)∫Ω‍|∇um|2(1+|um|)s+1dx≤1(a#)2∥f∥2,Ω2+4sa#(∥f∥1,Ω+∥h∥1,Γ).
Then, we argue as in the above case, concluding (18) with ϰ=4.
For both cases, we can extract a subsequence ofum, still denoted by um, such that it weakly converges to u in W1,q(Ω), where u∈Vq solves the limit problem (11) for all v∈Vq′.Remark 6.
In terms of Proposition5, the terms on the right-hand side of (11) have sense, since v∈W1,q′(Ω)↪C(Ω-) for q′>n, that is, q<n/(n-1).Remark 7.
The existence of a solution, which is given at Proposition5, 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 Proposition5, which will be required in Section 5, with datum belonging to the space of all signed measures with finite total variation ℳ(Ω)=(C0(Ω))′.Proposition 8.
Letg=0 on ΓD (possibly empty), let a∈L∞(Ω) satisfy 0<a#≤a≤a# a.e. in Ω, and, for each x∈Ω, let δx∈ℳ(Ω) be the Dirac delta function. For any 1≤q<n/(n-1) there exists u∈Vq solving
(33)∫Ω‍a∇u·∇vdx=〈δx,v〉ℳ(Ω)×C0(Ω),∀v∈C0(Ω)∩V1,
for every i=1,…,n. Moreover, one has the following estimate:
(34)∥∇u∥q,Ω≤C1(Ω,n,q)ϰa#+C2(n,q,ϰa#),
where the constants C1(Ω,n,q), C2(n,q,A), and ϰ are determined in Proposition 5.Proof.
Since the Dirac delta functionδx∈ℳ(Ω) can be approximated by a sequence {fm}m∈ℕ⊂L∞(Ω) such that
(35)∥fm∥1,Ω=1,lim⁡m→∞∫Ω‍fmφdx=〈δx,φ〉ℳ(Ω)×C0(Ω),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi∀φ∈C0(Ω),
identity (21) holds, with f being replaced by fm, 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 |ΓD|>0.Proposition 9.
Letp>n≥2, |ΓD|>0, and u∈H1(Ω) be any weak solution to (1)–(3) in accordance with Definition 1. If g∈L∞(ΓD), f∈Lp(Ω), f∈Lnp/(p+n)(Ω), and h∈L(n-1)p/n(Γ), then one has
(36)esssup⁡Ω|u|≤esssup⁡ΓD|g|+Cna#|Ω|1/n-1/p×(∥f∥p,Ω+Sp′∥f∥np/p+n,Ω+Kp′∥h∥(n-1)p/n,Γ),
where Cn=2n(p-2)/[2(p-n)]S2 if n>2 and C2=2(3p-2)/[2(p-2)].Proof.
Letk≥k0=esssup⁡{|g(x)|:x∈ΓD}. Choosing v=sign⁡(u)(|u|-k)+  =  sign⁡(u)max⁡{|u|-k,0}∈HΓD1(Ω) as a test function in (11), then ∇v=∇u∈L2(A(k)), and we deduce
(37)a#∫A(k)‍|∇u|2dx≤∥f∥2,A(k)∥∇u∥2,A(k)+∥f∥np/(p+n),Ω∥(|u|-k)+∥np/(np-n-p),Ω+∥h∥(n-1)p/n,Γ∥(|u|-k)+∥(n-1)p/(np-n-p),Γ,
where A(k)={x∈Ω:|u(x)|>k}. Using the Hölder inequality, it follows that
(38)∥f∥2,A(k)≤∥f∥p,Ω|A(k)|1/2-1/p(p>2).
Making use of (7)-(8) and (|u|-k)+∈WΓD1,q(Ω) with q=p′<n and the Hölder inequality, we get
(39)∥(|u|-k)+∥np/(n(p-1)-p),Ω≤Sp′∥∇u∥2,A(k)|A(k)|1/p′-1/2;∥(|u|-k)+∥(n-1)p/(n(p-1)-p),Γ≤Kp′∥∇u∥2,A(k)|A(k)|1/p′-1/2,
if provided by p′<2≤n. Inserting (38)-(39) into (37) we obtain
(40)∥∇u∥2,A(k)≤Cn,p|A(k)|1/2-1/p,
where the positive constant Cn,p is
(41)Cn,p=∥f∥p,Ω+Sp′∥f∥np/(p+n),Ω+Kp′∥h∥(n-1)p/n,Γa#,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi∀n≥2.
Taking into account that A(h)⊂A(k) when h>k>k0, we find
(42)(h-k)|A(h)|1/α≤∥(|u|-k)+∥α,A(h)≤∥(|u|-k)+∥α,A(k)∶=I,∀α≥1.
