Harnack inequality for non-divergence structure semi-linear elliptic equations

Abstract In this paper we establish a Harnack inequality for non-negative solutions of L ⁢ u = f ⁢ ( u ) {Lu=f(u)} where L is a non-divergence structure uniformly elliptic operator and f is a non-decreasing function that satisfies an appropriate growth conditions at infinity.


Introduction
Let Ω ⊆ ℝ n be a bounded domain. We consider a second order differential operator in non-divergence form given by where the coefficients a ij , b j and c are assumed to be measurable on Ω. We also suppose that the coefficient matrix A(x) := (a ij (x)) is an n × n real symmetric matrix and that L is uniformly elliptic in Ω in the sense that λ|ξ | ≤ ⟨A(x)ξ, ξ ⟩ ≤ Λ|ξ | for all (x, ξ ) ∈ Ω × ℝ n , for some < λ ≤ Λ. We will use the notation L for the principal part of L, that is, Our purpose in this work is to establish a Harnack inequality for non-negative solutions of Lu = f(u) in Ω, (1.2) where the non-linearity f is an increasing function on ℝ + = ( , ∞) that satisfies appropriate growth conditions at infinity. The Harnack inequality is an important tool in the investigation of qualitative properties of solutions to second order elliptic as well as parabolic PDEs. The most recent and remarkable application was demonstrated in Perelman's use of a version of the Harnack inequality for the Ricci flow to settle the century old Poincaré's conjecture which states that any closed -manifold with trivial fundamental group is diffeomorphic to S (see [10]). We refer the reader to the paper [7] for a comprehensive discussion of the Harnack inequality and its history.
Let us now review some works on the Harnack inequality on semi-linear elliptic equations that are closely related to the content of this paper. In [4], Finn and McOwen establish a Harnack inequality for non-negative solutions of ∆u = u q for q > . In [2], Dindoš extends the results of Finn and McOwen by replacing the nonlinearity f(t) = t q , q > , by a strictly convex function f that satisfies conditions (f1) and (f2) stated in Section 2 below. The condition (f2), which was introduced by Dindoš in the context of Harnack inequalities, appears to play a particularly crucial role in the study of the Harnack inequality for non-negative solutions of the semilinear equation ∆u = f(u). Our main objective in this paper is to extend Dindoš' Harnack inequality results to non-negative solutions of (1.2) in Ω, where L is as in (1.1) and f is a continuous function which satisfies conditions that are weaker than some of those used in [2]. Finally, we provide an example to show that Dindoš' condition (f2) cannot be relaxed for the stated Harnack inequality to hold.
The paper is organized as follows. In Section 2 we recall some useful facts and fix notations. An L ∞ -bound on non-negative solutions of Lu = f(u) in Ω is established in Section 3. In Section 4 we state and prove our main result: the Harnack inequality for non-negative solutions of Lu = f(u) in Ω. We also demonstrate that Dindoš' condition (f2) cannot be relaxed for this Harnack inequality to hold. Finally, we include an Appendix where we state and prove some useful technical results that are used in the proof of the Harnack inequality.

