Explicit Estimates for Solutions of Mixed Elliptic Problems

Abstract

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝ^𝑛 (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 ℝ^𝑛(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 𝜕Ω=Γ_𝐷∪Γ. 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: ()()−∇⋅𝑎∇𝑢=𝑓−∇⋅𝐟,inΩ;𝑎∇𝑢−𝐟⋅𝐧=ℎ,onΓ;𝑢=𝑔,onΓ_𝐷, (1)(2)(3) where 𝐧 is the unit outward normal to the boundary 𝜕Ω.

Set for any 𝑞≥1𝑉_𝑞=𝑊_Γ^1,𝑞_𝐷(Ω)=𝑣∈𝑊^1,𝑞(Ω)∶𝑣=0onΓ_𝐷,ifΓ_𝐷𝑉>0,_𝑞(𝜕Ω)=𝑣∈𝑊^1,𝑞(Ω)∶∫_𝜕Ω,𝑉𝑣d𝑠=0_𝑞(Ω)=𝑣∈𝑊^1,𝑞(Ω)∶∫_Ω𝑣d𝑥=0,otherwise, (4) 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 ℝ^𝑛. The significance of |⋅| depends on the kind of the set.

Defining the 𝑊^1,𝑞 norm by ‖𝑣‖_1,𝑞,Ω1∶=𝐶_∗‖𝑣‖+1^𝑞,Ω+‖∇𝑣‖_𝑞,Ω, (5) with 𝐶_∗ being any of the Poincaré constants, 𝑣−_Σ𝑣d𝑠_𝑞,Ω≤𝐶_∗‖∇𝑣‖_𝑞,Ω,∀𝑣∈𝑊^1,𝑞(Ω);𝑣−_Ω𝑣d𝑥_𝑞,Ω≤𝐶_∗‖∇𝑣‖_𝑞,Ω,∀𝑣∈𝑊^1,𝑞(Ω), (6) where Σ⊂𝜕Ω and ⨍_𝐴 means the integral average over the set 𝐴 of positive measure, the Sobolev and trace inequalities read ‖𝑣‖_𝑞∗,Ω≤𝑆_𝑞‖∇𝑣‖_𝑞,Ω;‖𝑣‖_{𝑞∗,𝜕Ω}≤𝐾_𝑞‖∇𝑣‖_𝑞,Ω,∀𝑣∈𝑉_𝑞. (7)(8) 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 𝑊^1,𝑞(Ω)↪𝐿^𝑞∗(Ω) and 𝑊^1,𝑞(Ω)↪𝐿^𝑞∗(𝜕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]) 𝑆_𝑞=𝜋^−1/2𝑛^−1/𝑞𝑞−1𝑛−𝑞^1−1/𝑞×Γ(1+𝑛/2)Γ(𝑛)Γ𝑛/𝑞Γ1+𝑛−𝑛/𝑞^1/𝑛;𝐾_𝑞=𝜋^(1−𝑞)/2𝑞−1𝑛−𝑞^𝑞−1×Γ𝑞(𝑛−1)×2𝑞−1⁻¹Γ(𝑛−1)×2𝑞−1^{−1(𝑞−1)/(𝑛−1)}. (9) 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 ℝ^𝑛; 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)/2}𝑛(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 ‖𝑢‖_{𝑛/(𝑛−1),Ω}≤𝑛^−1/2‖∇𝑢‖_1,Ω. (10) 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 _Ω𝑎∇𝑢⋅∇𝑣d𝑥=_Ω+𝐟⋅∇𝑣d𝑥_Ω𝑓𝑣d𝑥+_Γℎ𝑣d𝑠,∀𝑣∈𝑉₂, (11) where 𝑔∈𝐿²(Γ_𝐷), 𝐟∈𝐋²(Ω), and 𝑓∈𝐿^𝑡(Ω), with 𝑡=(2^∗)^′; 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 𝑔∈𝐻¹(Ω) as an extension of 𝑔∈𝐿²(Γ_𝐷) (i.e., it is such that 𝑔=𝑔 a.e. on Γ_𝐷), the following estimate holds: ‖∇𝑢‖_2,Ω≤𝑎^#𝑎_#+1∇𝑔_2,Ω+1𝑎_#‖𝐟‖_2,Ω+𝐶_𝑛𝑓_𝑡,Ω,‖ℎ‖_𝑠,Γ, (12) where 𝐶_𝑛(𝐴,𝐵)=𝑆₂𝐴+𝐾₂𝐵 if 𝑛>2, 𝐶₂(𝐴,𝐵)=|Ω|^1/𝑡′𝑆_{2𝑡/(3𝑡−2)}𝐴+|Ω|^1/(2𝑠′)𝐾_{2𝑠/(2𝑠−1)}𝐵 if 𝑡<2, and 𝐶₂(𝐴,𝐵)=|Ω|^1/𝑡′𝐴/√2+|Ω|^1/(2𝑠′)𝐾_{2𝑠/(2𝑠−1)}𝐵 if 𝑡≥2. In particular, 𝑢−𝑔∈𝐻¹_Γ𝐷(Ω) is unique.