Case
(
n
>
2
)
. Take α=2*=2n/(n-2) in (42). Making use of (7) and (|u|-k)+∈WΓD1,q(Ω) with q=2 and inserting (40), we deduce
(43)I≤S2∥∇u∥2,A(k)≤S2Cn,p|A(k)|1/2-1/p.
Therefore, we conclude
(44)|A(h)|≤(S2Cn,ph-k)2*|A(k)|β,β=2*(12-1p),
where β>1 if and only if p>n. By appealing to [13, Lemma 4.1] we obtain
(45)|A(k0+S2Cn,p|Ω|(β-1)/α2β/(β-1))|=0.
This means that the essential supremum does not exceed the well-determined constant M∶=k0+S2Cn,p|Ω|(β-1)/α2β/(β-1).
Case
(
n
=
2
). Choose α=2 in (42). Using (10) for (|u|-k)+∈WΓD1,2(Ω) followed by the Hölder inequality and inserting (40), we obtain
(46)I≤12∥∇u∥2,A(k)|A(k)|1/2≤C2,p2|A(k)|1-1/p.
Therefore, we find
(47)|A(h)|≤(C2,p/2h-k)2|A(k)|β,β=2(1-1p),
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 Proposition9.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 Proposition9 up to the boundary.Proposition 11.
Under the conditions of Proposition9, any weak solution u∈HΓD1(Ω) to (1)–(3) satisfies, for p>2(n-1) if n>2,
(48)esssup⁡Ω∪Γ|u|≤esssup⁡ΓD|g|+2(n-1)(p-2)/[p-2(n-1)]a#×|Ω|1/[2(n-1)]-1/p×[(|Ω|(n-2)/[2(n-1)2]S2Kb′+K2Kp′)(|Ω|(n-2)/[2n(n-1)]S2+K2)∥f∥p,Ω+(|Ω|(n-2)/[2n(n-1)]S2Sb′+K2Sp′)∥f∥np/(p+n),Ω+(|Ω|(n-2)/[2(n-1)2]S2Kb′+K2Kp′)∥h∥(n-1)p/n,Γ],
with 1/b=1/p+(n-2)/[2n(n-1)]. For n=2, p>2α/(α-1), and α>1, then any weak solution u∈HΓD1(Ω) to (1)–(3) satisfies
(49)esssup⁡Ω∪Γ|u|≤esssup⁡ΓD|g|+2(p(α+1)-2α)/(p(α-1)-2α)a#×|Ω|(α-1)/(2α)-1/p×[(|Ω|1/(2α)S2α/(α+2)+K2α/(α+1))∥f∥p,Ω+(|Ω|1/2+1/αS2α/(α+2)S2αp/[p(2α-1)-2α]+K2α/(α+1)Sp′)×∥f∥2p/(p+2),Ω+(|Ω|1/αS2α/(α+2)K2αp/[p(2α-1)-2α]+K2α/(α+1)Kp′)×∥h∥p/2,Γ].Proof.
Letk≥k0=esssup⁡{|g(x)|:x∈ΓD}. For each b>2, f∈Lb(Ω), f∈Lnb/(b+n)(Ω), and h∈L(n-1)b/n(Γ), (40) reads
(50)∥∇u∥2,A(k)∩Ω≤Cn,b|A(k)|1/2-1/b,
where A(k)={x∈Ω-:|u(x)|>k}. With this definition, the integral I from the proof of Proposition 9 reads
(51)I=∥(|u|-k)+∥α,A(k)∩Ω+∥(|u|-k)+∥α,A(k)∩Γ,
and, for h>k>k0, we have
(52)(h-k)|A(h)|1/α≤I,∀α≥1.
Case
(
n
>
2
)
. Take α=2(n-1)/(n-2)<2*=2n/(n-2). Making use of (7)-(8) and (|u|-k)+∈WΓD1,q(Ω) with q=2, we deduce
(53)I≤S2|A(k)|1/α-1/2*∥∇u∥2,A(k)∩Ω+K2∥∇u∥2,A(k)∩Ω.
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
(54)I≤(S2Cn,b+K2Cn,p)|A(k)|1/2-1/p,1b=1p+n-22n(n-1).
Therefore, we conclude(55)|A(h)|≤(S2Cn,b+K2Cn,ph-k)α|A(k)|β,hhhhhhhhhhhhhhhhiβ=α(12-1p),
where β>1 if and only if p>2(n-1). Notice that
(56)Cn,b≤|Ω|(n-2)/[2n(n-1)]×(∥f∥p,Ω+Sb′∥f∥np/(p+n),Ω)+|Ω|(n-2)/[2(n-1)2]Kb′∥h∥(n-1)p/n,Γ.
Case
(
n
=
2
). Using (7) with q=2α/(α+2)<2, (8) with q=2α/(α+1)<2, and the Hölder inequality, we have
(57)∥(|u|-k)+∥α,A(k)∩Ω≤S2α/(α+2)∥∇u∥2,A(k)∩Ω|A(k)|1/α;∥(|u|-k)+∥α,A(k)∩Γ≤K2α/(α+1)∥∇u∥2,A(k)∩Ω|A(k)|1/(2α).