Preliminaries
One of the most celebrated results in the area of second order elliptic and parabolic equations in nondivergence form is the Harnack inequality of Krylov and Safonov for non-negative solutions of Lu = g in Ω, which we recall below, in the form stated in [1,Theorem 4.1]. The reader is referred to [6] for a proof. Theorem 2.1 (Krylov and Safonov). Given z ∈ Ω and R > with B(z, R) ⊆ Ω, suppose g ∈ L n (B(z, R)) and that there are constants α, β such that ‖b(x)‖ ≤ β and |c(x)| ≤ α for all x ∈ B(z, R). Let u ∈ W ,n (B(z, R)) satisfy u ≥ in B(z, R) and Lu = g in B(z, R). Then where C := C(n, Λ/λ, βR, αR ).
For the remainder of this paper, we will suppose that there are some positive constants Θ * , Θ * such that Going back to (1.2) we will assume the following on f , defined on ℝ + := ( , ∞): (f1) f : ℝ + → ℝ + is non-decreasing and continuous such that f(t) > for t > . (f2) There exists θ > such that Condition (f2) was introduced by Dindoš, in [2], in his investigation of the Harnack inequality for nonnegative solutions of ∆u = f(u), where f , in addition to satisfying (f1) and (f2), is assumed to be strictly convex.
Let us record two lemmas that will be useful for us. We provide proofs in the Appendix.
The function Ψ is continuous on ( , ∞), and the change of variable ξ = F(s) − F(t) shows that it is also decreasing. Moreover, Ψ(t) → as t → ∞ (see estimate (A.8) in the Appendix where the proof of Lemma 2.4 is given). We will use Φ to denote the inverse of Ψ so that The following lemma provides a useful ingredient in the proof of the Harnack inequality for non-negative solutions of (1.2).
Let us recall some more results due to Keller [8] and Osserman [11] that will prove to be useful for us later. Suppose f satisfies the Keller-Osserman condition (2.2). Given R > and z ∈ ℝ n , let B := B(z, R) be the ball of radius R in ℝ n centered at z. If κ is a positive constant, then the boundary value problem [5,8,11]). In fact, for r = |x − z|, φ satisfies the ODE and the following inequalities: Moreover, as a consequence of (2.6), we have the following: We should point out that φ is a strictly convex function on [ , R), and hence the Hessian For an easy reference we summarize the above results in the following lemma.

An L ∞ -bound for solutions
We begin this section by recalling an inequality from the theory of matrices. See, for instance, [3] .
Given any n × n symmetric real matrix A and an n × n positive semi-definite matrix B the following inequality holds: Here λ min (A) and λ max (A) are the minimum and maximum eigenvalues of A. In the following lemma we will use the right-hand side inequality to obtain a super-solution for equation (1.2). Lemma 3.1. Suppose f satisfies (f1) and (f2). There exists R > such that for any given z ∈ Ω and < R < R with B := B(z, R) ⊆ Ω, the equation admits a super-solution. Moreover, the following estimate holds: Then Thus, Recalling that κ = ( Λ) − , we further restrict R so that This completes the proof of the lemma.
One last condition needed on f is the following: (f3) The function t → f(t)/t is non-decreasing on ( , ∞).
Given x ∈ Ω, we will use the notation d(x) for dist(x, ∂Ω). The following estimate on solutions of Lu = f(u) on Ω is another crucial result used for deriving the Harnack inequality.
Proof. Let us fix t such that f(t) ≥ Θ * t for t ≥ t . Moreover, we take R > as in the proof of Lemma 3.1 so To construct a non-increasing function η such that (3.2) holds for any non-negative solution u of Lu = f (u) in Ω, we consider the sets Ω R and Ω \ Ω R separately. To this end, let u be any non-negative solution of Lu = f(u) in Ω.
Consequently, we see that Let us consider the following differential operator: As a consequence of (f3) we see thatL Since u ≤ w on ∂O, we conclude by the (Alexandroff-Bakelman-Pucci) maximum principle that u ≤ w on O, which is a contradiction. Therefore, u ≤ w in B(z, R). In particular, Letting R → d(z), we conclude that Since v ϵ ≥ t on Ω ϵ , we also note that Therefore, estimate (3.3) holds in Ω ϵ with w replaced by v ϵ . Now if we set then, usingL ϵ in place ofL, we can argue as in Case 1 (with v ϵ in place of w) to show that We then let ϵ → + and conclude u ≤ Φ(R / Λ) in Ω \ Ω R . Unauthenticated Download Date | 6/14/16 6:43 AM Putting the two cases together, we conclude that u(x) ≤ η(d(x)) in Ω, where This completes the proof.

Remark 3.4.
We wish to emphasize that only the bound c(x) ≤ Θ * in Ω from above is used in the proofs of Lemma 3.1 and Theorem 3.3.