Proof. For 𝑔∈𝐿²(Γ_𝐷) there exists an extension 𝑔∈𝐻¹(Ω) such that 𝑔=𝑔 a.e. on Γ_𝐷. The existence and uniqueness of a weak solution 𝑤∈𝐻¹_Γ𝐷(Ω) are well known via the Lax-Milgram lemma, to the variational problem: _Ω𝑎∇𝑤⋅∇𝑣d𝑥=_Ω𝑔+𝐟−𝑎∇⋅∇𝑣d𝑥_Ω𝑓𝑣d𝑥+_Γℎ𝑣d𝑠, (13) for all 𝑣∈𝐻¹_Γ𝐷(Ω). Therefore, the required solution is given by 𝑢=𝑤+𝑔.
If 𝑔=0, 𝑔=0 and then 𝑢≡𝑤.
Taking 𝑣=𝑤∈𝐻¹_Γ𝐷(Ω) as a test function in (13), applying the Hölder inequality, and using the lower and upper bounds of 𝑎, we obtain 𝑎_#‖∇𝑤‖²_2,Ω≤‖𝐟‖_2,Ω+𝑎^#∇𝑔_2,Ω‖∇𝑤‖_2,Ω+𝑓_𝑡,Ω‖𝑤‖_𝑡′,Ω+‖ℎ‖_𝑠,Γ‖𝑤‖_𝑠′,Γ. (14)
For 𝑛>2, this inequality reads 𝑎_#‖∇𝑤‖_2,Ω≤‖𝐟‖_2,Ω+𝑎^#∇𝑔_2,Ω+𝑆₂𝑓_{2𝑛/(2+𝑛),Ω}+𝐾₂‖ℎ‖_{2(𝑛−1)/𝑛,Γ}, (15) 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 ‖𝑤‖_𝑡′,Ω≤|Ω|^{1/𝑡′−1/2}‖𝑤‖_2,Ω≤1√2|Ω|^{1/𝑡′−1/2}‖∇𝑤‖_1,Ω≤1√2|Ω|^1/𝑡′‖∇𝑤‖_2,Ω,();‖𝑤‖𝑡≥2_𝑡′,Ω≤𝑆_{2𝑡/(3𝑡−2)}‖∇𝑤‖_{2𝑡/(3𝑡−2),Ω}≤𝑆_{2𝑡/(3𝑡−2)}|Ω|^1/𝑡′‖∇𝑤‖_2,Ω,();‖𝑤‖𝑡<2_𝑠′,Γ≤𝐾_{2𝑠/(2𝑠−1)}‖∇𝑤‖_{2𝑠/(2𝑠−1),Ω}≤𝐾_{2𝑠/(2𝑠−1)}|Ω|^1/(2𝑠′)‖∇𝑤‖_2,Ω. (16) 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: ‖∇𝑢‖_2,Ω≤1𝑎_#‖𝐟‖_2,Ω+𝐶_𝑛𝑓_𝑡,Ω,‖ℎ‖_𝑠,Γ, (17) 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 𝑣∈𝑉_𝑞′. Moreover, one has the following estimate: ‖∇𝑢‖_𝑞,Ω≤𝐶₁‖𝐟‖Ω,𝑛,𝑞^2,Ω𝑎_#+𝜘𝑓_1,Ω+‖ℎ‖_1,Γ𝑎_#+𝐶₂‖𝐟‖𝑛,𝑞,^2,Ω𝑎_#+𝜘𝑓_1,Ω+‖ℎ‖_1,Γ𝑎_#, (18) with 𝜘=2 if |Γ_𝐷|>0, 𝜘=4 if |Γ_𝐷|=0, and 