Thus, we deduce
(58)I≤S2α/(α+2)|A(k)|1/α∥∇u∥2,A(k)∩Ω+K2α/(α+1)|A(k)|1/(2α)∥∇u∥2,A(k)∩Ω.
Applying (50) twice (b=2αp/(p+2α)>2 and b=p>2α/(α-1)>2), we conclude
(59)|A(h)|≤(S2α/(α+2)C2,2αp/(p+2α)+K2α/(α+1)C2,ph-k)α|A(k)|β,
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 BR(x)⊂⊂Ω and η∈W01,∞(BR(x)) 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.
Letn≥2, |ΓD|>0, f=0 in Ω, f,g,h=0, respectively, in Ω, on ΓD, 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(60)∫Ω‍η2|∇u|2dz≤4a#a#∫Ω‍u2|∇η|2dz,for anyη∈W1,∞(ℝn).(2)
For arbitraryx∈Ω,R>0, andk0≥0,(61)ess
supΩ(x,R/2)|u|≤k0+23n+2+(3n)2/4c3n/2ωn×(R-n∫Ω(x,R)‍(|u|-k0)2dz)1/2,wherec=S2n/(n+2)(1+2a#/a#)andΩ(x,r)=Ω∩Br(x)for anyr>0.Proof.
(
1
) Let us choose v=uη2∈HΓD1(Ω) as a test function in (11). Thus, applying the Hölder inequality we deduce
(62)∫Ω‍aη2|∇u|2dz=-2∫Ω‍aη∇u·∇ηudz≤12∫Ω‍aη2|∇u|2dz+2∫Ω‍au2|∇η|2dz.
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,
(63)(h-k)|A(h,r)|1/2≤∥|u|-k∥2,A(h,r)≤∥|u|-k∥2,A(k,r),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi∀r>0.
Fix0<r<R≤R0, and let us take v=sign⁡(u)(|u|-k)+η2∈HΓD1(Ω) as a test function in (11), where η∈W1,∞(ℝn) is the cut-off function defined by η≡1 in Br(x), η≡0 in ℝn∖BR(x), and η(y)=(R-|y-x|)/(R-r) for all y∈BR(x)∖Br(x). Thus, we have that 0≤η≤1 in ℝn and |∇η|≤1/(R-r) a.e. in BR(x) and that (60) reads
(64)∫A(k,R)‍η2|∇u|2dz≤4a#a#∫A(k,R)‍|∇η|2(|u|-k)2dz.
Making use of (7) and η(|u|-k)+∈WΓD1,q(Ω) with exponent q=2n/(n+2)<2≤n and the Hölder inequality, we have
(65)∥|u|-k∥2,A(k,r)≤∥η(|u|-k)+∥2,Ω≤S2n/(n+2)×∥∇(η(|u|-k)+)∥2n/(n+2),Ω≤S2n/(n+2)×∥(|u|-k)+∇η+η∇u∥2,A(k,R)|A(k,R)|1/n.
Applying the properties ofη, inserting (64) into (65), and gathering the second inequality from (63), we get
(66)∥|u|-k∥2,A(h,r)≤|A(k,R)|1/ncR-r∥|u|-k∥2,A(k,R).
In order to apply [13, Lemma 5.1] that leads to
(67)ϕ(k0+2(α+β)β/[α(β-1)]c1/α(σR0)γ/α×[ϕ(k0,R0)](β-1)/α,R0-σR0k0+2(α+β)β/α(β-1)c1/α(σR0)γ/α)=0,∀σ∈]0,1[
with γ=1, α=2/(3n), and β=1+2/(3n)>1, we use (66) and inequality (63) with r replaced by R, obtaining
(68)ϕ(h,r)∶=|A(h,r)|∥|u|-h∥2,A(h,r)≤|A(k,R)|1+1/n-α/2cR-r1(h-k)α∥|u|-k∥2,A(k,R)1+α=c(R-r)(h-k)2/(3n)ϕ(k,R)1+2/(3n).
Then, taking R=R0 and σ=1/2, (61) holds.
Therefore, the proof of Proposition12 is finished.Remark 13.
The cut-off function explicitly given in Proposition12 does not belong to C1(BR(x)).
Let us prove the corresponding Neumann version of Proposition12.Proposition 14.
Letn≥2, |ΓD|=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 c=S2n/(n+2)(R+1+2a#/a#).Proof.
Fixk0∈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 ofη, the W1,q-Sobolev inequality for η(|u|-k)+∈W1,q(Ω) with exponent q=2n/(n+2)<2≤n, and the Hölder inequality, we have
(69)∥|u|-k∥2,A(k,r)≤∥η(|u|-k)+∥2,Ω≤S2n/(n+2)∥η(|u|-k)+∥1,2n/(n+2),A(k,R)≤S2n/(n+2)((1+1R-r)∥|u|-k∥2,A(k,R)hhhhhhhhi+∥η∇u∥2,A(k,R)(1+1R-r))|A(k,R)|1/n.
