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A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.


Introduction
We consider the prescribed mean curvature equation of the form div Du 1 + |Du| 2 = N H in Ω, (1.1) where Ω is a domain in R N and N ≥ 2. The function u : Ω → R is unknown, Du denotes the gradient of u in spatial variables x and H : Ω → R is a locally Lipschitz continuous function in Ω. When the solution u is Lipschitz continuous, equation (1.1) is regarded as uniformly elliptic. However, when u is only semicontinuous, equation (1.1) can be non-uniformly elliptic.
Our goal is to prove a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation. Here, our strong comparison principle is stated as follows: if a lower semicontinuous viscosity supersolution u and an upper semicontinuous viscosity subsolution v satisfy that u ≥ v in Ω and u(x 0 ) = v(x 0 ) at some point x 0 ∈ Ω, then u ≡ v in Ω.
It is well known that for linear elliptic equations the strong comparison principle is equivalent to the strong maximum principle since the difference of two solutions is still a solution. Here, the strong maximum principle is the following: if a subsolution u satisfies that u ≤ m with some constant m and u(x 0 ) = m at some point x 0 ∈ Ω, then u ≡ m in Ω. Evidently the strong comparison principle implies the strong maximum principle provided that the constant m is a supersolution. The strong maximum principle for classical solutions of linear and nonlinear elliptic equations has been well studied (cf.
[ GT], [PW] There are some results on the strong maximum principle for weak solutions in the viscosity sense. For notations of viscosity solutions we refer to the literature [CIL] and [Ko].
The strong maximum principle for semicontinuous viscosity solutions has been proved by [KaKu], [BD], [BB], [KoKo], and [BGI]. for uniformly elliptic equations of the form F (D 2 u) = 0, but it does not work for nonuniformly elliptic equations of the form F (Du, D 2 u) = f (x) such as (1.1).
In the present paper we consider lower semicontinuous viscosity supersolutions and upper semicontinuous viscosity subsolutions of (1.1). Therefore, we have to deal with nonuniformly elliptic equations. Our proof is different from usual one. After being reduced to the case where both the supersolution u and the subsolution v are bounded, by virtue of Jensen, Lions and Souganidis [JLS], we introduce the inf and sup convolutions of u and v respectively, where those convolutions are continuous functions and moreover they are monotone with respect to the parameter. Then we consider the Dirichlet problems for (1.1) in every sufficiently small ball centered at a point x 0 , where u touches v. We choose the continuous boundary data as the inf and sup convolutions of u and v, respectively. Since H is locally Lipschitz continuous, by the theory of quasilinear elliptic equations (see [GT]), the gradient estimates of classical solutions are available and these problems have unique classical solutions provided that the ball is sufficiently small. Here the strong comparison principle is applicable to these two classical solutions and also a weak comparison principle is applicable to compare u and v with these two classical solutions, respectively. Eventually, these comparisons yield that u and v coincide with each other on the boundary of each small ball centered at a point x 0 , and hence u and v coincide with each other in a small ball centered at a point x 0 . Then the conclusion follows from the connectedness of the domain.
The present paper is organized as follows. In section 2 we state our main theorem and prove it. In section 3 we give a weak comparison principle as a corollary of our strong comparison principle. In Appendix we prove a weak comparison principle for (1.1) which compares a lower semicontinuous viscosity supersolution with a classical solution, or an upper semicontinuous viscosity subsolution with a classical solution.