𝐶₁=|Ω|Ω,𝑛,𝑞_1/𝑞−1/2𝑛−𝑞^3/2𝑞(𝑛−2)𝑛+𝑞−𝑛𝑞^1/2×2^{1/𝑞+(3𝑛−𝑞(𝑛+1))/[2(𝑛−𝑞)]},𝑖𝑓𝑛>2,2−𝑞^−1/22^{(6−𝑞)/2−1/𝑞}𝐶,𝑖𝑓𝑛=2,₂=𝐴𝑛,𝑞,𝐴_{2(𝑛−𝑞)/[𝑞(𝑛−2)]}𝑛−𝑞𝑛+𝑞−𝑛𝑞^{(𝑛−𝑞)/[𝑞(𝑛−2)]}×2^{(2−𝑞)(𝑛𝑞−𝑛+𝑞)/[𝑞2(𝑛−2)]}𝑆_𝑞^{𝑛(2−𝑞)/[𝑞(𝑛−2)]}𝐴,𝑖𝑓𝑛>2_2/(𝑞−1)|Ω|^1/𝑞−1/2×(𝑞−1)3−𝑞^{(3−𝑞)/(𝑞−1)(2−𝑞)/(2𝑞)}2−𝑞^1/(𝑞−1)×2^ℓ𝑆_𝑞^{(3−𝑞)/(𝑞−1)},𝑖𝑓𝑛=2, (19) where ℓ∈]2,+∞[ is explicitly given in (30).

Proof. For each 𝑚∈ℕ, take 𝑓_𝑚=𝑚𝑓𝑚+𝑓∈𝐿^∞(Ω),ℎ_𝑚=𝑚ℎ𝑚+|ℎ|∈𝐿^∞(Γ). (20) Applying Propositions 2 and 3, there exists a unique solution 𝑢_𝑚∈𝑉₂ to the following variational problem: _Ω𝑎∇𝑢_𝑚⋅∇𝑣d𝑥=_Ω𝐟⋅∇𝑣d𝑥+_Ω𝑓_𝑚𝑣d𝑥+_Γℎ_𝑚𝑣d𝑠,∀𝑣∈𝑉₂. (21) In particular, (21) holds for all 𝑣∈𝑉_𝑞′ (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 𝑣=sign𝑢_𝑚11−1+𝑢_𝑚^𝑠∈𝑊_Γ^1,2_𝐷(Ω)∩𝐿^∞(Ω),for𝑠>0, (22) as a test function in (21). Hence it follows that 𝑎_#Ω𝑠∇𝑢_𝑚²1+𝑢_𝑚^𝑠+1‖𝐟‖d𝑥≤𝑠_2,Ω∇𝑢_𝑚1+𝑢_𝑚^(𝑠+1)/2_2,Ω+𝑓_1,Ω+‖ℎ‖_1,Γ, (23) and consequently _Ω∇𝑢_𝑚²1+𝑢_𝑚^𝑠+11d𝑥≤𝑎_#²‖𝐟‖²_2,Ω+2𝑠𝑎_#𝑓_1,Ω+‖ℎ‖_1,Γ. (24)
By the Hölder inequality with exponents 2/𝑞 and 2/(2−𝑞)>1, we have _Ω∇𝑢_𝑚^𝑞d𝑥≤_Ω∇𝑢_𝑚²1+𝑢_𝑚^𝑠+1d𝑥^𝑞/2×_Ω1+𝑢_𝑚^{(𝑠+1)𝑞/(2−𝑞)}d𝑥^(2−𝑞)/2. (25) Set 𝑀(𝑠)‖𝐟‖∶=^2,Ω𝑎_#+2𝑓_1,Ω+‖ℎ‖_1,Γ𝑎_#𝑠^1/2. (26)