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 setΩ(x,R) is open and bounded but may be neither convex nor connected (see Figure 1).Figure 1
2D schematic representations of a Lipschitz domainΩ (with |Ω|=1) representing an electrolytic cell, Γ being the union of the recipient and air contact boundaries, ΓD representing the surface of two electrodes submerged in the electrolyte, and Ω(x,r)=Ω∩Br(x) denoting the subset centered at three different points.Finally, we state the following local version that will be required in Section5. 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.
Letn≥2, a∈L∞(Ω) satisfy 0<a#≤a≤a# a.e. in Ω, x∈Ω, and let R>0 be such that |Ω∩∂BR(x)|>0. If u∈H1(Ω(x,R)) solves the local variational formulation
(70)∫Ω(x,R)‍a∇u·∇vdz=0,∀v∈HΩ∩∂BR(x)1(Ω(x,R)),
then one has
(71)ess
supΩ(x,R/2)|u|≤23n+2+(3n)2/4S2n/(n+2)3n/2(R+1+2a#a#)3n/2×ωnRn/2(∫Ω(x,R)‍u2dz)1/2.Proof.
First we argue as in Proposition12, with k0=0. The validity of properties (63) and (64) remains. The application of the W1,2n/(n+2)-Sobolev inequality is available for η(|u|-k)+∈W1,2n/(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 eachx∈Ω, one says that E is a Green kernel associated with (1)–(3), if it solves
(72)∇·(a∇E(x,·))=δx,
where δx is the Dirac delta function at the point x, in the following sense: there is q>1 such that E verifies the variational formulation:
(73)∫Ω‍a(y)∇yE(x,y)·∇yv(y)dy=v(x),∀v∈Vq.
If |ΓD|>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 functionG verifying
(74)G(x,y)=lim⁡ρ→0Gρ(x,y),∀x∈Ω,a.e.y∈Ω,x≠y,
is standard if n>2 (see, e.g., [32, 34]), with Gρ(x,·)∈H01(Ω) being the unique solution to
(75)∫Ω‍a∇Gρ·∇vdy=1|Bρ(x)|∫Bρ(x)‍vdy,
for all v∈H01(Ω), for any x∈Ω, and for ρ>0 such that Bρ(x)⊂Ω. Moreover, G satisfies, for some positive constant C(n) and n>2 [32, Theorem 1.1],
(76)G(x,y)≤C(n)(1+log⁡(a#a#))(a#)(n-2)/2(a#)n/2|x-y|n-2.
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.
Letn≥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 G=G(x,·)∈WΓD1,q(Ω)∩H1(Ω∖Br(x)) according to Definition 17 and enjoying the following estimates:
(77)∥∇G∥q,Ω≤C1(Ω,n,q)A+C2(n,q,A);(78)∥G∥qn/(n-q),Ω≤Sq(C1(Ω,n,q)A+C2(n,q,A)),
with A=2/a# 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
(79)|G(x,y)|≤C(a#)(δ(Ω)2+1+2a#a#)3n/2|x-y|1-n/q,
for a.e. x,y∈Ω such that x≠y, where
(80)C(a#)=23n+1+n/q+(3n)2/4S2n/(n+2)3n/2ωn3/2+1/n-1/q×Sq(C1(Ω,n,q)A+C2(n,q,A)).Proof.
For anyx∈Ω and ρ>0 such that Bρ(x)⊂⊂Ω, the existence and uniqueness of Gρ=Gρ(x,·)∈HΓD1(Ω) solving (75), for all v∈HΓD1(Ω), are due to Proposition 2 with f=0 a.e. in Ω, g,h=0 a.e. on, respectively, ΓD and Γ, and f=χBρ(x)/|Bρ(x)| belonging to L2n/(n+2)(Ω) if n>2 and to L2(Ω) if n=2. Moreover, (12) reads
(81)∥∇Gρ∥2,Ω≤1a#×{S2ωn1/n-1/2ρ1-n/2,ifn>2,|Ω|1/2(2ω2ρ2)-1/2,ifn=2.
Therefore, for any r>0 such that Br(x)⊂⊂Ω, there exists G∈H1(Ω∖Br(x)) such that
(82)Gρ⇀GinH1(Ω∖Br(x))asρ⟶0+.
In order forG to correspond to the well defined one in (74), the W1,q estimate (77) is true for Gρ due to (18) with ϰ=2, by applying Proposition 5 with f=0, g,h=0, and f=χBρ(x)/|Bρ(x)|∈L1(Ω). Then, we can extract a subsequence of Gρ, still denoted by Gρ, weakly converging to G in W1,q(Ω) as ρ tends to 0, with G∈Vq solving (73) for all v∈Vq′. 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(83)∫Ω‍a|∇(Gρ-|Gρ|)|2dy=2|Bρ(x)|(∫Bρ(x)‍Gρdy-∫Bρ(x)‍|Gρ|dy)≤0.