Main theorem
Let Ω be a domain in R N , N ≥ 2 and let u : Ω → R. For functions u we set Here Du denotes the gradient of u in spatial variables x. Let H : Ω → R be a locally Lipschitz continuous function in Ω. Then equation (1.1) is written as Our main theorem concerns an extension of the strong comparison theorem to semicontinuous viscosity supersolutions and subsolutions of (2.1). We will use the following notations: Also, USC(Ω), LSC(Ω) are defined similarly.
Theorem 2.1 Let u ∈ LSC(Ω) be a viscosity supersolution of (2.1), that is, in the viscosity sense, and let v ∈ USC(Ω) be a viscosity subsolution of (2.1), that is, Remark 2.2 A continuous viscosity solution u of (2.1) means that u ∈ C(Ω) is both a viscosity supersolution and subsolution of (2.1). Combining the results of [B] and [T] yields the strong comparison principle for continuous viscosity solutions of (2.1). Indeed, it is shown in [B] that continuous viscosity solutions of (2.1) are Lipschitz continuous. Then, equation (2.1) is regarded as uniformly elliptic, and hence thanks to Trudinger's results in [T] we see that the strong comparison principle for continuous viscosity solutions of (2.1) holds.
The following weak comparison principle, which is proved in more general form in [KaKu, Theorem 3, p. 475], plays a key role in the present paper. Therefore we give a simple proof directly by using the implicit function theorem and the definition of viscosity solutions in the Appendix.
Now we are in a position to prove Theorem 2.1.
Proof of Theorem 2.1.
1st step: Reduction to the case where both u and v are bounded: Let E be a bounded domain with E ⊂ Ω and x 0 ∈ E. Since u is lower semicontinuous and v is upper semicontinuous, there exists K > 0 such that u > −K and v < K in E.
We use a notation B(x, r) as an open ball in R N of radius r > 0 centered at x ∈ R N . For simplicity we write in particular B r := B(x 0 , r) for every r > 0. Choose a positive number Then u R ∈ LSC(B R ) and v R ∈ USC(B R ) are bounded in B R . Moreover, u R is a viscosity supersolution of (2.1) in B R and v R is a viscosity subsolution of (2.1) in B R . Since Thus we may assume that u and v are bounded in B R .
Notice that u ε , v ε ∈ C(B R ), and at each point x ∈ B R the inf convolution u ε (x) increases to u(x) and the sup convolution v ε (x) decreases to v(x) as ε decreases to 0.
Proof. By setting ρ = R 2 , we observe that for every 0 where ω N denotes the Lebesgue measure of the unit ball in R N .
Fix ε 0 > 0 arbitrarily. Let us show that For each ε ∈ (0, ε 0 ], we set we know δ ε ≤ 0. Since u ε (x) increases to u(x) and v ε (x) decreases to v(x) as ε decreases to 0 at each x ∈ B R , δ ε is monotone increasing as ε decreases to 0. Let us show a lemma.
We return to the proof of Proposition 2.4. In order to prove that let us show that for each r with 0 < r ≤ ρ, Suppose that there exist r with 0 < r ≤ ρ and a point x 1 ∈ ∂B r so that is continuous, there exist r 1 with 0 < r 1 < r 4 and β > 0 such that (2.6) We divide ∂B r into two pieces: Clearly ∂B r is a disjoint union of Γ + and Γ 0 .
Note that (2.6) gives in particular Then for every 0 < ε ≤ ε 0 , by the monotonicity of u ε and v ε and by the definition of δ ε we see that Since H is Lipschitz continuous in B r , we see that the interior estimates of [GT,Corollary 16.7,p. 407] are available. Therefore it follows from (2.2) and the theory of the prescribed mean curvature equation ( [GT,Theorem 16.10,p. 408 ]) that there exist Also,û ε increases andv ε decreases as ε decreases to 0 by the monotonicity of u ε and v ε .
Eventually, we conclude that v ε 0 ≥ u ε 0 on ∂B r . Since this holds for every 0 < r ≤ ρ, 3rd step: Completion of the proof of Theorem 2.1: By Proposition 2.4, letting Since Ω is connected, we conclude that u ≡ v in Ω.

A weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation
The strong comparison principle proved in section 2 yields a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.
Theorem 3.1 Let Ω be a bounded domain in R n . Let u ∈ LSC(Ω) and v ∈ USC(Ω) be viscosity super and subsolutions of (2.1), respectively. Assume that u ≥ v on ∂Ω. Then u ≥ v in Ω, and hence either u ≡ v in Ω or u > v in Ω.
Note that v + θ is also a viscosity subsolution of (2.1). By Theorem 2.1 we have that u ≡ v + θ in Ω, which contradicts the assumption that u ≥ v on ∂Ω. Therefore we see Moreover, if there exists a point x 2 ∈ Ω so that u(x 2 ) = v(x 2 ), then u ≡ v in Ω by Theorem 2.1, which concludes that either u ≡ v in Ω or u > v in Ω.
Although Proposition 2.3 was already proved in [KaKu,Theorem 3,p. 457], for convenience we will give a simple proof directly by using the implicit function theorem and the definition of viscosity solutions.
Proof of Proposition 2.3. By the argument in [MS,Theorem A.1,p. 253], which applies Sard's theorem to a smooth function being comparable to the distance function to the closed set R N \ Ω due to Calderón and Zygmund [Z,Lemma 3.6.1,p. 136] (see also [CZ, Lemma 3.2, p. 185]), we observe that for each small ε > 0 there exists a smooth open set Since Ω is bounded, we notice that Ω ε is a union of a finite number of smooth domains.
Let u ∈ LSC(Ω) be a viscosity supersolution of (2.1) and let v ∈ C 2 (Ω) ∩ C(Ω) be a classical solution of (2.1). Assume that u ≥ v on ∂Ω. Since H is locally Lipschitz continuous, we see that the interior estimates of [GT,Corollary 16.7,p. 407] are available.
Hence it follows from (3.1) that Thus, letting ε → 0 yields that u ≥ v in Ω, which completes the proof of Proposition 2.3.