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 ∇𝑢_𝑚𝑞,Ω≤𝑀(𝑠)2^{(2−𝑞)(𝑛𝑞/(𝑛−𝑞)−1)/(2𝑞)}×|Ω|^1/𝑞−1/2+𝑆_𝑞∇𝑢_𝑚𝑞,Ω^{𝑛(2−𝑞)/[2(𝑛−𝑞)]}≤𝑀𝑛+𝑞−𝑛𝑞2𝑛−𝑞^{(2−𝑞)(𝑛𝑞/(𝑛−𝑞)−1)/(2𝑞)}|Ω|^1/𝑞−1/2+𝑛2−𝑞2𝑛−𝑞∇𝑢_𝑚𝑞,Ω+𝑞(𝑛−2)2×𝑀𝑛−𝑞𝑛+𝑞−𝑛𝑞𝑛−𝑞^{2(𝑛−𝑞)/[𝑞(𝑛−2)]}×2^{(2−𝑞)(𝑛𝑞−𝑛+𝑞)/[𝑞2(𝑛−2)]}𝑆_𝑞^{𝑛(2−𝑞)/[𝑞(𝑛−2)]}, (27) 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 1+𝑢_𝑚^{(𝑠+1)𝑞/(2−𝑞)}≤2^{(𝑠+1)𝑞/(2−𝑞)−1}1+𝑢_𝑚^{(𝑠+1)𝑞/(2−𝑞)}≤2^{((𝑠+2)𝑞−2)/(2−𝑞)}+1−𝑠2𝑠+12𝜖^{(𝑠+1)/(1−𝑠)}×2^{2[(𝑠+2)𝑞−2]/[(2−𝑞)(1−𝑠)]}+𝜖𝑢_𝑚^𝑞∗. (28) Let us choose, for instance, 𝑠=2−𝑞<1, and 𝜖=[2𝑆_𝑞𝑀(𝑠)]^{−2𝑞/(2−𝑞)}. Then, we obtain ∇𝑢_𝑚𝑞,Ω≤𝑀(𝑠)𝜖^{(2−𝑞)/(2𝑞)}𝑆_𝑞∇𝑢_𝑚𝑞,Ω+|Ω|^1/𝑞−1/2×2^{((𝑠+2)𝑞−2)/(2−𝑞)}+1−𝑠2𝑠+12𝜖^{(𝑠+1)/(1−𝑠)}×2^{2[(𝑠+2)𝑞−2]/[(2−𝑞)(1−𝑠)](2−𝑞)/(2𝑞)}≤12∇𝑢_𝑚𝑞,Ω+𝑀2−𝑞|Ω|^1/𝑞−1/2×2^{((4−𝑞)𝑞−2)/(2𝑞)}+12𝑞−13−𝑞^{(3−𝑞)/(𝑞−1)(2−𝑞)/(2𝑞)}×2^ℓ𝑀2−𝑞^{(3−𝑞)/(𝑞−1)}𝑆_𝑞^{(3−𝑞)/(𝑞−1)}, (29) where ℓ is given by 2)−()ℓ=(𝑠+2𝑞−22−3𝑞𝑠+1()=2𝑞1−𝑠−5𝑞²+19𝑞−102𝑞𝑞−1⟶2⁺, (30) as 𝑞→2⁻. Hence, we find (18) with 𝜘=2.
Case (|Γ_𝐷|=0). We choose, for 𝑠>0, 𝑢𝑣=−sign_𝑚1+𝑢_𝑚^𝑠+_𝜕Ωsign𝑢_𝑚1+𝑢_𝑚^𝑠d𝑠∈𝑉₂();𝑢𝜕Ω𝑣=−sign_𝑚1+𝑢_𝑚^𝑠+_Ωsign𝑢_𝑚1+𝑢_𝑚^𝑠d𝑥∈𝑉₂(Ω), (31) as a test function in (21). Since |𝑣|≤2 a.e. in Ω, it follows that _Ω∇𝑢_𝑚²1+𝑢_𝑚^𝑠+11d𝑥≤𝑎_#²‖𝐟‖²_2,Ω+4𝑠𝑎_#𝑓_1,Ω+‖ℎ‖_1,Γ. (32) 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 𝑣∈𝑉_𝑞′.

Remark 6. In terms of Proposition 5, the terms on the right-hand side of (11) have sense, since 𝑣∈𝑊^1,𝑞′(Ω)↪𝐶(Ω) 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 ℳ(Ω)=(𝐶₀(Ω))^′.