Then, Gρ=|Gρ|, and, by passing to the limit as ρ tends to 0, the nonnegativeness claim holds.
For eachx,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
(84)|G(x,y)|≤CRn/2(∫Ω(y,R)‍G2(x,z)dz)1/2≤Cωn1/2+1/n-1/qRn(1/2+1/n-1/q)-n/2∥G∥qn/(n-q),Ω,
with C=23n+2+(3n)2/4(∂(Ω)/2+1+2a#/a#)3n/2S2n/(n+2)3n/2ωn.
Hence, using (78) we conclude (79), which completes the proof of Proposition 18.Remark 19.
Sinceqn/(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∈Lp(Ω) for any p∈[1,n/(n-2)[.For eachx∈Ω, the Neumann function is defined as N=G+w being the solution of the regularity problem [33, Definition 2.5], where G∈H01(Ω) 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]:
(85)∫Ω‍a∇w·∇vdy=⨍∂Ωvdsy-⨍∂Ωvdωyx,∀v∈V2(∂Ω).
Here, ωx is the L-harmonic measure [38]; that is, it is unique probability measure on ∂Ω such that
(86)Lg(x)=∫∂Ω‍gdωyx,
due to the Riesz representation theorem applied to the continuous linear functional L:g∈C(∂Ω)→Lg(x)∈C(Ω-), where Lg(x) is the solution to the Dirichlet problem (1) with f=0 and f=0 and (3) with ΓD=∂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 eachx∈Ω, N(x,·) admits an extension N~(x,·) across ∂Ω (cf. [33, Lemmas 2.9 and 2.11]) to the domain Ω~ which is such that
(87)y∈∂Ω~⟺y∈ℝn∖Ω-:y=y*+(y*-𝒯(y*)),hhhhhhhhhhhhhhhhhhhhhiforsomey*∈∂Ω,
where 𝒯 is the homothety function that reduces ∂Ω into its half, that is, the homothetic boundary with measure |∂Ω|/2. That is, each y∈Ω~∖Ω- is the reflection of 𝒯(yδ) across ∂Ω in the following sense:
(88)y=y*+(y*-yδ)y*-𝒯(y*)|y*-𝒯(y*)|,
where yδ∈Ω is such that yδ=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.
Letn≥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,·)∈Vq solving (72) that satisfies (77)-(78) and (79), with A=2/a#.Proof.
For eachx∈Ω and ρ>0 such that Bρ(x)⊂Ω, the existence of a unique Neumann function Nρ(x,·)∈V2 solving (75), for all v∈V2, is consequence of Proposition 3 with f=0, g,h=0, and f=χBρ(x)/|Bρ(x)|∈Lt(Ω) for t=2n/(n+2) if n>2 and any t<2 if n=2. Arguing as in the proof of Proposition 18, Nρ belongs to W1,q(Ω), uniformly for x∈Ω, according to (18) with ϰ=4. Therefore, we may pass to the limit as ρ→0, finding N∈Vq, solving (72). The remaining estimates (78)-(79), under A=2/a#, are obtained exactly as in the proof of Proposition 14.Hereafter,∂xi denotes the partial derivative ∂/∂xi.Proposition 22.
Letn≥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,  ∂xiE(x,·)∈Vq∩L∞(Ω) is uniformly bounded for x∈Ω. In particular, it satisfies (77)-(78) and (79), where A=ϰ/a#, with ϰ=2 if |ΓD|>0 and ϰ=4 if |ΓD|=0.Proof.
For eachx∈Ω, we may approximate ∂xiE(x,·) by {∂xiEρ}ρ>0, where Eρ=Eρ(x,·)∈V2 solves (75) for every ρ>0 such that Bρ(x)⊂Ω. Since ∂xi[χBρ(x)/(ρn|B1(0)|)] is a Dirac delta function, Proposition 8 ensures that ∂xiEρ verifies (77) and also (78) by the Sobolev inequality (7), with A=ϰ/a#, where ϰ=2 if |ΓD|>0 and ϰ=4 if |ΓD|=0. Consequently, (77)-(78) hold, by passage to the weak limit.
To prove estimate (79) for ∂xiE(x,·), let us take y∈Ω such that R=|x-y|/2>0. Thus, E(x,·)∈H1(Ω(y,R))∩Vq verifies ∇·(a∇∂xiE)=0 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 ∂xiE.Remark 23.