Harnack inequality for semi-linear equations
Given x ∈ Ω, we use the notation where the constant C depends on n, Λ λ , Θ * d(z) and d (z)(max x∈B(z,δ (z)) | c(x)|). Since |c | ≤ |c| + V, to show that the constant C in (HI ϵ ) depends only on n, Λ/λ, Θ * d(z) and Θ * d (z), it suffices to show that d (z)M(z) is uniformly bounded on Ω, independently of ϵ, where To this end, let us begin by noting that condition (f3) and (4.2) show that . Therefore, for any x ∈ B(z, δ (z)), recalling that η is nonincreasing and using (f3) once again, we estimate Hence, by Lemma 2.4, for any z ∈ Ω we have Therefore, we have shown that the constant C in (HI ϵ ) is independent of z ∈ Ω, ϵ > and the solution u. Now we let ϵ → in (HI ϵ ) to get the desired Harnack inequality.
But then sup Thus, w ≡ on B(z, d(z)/ ). Of course this is not possible if f( ) > .
However, we can relax the hypothesis (f3) on f and still obtain the Harnack inequality for solutions of Lu = f (u) in Ω that are uniformly bounded away from zero with the constant in the inequality that now depends on the uniform lower bound. To be precise, we consider the following condition on f : (f3) * There exists τ > such that t → f(t)/t is non-decreasing on (τ, ∞). Given δ > , let in Ω such that u ≥ δ on Ω .
Proof. Let us first note that condition (f3) has been used only in the proofs of Theorem 3.3 and Theorem 4.1.
We thus need to show how these proofs need to be modified when condition (f3) * replaces condition (f3).
In the proof of Theorem 3.3, we only need to choose t large enough such that t > τ. The same proof then shows that Theorem 3.3 holds for all solutions of Lu = f(u) in Ω. In the proof of Theorem 4.1, condition (f3) * can be used to estimate V(x) as follows. Let u ∈ H δ be arbitrary. First we note that estimate (4.3), and hence (4.4) hold whenever u(x) ≥ τ. If u(x) < τ, then Therefore, for any z ∈ Ω we have which is uniformly bounded in Ω by a constant independent of u and ϵ. As a consequence, the constant C in (HI ϵ ) is independent of u and ϵ as well. Letting ϵ → in (HI ϵ ) completes the proof.  We now show that this form of the Harnack inequality fails if Dindoš' condition (f2) does not hold. In other words, we exhibit an example of f that satisfies conditions (f1), (2.2) and (f3) but not (f2) such that inequality (4.5) cannot hold with a constant C, depending on dimension only. For this, we consider the following nonlinearity: It is easy to see that f satisfies conditions (f1), (2.2) and (f3), but not (f2). Below we produce a positive solution is unbounded over z ∈ B( , R), thus showing that inequality (4.5) cannot hold for a constant C independent of z ∈ B( , R).
Since f( ) = and f satisfies (f1) as well as the Keller-Osserman condition (2.2), for a given R > there exists a radially symmetric non-negative solution u of By the strong maximum principle of Vasquez [12], we note that u(x) > for |x| < R. Let v(r) = u(|x|) for r = |x|. Then If we multiply by v ὔ and integrate over ( , r), we find Since v ὔ (r) → ∞ as r → R and v ὔ is increasing, we have (see [9,Lemma 2 Therefore, for r near R we have Integration over (r, R) yields Now, if r < r < R with r close to R, then we have Then, from (4.7), we find Therefore, we see that (4.6) becomes arbitrarily large as d(z) → .

A Appendix
Proof of Lemma 2.2. By condition (f2) we pick ϱ such that Then there exists M ϱ > such that By iterating (A.2), we see that for any positive integer k. Thus, for any positive integer k with t = M ϱ , inequality (A. 3) becomes Let t ≥ M ϱ , and let k ∈ ℕ ∪ { } with Let also q := ln(θϱ) ln θ = + ln ϱ ln θ .
We note that q > , and ϱθ = θ q . Using (A.4), we obtain the following chain of inequalities: ϱθ . Since q > , we conclude that f satisfies condition (2.1), thus completing the proof of Lemma 2.2. Proof of Lemma 2.4. We start by making some preparatory observations. Note that For s ≥ t we see that Thus, for s ≥ t we have − F(t) F(s) ≥ .
Using this, we find that Therefore, for s ≥ t > the following holds:
Using this last inequality in (A.9), we find that Let t > be such that Φ(t) ≥ M ϱ for all < t < t . Then from (A.11), we conclude that there exists C > such that for all < t < t , t f(Φ(t)) Φ(t) ≤ C.