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 𝑎∇𝑢⋅∇𝑣d𝑥=𝛿_Ω𝑥,𝑣_{ℳ(Ω)×𝐶0(Ω)},∀𝑣∈𝐶₀(Ω)∩𝑉₁, (33) for every 𝑖=1,…,𝑛. Moreover, one has the following estimate: ‖∇𝑢‖_𝑞,Ω≤𝐶₁Ω,𝑛,𝑞𝜘𝑎_#+𝐶₂𝑛,𝑞,𝜘𝑎_#, (34) 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 𝑓_𝑚1,Ω=1,lim_𝑚→∞Ω𝑓_𝑚𝜑d𝑥=𝛿_𝑥,𝜑_{ℳ(Ω)×𝐶0(Ω)},∀𝜑∈𝐶₀(Ω), (35) 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 esssup_Ω|𝑢|≤esssup_Γ𝐷𝑔+𝐶_𝑛𝑎_#|Ω|^{1/𝑛−1/𝑝}×‖𝐟‖_𝑝,Ω+𝑆_𝑝′𝑓_{𝑛𝑝/𝑝+𝑛,Ω}+𝐾_𝑝′‖ℎ‖_{(𝑛−1)𝑝/𝑛,Γ}, (36) 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=esssup{|𝑔(𝑥)|∶𝑥∈Γ_𝐷}. Choosing 𝑣=sign(𝑢)(|𝑢|−𝑘)⁺  =  sign(𝑢)max{|𝑢|−𝑘,0}∈𝐻¹_Γ𝐷(Ω) as a test function in (11), then ∇𝑣=∇𝑢∈𝐋²(𝐴(𝑘)), and we deduce 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}, (37) where 𝐴(𝑘)={𝑥∈Ω∶|𝑢(𝑥)|>𝑘}. Using the Hölder inequality, it follows that ‖𝐟‖_2,𝐴(𝑘)≤‖𝐟‖_𝑝,Ω|𝐴(𝑘)|^1/2−1/𝑝𝑝>2. (38)
Making use of (7)-(8) and (|𝑢|−𝑘)⁺∈𝑊_Γ^1,𝑞_𝐷(Ω) with 𝑞=𝑝^′<𝑛 and the Hölder inequality, we get (|𝑢|−𝑘)⁺_{𝑛𝑝/(𝑛(𝑝−1)−𝑝),Ω}≤𝑆_𝑝′‖∇𝑢‖_2,𝐴(𝑘)|𝐴(𝑘)|^{1/𝑝′−1/2};(|𝑢|−𝑘)⁺_{(𝑛−1)𝑝/(𝑛(𝑝−1)−𝑝),Γ}≤𝐾_𝑝′‖∇𝑢‖_2,𝐴(𝑘)|𝐴(𝑘)|^{1/𝑝′−1/2}, (39) if provided by 𝑝^′<2≤𝑛. Inserting (38)-(39) into (37) we obtain ‖∇𝑢‖_2,𝐴(𝑘)≤𝐶_𝑛,𝑝|𝐴(𝑘)|^1/2−1/𝑝, (40) where the positive constant 𝐶_𝑛,𝑝 is 𝐶_𝑛,𝑝=‖𝐟‖_𝑝,Ω+𝑆_𝑝′𝑓_{𝑛𝑝/(𝑝+𝑛),Ω}+𝐾_𝑝′‖ℎ‖_{(𝑛−1)𝑝/𝑛,Γ}𝑎_#,∀𝑛≥2. (41) Taking into account that 𝐴(ℎ)⊂𝐴(𝑘) when ℎ>𝑘>𝑘₀, we find (ℎ−𝑘)|𝐴(ℎ)|^1/𝛼≤(|𝑢|−𝑘)⁺_{𝛼,𝐴(ℎ)}≤(|𝑢|−𝑘)⁺_{𝛼,𝐴(𝑘)}∶=𝐼,∀𝛼≥1. (42)
Case (𝑛>2). Take 𝛼=2^∗=2𝑛/(𝑛−2) in (42). Making use of (7) and (|𝑢|−𝑘)⁺∈𝑊_Γ^1,𝑞_𝐷(Ω) with 𝑞=2 and inserting (40), we deduce 𝐼≤𝑆₂‖∇𝑢‖_2,𝐴(𝑘)≤𝑆₂𝐶_𝑛,𝑝|𝐴(𝑘)|^1/2−1/𝑝. (43) Therefore, we conclude |𝐴(ℎ≤𝑆)|²𝐶_𝑛,𝑝ℎ−𝑘^2∗|𝐴(𝑘)|^𝛽,𝛽=2^∗12−1𝑝, (44) where 𝛽>1 if and only if 𝑝>𝑛. By appealing to [13, Lemma 4.1] we obtain 𝐴𝑘₀+𝑆₂𝐶_𝑛,𝑝|Ω|^{(𝛽−1)/𝛼}2^{𝛽/(𝛽−1)}=0. (45) This means that the essential supremum does not exceed the well-determined constant 𝑀∶=𝑘₀+𝑆₂𝐶_𝑛,𝑝|Ω|^{(𝛽−1)/𝛼}2^{𝛽/(𝛽−1)}.