Notice thatq′>n implies that E is not an admissible test function in
(89)∫Ω‍a(y)∇∂xiE(x,y)·∇v(y)dy=∂xiv(x),∀v∈Vq′,
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 xi. We emphasize that, for each x∈Ω and any r>0 such that r<dist⁡(x,∂Ω), the symmetric function E(x,·)∈Vq∩H1(Ω∖Br(x)) verifies, by construction, the limit system of identities
(90)∫Ω‍aφ2∇(∇xE)(x,·)·∇vdy=-2∫Ω‍aφv∇(∇xE)(x,·)·∇φdy,
for any φ∈W1,∞(ℝn) such that supp⁡(φ)⊂ℝn∖B∂(Ω)(x)∪Br(x)¯, 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.
Leta∈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,
(91)|a(x)-a(y)|≤ω(|x-y|),0<Ca∶=∫01‍ω(t)tdt<∞,
then for each x∈Ω and R>0, any function u∈W1,1(Ω) solving
(92)∇·(a∇u)=0inΩ(x,R),
in the sense of distributions, enjoys, a.e. y∈Ω,
(93)|∇u(y)|≤a#δ(Ω)Canωn(4π3+n)1|x-y|×∫Bd(y)‍|u(z)||y-z|ndz,
where
(94)d={|x-y|2,if|x-y|<2r,|x-y|ν,if|x-y|=νr
for some 2≤ν<δ(Ω)/r and 0<r<min⁡{1,δ(y)} with δ(y):=dist⁡(y,∂Ω).Proof.
By density, sinceu∈W1,1(Ω) there exists a sequence {um}m∈ℕ⊂C1(Ω-) such that um→u in W1,1(Ω). In particular, um→u in L1(Ω) and ∇um→∇u a.e. in Ω. Thus, it is sufficient to prove estimate (93), under the assumption u∈C1(Ω-).
Fixx∈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
(95)sup⁡z∈Br(y)|x-z||∇u(z)|=M,|x-y||∇u(y)|>2bM,
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 Bd(y)⊂⊂Ω.
In order to determine the final constant in (93), let η∈C01(Bd(y))∩W2,∞(Ω) be the cut-off function explicitly given by
(96)η(z)={1,ifz∈Bd/2(y),2-1(1+cos⁡[4π3d2(|z-y|2-d24)]),ifd2≤|z-y|<d,0,ifz∈Ω∖Bd(y).
Thus, η satisfies 0≤η≤1,
(97)|∇η(z)|≤c1d≤c1ν|x-y|-1,∀z∈Ω(c1=4π3),|Δη(z)|≤c1c2d2≤c1c2ν2|x-y|-2,a.e.z∈Ω(c2=c1+n).
For w∈B:=Bd(y), we multiply (92) by GL(w,·)η/a(y) where GL is the fundamental solution of Laplace equation:
(98)GL(w,z)=CL{|w-z|2-n2-n,ifn>2,ln⁡[|w-z|],ifn=2
with
(99)CL∶=ba#r(c1ν+2(n+1))Canωn,
and we integrate over B to get
(100)0=∫B‍a(z)a(y)∇zu(z)·∇z(η(z)GL(w,z))dz=∫B‍a(z)-a(y)a(y)∇zu(z)·∇z(η(z)GL(w,z))dz-∫B‍u(z)Δzη(z)GL(w,z)dz-2∫B‍u(z)∇zη(z)·∇zGL(w,z)dz-u(w)η(w),
taking into account the use of integration by parts. Differentiating the above identity with respect to w and setting w=y it results in
(101)∇u(y)=I1+I2,
where
(102)I1=∫Ba(z)-a(y)a(y)‍×(∇yGL(y,z)∇zη(z)+η(z)∇y∇zGL(y,z))·∇zu(z)dz;I2=-∫B‍u(z)∇yGL(y,z)Δzη(z)dz-2∫B‍u(z)∇y∇zGL(y,z)·∇zη(z)dz.
Using the lower bound of a, the definition of GL, and the properties of η, we have
(103)I1≤CLa#∫B‍|a(z)-a(y)|hhhhhhi×(c1ν|y-z|n-1|x-y|+n+1|y-z|n)|∇zu(z)|dz;I2≤CLc1ν×∫B‍|u(z)|(c2ν|y-z|n-1|x-y|2+2(n+1)|y-z|n|x-y|)dz.
By appealing to (95), we obtain
(104)2bM<|x-y||∇u(y)|≤CLMa#∫B|a(z)-a(y)||x-z|‍hhhhhhhhi×(c1ν|y-z|n-1+(n+1)|x-y||y-z|n)dzhh+CLc1ν∫B‍|u(z)|(c2ν|y-z|n-1|x-y|+2(n+1)|y-z|n)dz.
Considering that, for allx,y∈Ω and z∈Bd(y),
(105)|y-z|≤|x-y|≤|x-z|+|z-y|
we obtain
(106)2bM<CLMa#(∫B|a(z)-a(y)|c1ν+n+1|y-z|n-1|x-z|dz‍hhhhhhhh+(n+1)∫B‍|a(z)-a(y)||y-z|ndz)hh+CLc1ν(c2ν+2(n+1))∫B‍|u(z)||y-z|ndz.
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∈Bd(y) we have |y-z|<|x-y|/ν. Hence, we find (ν-1)|y-z|<|x-z| and consequently
(107)1|y-z|n-1|x-z|<1(ν-1)|y-z|n≤1|y-z|n,ν≥2.
If d=|x-y|/2 and z∈Bd(y), clearly (107) holds denoting ν=2.
Returning to (106), substituting the value of CL from (99) with r≤1, and dividing by b>0, we write it as
(108)2M<MCanωn∫B‍|a(z)-a(y)||y-z|ndz+a#δ(Ω)c1Canωn(1+nc1)∫B|u(z)||y-z|n‍dz.
In ann-dimensional Euclidean space, the spherical coordinate system consists of a radial coordinate t and n-1 angular coordinates ϕ1,…,ϕn-2∈[0,π] and ϕn-1∈[0,2π[, and the Cartesian coordinates are z1=y1+tcos⁡(ϕ1), z2=y2+tsin(ϕ1)cos⁡(ϕ2),…,zn-1=yn-1+tsin(ϕ1)⋯sin(ϕn-2)cos⁡(ϕn-1), and zn=yn+tsin(ϕ1)⋯sin(ϕn-1). Since the Jacobian of this transformation is tn-1sinn-2(ϕ1)⋯sin(ϕn-2) and
(109)ωn=1n∫0π‍⋯∫0π‍∫02π‍sinn-2(ϕ1)hhhhhhhhhhhhh×sinn-3(ϕ2)⋯sin(ϕn-2)dϕ1⋯dϕn-1,
applying (91), we deduce
(110)∫B‍|a(z)-a(y)||y-z|ndz≤nωn∫0dω(t)t‍dt≤Canωn.
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
(111)lim⁡r→0sup⁡ρ≤r⨍Bρ|f(x)-(⨍Bρf(y)dy)|dx=0,
where Bρ 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
(112)BMO={sup⁡Bf∈Lloc⁡1(Ω):hhhsup⁡B⨍B∣f(x)-(⨍Bf(y)dy)dx<∞},
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.
Letn≥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,
(113)|∇yE(x,y)|≤C(Ω,n,q,a)|x-y|-n/q,
with
(114)C(Ω,n,q,a)=δ(Ω)Ca(4π3+n)23n+2n/q+(3n)2/4×S2n/(n+2)3n/2ωn1/n-1/q+3/2Sq×(C1(Ω,n,q)ϰa#+a#C2(n,q,ϰa#))×(δ(Ω)2+1+2a#a#)3n/2,
where ϰ=2 if |ΓD|>0, ϰ=4 if |ΓD|=0, and the constants C1(Ω,n,q) and C2(n,q,ϰ/a#) are explicitly given in Proposition 5.Proof.
Letx∈Ω be arbitrary. Using property (93) and applying (79), we get
(115)|∇yE(x,y)|≤δ(Ω)Ca(C1(Ω,n,q)ϰa#+a#C2(n,q,ϰa#))×C|x-y|(∂(Ω)2+1+2a#a#)3n/2×∫Bd(y)‍|x-z|1-n/q|y-z|ndz,
with
(116)C=(4π3+n)23n+1+n/q+(3n)2/4S2n/(n+2)3n/2ωn1/n-1/q+1/2nSq.
Considering that, for all x,y∈Ω and z∈Bd(y) with d≤|x-y|/2,
(117)|y-z|≤|x-y|2≤12(|x-z|+|z-y|)hhhhhi⟹{|y-z|≤|x-z|,|x-y|≤2|x-z|,
we compute
(118)∫Bd(y)‍|x-z|1-n/q|y-z|ndz≤2n/q∫Bd(y)‍|x-y|-n/q|y-z|n-1dz=2n/qdnωn|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 (113).


## 6.W1,p Constants (p>n)

Letp>n, g=0 on ΓD (possibly empty), and u∈Vp solve (11) for all v∈Vp′. 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 C1,1 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 W1,∞ and piecewise C1,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 classC1,1, W1,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 W1,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 W1,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 W1,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 solutionu∈Vp(p>n) and the Green kernel E associated with (1)–(3); that is, E∈Vp′ 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 v=E∈Vp′ and v=u∈Vp as test functions in (11) and (73), respectively, obtaining the Green representation formula:
(119)u(x)=𝒯(f)(x)+𝒮(f)(x)+𝒦(h)(x),x∈Ω,
where 𝒯, 𝒮, and 𝒦 are the layer potential operators defined by
(120)𝒯(f)=∑j=1n‍∫Ω‍fj(y)∂E∂yj(·,y)dy;𝒮(f)=∫Ω‍f(y)E(·,y)dy;𝒦(h)=∫∂Ω‍χΓ(y)h(y)E(·,y)dsy.For every0<λ<n, u∈Ls(ℝn), and v∈Lt(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:
(121)∫∫ℝnu(x)|x-y|-λv(y)dxdy≤C(n,s,λ)∥u∥s,ℝn∥v∥t,ℝn,
where the constant is sharp [41], if s=t=2n/(2n-λ), defined by
(122)C(n,λ)=πλ/2Γ((n-λ)/2)Γ(n-λ/2)[Γ(n)Γ(n/2)]1-λ/n.In the presence of the Hardy-Littlewood-Sobolev inequality, we prove the followingW1,p estimate.Proposition 28.