Case (𝑛=2). Choose 𝛼=2 in (42). Using (10) for (|𝑢|−𝑘)⁺∈𝑊_Γ^1,2_𝐷(Ω) followed by the Hölder inequality and inserting (40), we obtain 1𝐼≤√2‖∇𝑢‖_2,𝐴(𝑘)|𝐴(𝑘)|^1/2≤𝐶_2,𝑝√2|𝐴(𝑘)|^1−1/𝑝. (46) Therefore, we find |𝐴(ℎ≤𝐶)|^2,𝑝/√2ℎ−𝑘²|𝐴(𝑘)|^𝛽1,𝛽=21−𝑝, (47) 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 𝑢∈𝐻¹_Γ𝐷(Ω) to (1)–(3) satisfies, for 𝑝>2(𝑛−1) if 𝑛>2, esssup_Ω∪Γ|𝑢|≤esssup_Γ𝐷𝑔+2^{(𝑛−1)(𝑝−2)/[𝑝−2(𝑛−1)]}𝑎_#×|Ω|^{1/[2(𝑛−1)]−1/𝑝}×|Ω|^{(𝑛−2)/[2𝑛(𝑛−1)]}𝑆₂+𝐾₂‖𝐟‖_𝑝,Ω+|Ω|^{(𝑛−2)/[2𝑛(𝑛−1)]}𝑆₂𝑆_𝑏′+𝐾₂𝑆_𝑝′𝑓_{𝑛𝑝/(𝑝+𝑛),Ω}+|Ω|^{(𝑛−2)/[2(𝑛−1)2]}𝑆₂𝐾_𝑏′+𝐾₂𝐾_𝑝′‖ℎ‖_{(𝑛−1)𝑝/𝑛,Γ}, (48) 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∈𝐻¹_Γ𝐷(Ω) to (1)–(3) satisfies esssup_Ω∪Γ|𝑢|≤esssup_Γ𝐷𝑔+2^{(𝑝(𝛼+1)−2𝛼)/(𝑝(𝛼−1)−2𝛼)}𝑎_#×|Ω|^{(𝛼−1)/(2𝛼)−1/𝑝}×|Ω|^1/(2𝛼)𝑆_{2𝛼/(𝛼+2)}+𝐾_{2𝛼/(𝛼+1)}‖𝐟‖_𝑝,Ω+|Ω|^1/2+1/𝛼𝑆_{2𝛼/(𝛼+2)}𝑆_{2𝛼𝑝/[𝑝(2𝛼−1)−2𝛼]}+𝐾_{2𝛼/(𝛼+1)}𝑆_𝑝′×𝑓_{2𝑝/(𝑝+2),Ω}+|Ω|^1/𝛼𝑆_{2𝛼/(𝛼+2)}𝐾_{2𝛼𝑝/[𝑝(2𝛼−1)−2𝛼]}+𝐾_{2𝛼/(𝛼+1)}𝐾_𝑝′×‖ℎ‖_𝑝/2,Γ. (49)

Proof. Let 𝑘≥𝑘₀=esssup{|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 ‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}≤𝐶_𝑛,𝑏|𝐴(𝑘)|^1/2−1/𝑏, (50) where 𝐴(𝑘)={𝑥∈Ω∶|𝑢(𝑥)|>𝑘}. With this definition, the integral 𝐼 from the proof of Proposition 9 reads 𝐼=(|𝑢|−𝑘)⁺_{𝛼,𝐴(𝑘)∩Ω}+(|𝑢|−𝑘)⁺_{𝛼,𝐴(𝑘)∩Γ}, (51) and, for ℎ>𝑘>𝑘₀, we have (ℎ−𝑘)|𝐴(ℎ)|^1/𝛼≤𝐼,∀𝛼≥1. (52)
Case (𝑛>2). Take 𝛼=2(𝑛−1)/(𝑛−2)<2^∗=2𝑛/(𝑛−2). Making use of (7)-(8) and (|𝑢|−𝑘)⁺∈𝑊_Γ^1,𝑞_𝐷(Ω) with 𝑞=2, we deduce 𝐼≤𝑆₂|𝐴(𝑘)|^{1/𝛼−1/2∗}‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}+𝐾₂‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}. (53) 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 𝐼≤𝑆₂𝐶_𝑛,𝑏+𝐾₂𝐶_𝑛,𝑝|𝐴(𝑘)|^1/2−1/𝑝,1𝑏=1𝑝+𝑛−2().2𝑛𝑛−1 (54)
Therefore, we conclude |𝐴(ℎ≤𝑆)|²𝐶_𝑛,𝑏+𝐾₂𝐶_𝑛,𝑝ℎ−𝑘^𝛼|𝐴(𝑘)|^𝛽,1𝛽=𝛼2−1𝑝, (55) where 𝛽>1 if and only if 𝑝>2(𝑛−1). Notice that 𝐶_𝑛,𝑏≤|Ω|^{(𝑛−2)/[2𝑛(𝑛−1)]}×‖𝐟‖_𝑝,Ω+𝑆_𝑏′𝑓_{𝑛𝑝/(𝑝+𝑛),Ω}+|Ω|^{(𝑛−2)/[2(𝑛−1)2]}𝐾_𝑏′‖ℎ‖_{(𝑛−1)𝑝/𝑛,Γ}. (56)
Case (𝑛=2). Using (7) with 𝑞=2𝛼/(𝛼+2)<2, (8) with 𝑞=2𝛼/(𝛼+1)<2, and the Hölder inequality, we have (|𝑢|−𝑘)⁺_{𝛼,𝐴(𝑘)∩Ω}≤𝑆_{2𝛼/(𝛼+2)}‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}|𝐴(𝑘)|^1/𝛼;(|𝑢|−𝑘)⁺_{𝛼,𝐴(𝑘)∩Γ}≤𝐾_{2𝛼/(𝛼+1)}‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}|𝐴(𝑘)|^1/(2𝛼). (57) Thus, we deduce 𝐼≤𝑆_{2𝛼/(𝛼+2)}|𝐴(𝑘)|^1/𝛼‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}+𝐾_{2𝛼/(𝛼+1)}|𝐴(𝑘)|^1/(2𝛼)‖∇𝑢‖_{2,𝐴(𝑘)∩Ω}. (58)
Applying (50) twice (𝑏=2𝛼𝑝/(𝑝+2𝛼)>2 and 𝑏=𝑝>2𝛼/(𝛼−1)>2), we conclude |𝐴(ℎ≤𝑆)|^{2𝛼/(𝛼+2)}𝐶_{2,2𝛼𝑝/(𝑝+2𝛼)}+𝐾_{2𝛼/(𝛼+1)}𝐶_2,𝑝ℎ−𝑘^𝛼|𝐴(𝑘)|^𝛽, (59) 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 𝜂∈𝑊₀^1,∞(𝐵_𝑅(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. 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_Ω𝜂²|∇𝑢|²𝑎d𝑧≤4^#𝑎_#Ω𝑢²∇𝜂²d𝑧, (60) for any 𝜂∈𝑊^1,∞(ℝ^𝑛).(2) For arbitrary 𝑥∈Ω, 𝑅>0, and 𝑘₀≥0,esssup_{Ω(𝑥,𝑅/2)}|𝑢|≤𝑘₀+2^{3𝑛+2+(3𝑛)2/4}𝑐^3𝑛/2𝜔_𝑛×𝑅^−𝑛_{Ω(𝑥,𝑅)}|𝑢|−𝑘₀²d𝑧^1/2, (61) where 𝑐=𝑆_{2𝑛/(𝑛+2)}(1+2𝑎^#/𝑎_#) and Ω(𝑥,𝑟)=Ω∩𝐵_𝑟(𝑥) for any 𝑟>0.