Letp>1, f∈Lt(Ω) with t∈]pn/(p+n),p[, f=0 in Ω, g=0 on ΓD (possibly empty), h=0 on Γ, and let a∈L∞(Ω) satisfy 0<a#≤a≤a# a.e. in Ω and (91). If u∈Vp solves (11), for all v∈Vp′, then u satisfies
(123)∥∇u∥p,Ω≤C(n,p′,nq)C(Ω,n,q,a)∥f∥t,Ω,
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
(124)∥∇u∥p,Ω≤C(n,2np)C(Ω,n,p2,a)∥f∥p′,Ω.Proof.
Since∇u∈Lp(Ω), (119) holds. Differentiating it, for i=1,…,n, we deduce
(125)∂u∂xi(x)=∫Ω‍∂E∂xi(x,y)f(y)dy.
Let w∈Lp′(Ω) be arbitrary such that ∥w∥p′,Ω=1. Using (113) for any 1<q<n/(n-1) and applying the Fubini-Tonelli theorem, we find
(126)∫Ω‍∇u(x)·w(x)dx≤C(Ω,n,q,a)∫∫Ω|w(x)||x-y|-n/q|f(y)|dxdy.
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 choose1<q=p/2<n/(n-1) and we use (121) with s=t=p′.Having the results established in Section5 in mind, we find a W1,p estimate for weak solutions where regularity (91) of the leading coefficient is not a necessary condition.Proposition 29.
Letp>n, f∈Lp(Ω), f∈Lp(Ω), g=0 on ΓD (possibly empty), h∈Lp(Γ), a∈L∞(Ω) satisfy 0<a#≤a≤a# a.e. in Ω, and let u∈Vp solve (11), for all v∈Vp′. Then u satisfies
(127)∥∇u∥p,Ω≤|Ω|1/p(C1(Ω,n,p′)a#+C2(n,p′,1a#))×(∥f∥p,Ω+∥f∥p,Ω+|Γ|1/[p/n-1)]×Kp′∥h∥p,∂Ω∥f∥p,Ω+∥f∥p,Ω+|Γ|1/[p/n-1)]),
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
(128)∂u∂xi(x)=∑j=1n‍∫Ω‍∂2E∂xi∂yj(x,·)fjdy+∫Ω‍∂E∂xi(x,·)fdy+∫Γ‍∂E∂xi(x,·)hdsy.
Letw∈Lp′(Ω) be arbitrary such that ∥w∥p′,Ω=1; applying the Fubini-Tonelli theorem and next the Hölder inequality, it follows
(129)∫Ω‍∇u·wdx≤∥f∥p,Ω(∫Ω‍|∑i,j=1n‍∫Ω‍∂2E∂xi∂yj(·,y)widx|p′dy)1/p′+∥f∥p,Ω(∫Ω‍|∑i=1n‍∫Ω‍∂E∂xi(·,y)widx|p′dy)1/p′+∥h∥p,Γ(∫Γ‍|∑i=1n‍∫Ω‍∂E∂xi(·,y)widx|p′dsy)1/p′.
Let us estimate the last integral on the right-hand side in (129), since the two other integrals are similarly bounded:
(130)ℐ∶=(∫Γ‍|∫Ω‍∑i=1n‍∂E∂xi(x,y)wi(x)dx|p′dsy)1/p′≤|Ω|1/p(∫Ω‍(∫Γ‍|∇xE(x,y)|p′dsy)|w(x)|p′dx)1/p′,
where |Ω|1/p is due to the embedding Lp′(Ω)↪L1(Ω). Considering that ∂xiE(x,·)∈W1,p′(Ω) uniformly for x∈Ω (cf. Proposition 22) and consequently also ∇xE(x,·)∈W1,p′(Ω)↪Lp′(n-1)/(n-p′)(Γ)↪Lp′(Γ) uniformly in x∈Ω, then we obtain
(131)ℐ≤|Ω|1/p|Γ|1/[p(n-1)]Kp′×(C1(Ω,n,p′)a#+C2(n,p′,1a#)).
Finally, inserting the above inequality into (129), the proof of Proposition 29 is finished.



---

*Source: 845760-2014-03-31.xml*