Proof. (1) Let us choose 𝑣=𝑢𝜂²∈𝐻¹_Γ𝐷(Ω) as a test function in (11). Thus, applying the Hölder inequality we deduce _Ω𝑎𝜂²|∇𝑢|²d𝑧=−2_Ω≤1𝑎𝜂∇𝑢⋅∇𝜂𝑢d𝑧2_Ω𝑎𝜂²|∇𝑢|²d𝑧+2_Ω𝑎𝑢²∇𝜂²d𝑧. (62) 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, (ℎ−𝑘)|𝐴(ℎ,𝑟)|^1/2≤‖|𝑢|−𝑘‖_{2,𝐴(ℎ,𝑟)}≤‖|𝑢|−𝑘‖_{2,𝐴(𝑘,𝑟)},∀𝑟>0. (63)
Fix 0<𝑟<𝑅≤𝑅₀, and let us take 𝑣=sign(𝑢)(|𝑢|−𝑘)⁺𝜂²∈𝐻¹_Γ𝐷(Ω) as a test function in (11), where 𝜂∈𝑊^1,∞(ℝ^𝑛) is the cut-off function defined by 𝜂≡1 in 𝐵_𝑟(𝑥), 𝜂≡0 in ℝ^𝑛⧵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 ℝ^𝑛 and |∇𝜂|≤1/(𝑅−𝑟) a.e. in 𝐵_𝑅(𝑥) and that (60) reads _{𝐴(𝑘,𝑅)}𝜂²|∇𝑢|²𝑎d𝑧≤4^#𝑎_{#𝐴(𝑘,𝑅)}∇𝜂²(|𝑢|−𝑘)²d𝑧. (64)
Making use of (7) and 𝜂(|𝑢|−𝑘)⁺∈𝑊_Γ^1,𝑞_𝐷(Ω) with exponent 𝑞=2𝑛/(𝑛+2)<2≤𝑛 and the Hölder inequality, we have ‖|𝑢|−𝑘‖_{2,𝐴(𝑘,𝑟)}≤𝜂(|𝑢|−𝑘)⁺_2,Ω≤𝑆_{2𝑛/(𝑛+2)}×∇𝜂(|𝑢|−𝑘)⁺_{2𝑛/(𝑛+2),Ω}≤𝑆_{2𝑛/(𝑛+2)}×(|𝑢|−𝑘)⁺∇𝜂+𝜂∇𝑢_{2,𝐴(𝑘,𝑅)}|𝐴(𝑘,𝑅)|^1/𝑛. (65)
Applying the properties of 𝜂, inserting (64) into (65), and gathering the second inequality from (63), we get ‖|𝑢|−𝑘‖_{2,𝐴(ℎ,𝑟)}≤|𝐴(𝑘,𝑅)|^1/𝑛𝑐‖‖𝑅−𝑟|𝑢|−𝑘_{2,𝐴(𝑘,𝑅)}. (66)
In order to apply [13, Lemma 5.1] that leads to 𝜙𝑘₀+2^{(𝛼+𝛽)𝛽/[𝛼(𝛽−1)]}𝑐^1/𝛼𝜎𝑅₀^𝛾/𝛼×𝜙𝑘₀,𝑅₀^{(𝛽−1)/𝛼},𝑅₀−𝜎𝑅₀=0,∀𝜎∈]0,1[ (67) with 𝛾=1, 𝛼=2/(3𝑛), and 𝛽=1+2/(3𝑛)>1, we use (66) and inequality (63) with 𝑟 replaced by 𝑅, obtaining 𝜙(ℎ,𝑟)∶=|𝐴(ℎ,𝑟)|‖|𝑢|−ℎ‖_{2,𝐴(ℎ,𝑟)}≤|𝐴(𝑘,𝑅)|^{1+1/𝑛−𝛼/2}𝑐1𝑅−𝑟(ℎ−𝑘)^𝛼‖|𝑢|−𝑘‖^1+𝛼_{2,𝐴(𝑘,𝑅)}=𝑐(𝑅−𝑟)(ℎ−𝑘)^2/(3𝑛)𝜙(𝑘,𝑅)^1+2/(3𝑛). (68) 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 𝑐=𝑆_{2𝑛/(𝑛+2)}(𝑅+1+2𝑎^#/𝑎_#).

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 ‖|𝑢|−𝑘‖_{2,𝐴(𝑘,𝑟)}≤𝜂(|𝑢|−𝑘)⁺_2,Ω≤𝑆_{2𝑛/(𝑛+2)}𝜂(|𝑢|−𝑘)⁺_{1,2𝑛/(𝑛+2),𝐴(𝑘,𝑅)}≤𝑆_{2𝑛/(𝑛+2)}11+𝑢|‖𝑅−𝑟‖|−𝑘_{2,𝐴(𝑘,𝑅)}+𝜂∇𝑢_{2,𝐴(𝑘,𝑅)}|𝐴(𝑘,𝑅)|^1/𝑛. (69) 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).

Figure 1 

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, 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 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,