A Comparison Theorem for Nonsmooth Nonlinear Operators

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity f is Lp function with p > 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.


Introduction
Let be a domain in R n , n ≥ 2. We will consider super-and sub-solutions of the equation where f is a real valued function from L p loc (R) with some p > 1. To make the term f (u) well-defined (measurable and belonging to L p loc ( )) we will assume that u ∈ C 1 ( ) and ∇u = 0 in . Usually, f is supposed to be continuous in almost all papers dealing with Eq. 1 and its generalisations (see, for example, [6] and [11] and numerous papers citing these notes).
It was shown in [4,Remark 2.3] that the strong maximum principle may fail if the function f is only Hölder continuous with an exponent less than 1. Optimal conditions on smoothness of f for validity of the strong maximum principle can be found in [12]. The main difference in our approach is that we compare functions in a neughborhood of a point Vladimir Kozlov vladimir.kozlov@liu.se where the gradients are not vanishing. This allows to remove any smoothness assumptions on f .
One of the main results of this paper is the following assertion: Theorem 1.1 Let f ∈ L p loc ( ), p > 1. Also let u 1 , u 2 ∈ C 1 ( ) have non-vanishing gradients in and satisfy the inequalities u 1 + f (u 1 ) ≥ 0 and u 2 + f (u 2 ) ≤ 0 (2) in the weak sense. If u 1 ≤ u 2 and u 1 (x 0 ) = u 2 (x 0 ) for some x 0 ∈ then u 1 = u 2 in the whole .
We note that the theorem is not true without assumptions that the gradients do not vanish, which follows from [4] (see [7]).
In the case p > n this theorem was proved in [7]. The proof there was based on a weak Harnack inequality for non-negative solutions to the second order equation in divergence form Lu and closely connected with L p properties of the coefficients b j . Therefore one of our main concerns is a strong maximum principle for solutions to Eq. 3. We always assume that the matrix (a ij ) is symmetric and uniformly elliptic: It was proved in [14] that if |b| ∈ L p loc ( ) (here b = (b 1 , . . . , b n )) with p > n then a nonnegative weak solution to Eq. 3 satisfies (here B ρ (x) stands for the ball of radius ρ centered at where B 3ρ (x 0 ) ⊂ and γ ∈ (1, n/(n − 2)). So the restriction p > n in this assertion inherits in the theorem in [7]. 1 For our purpose another type of assumptions on the coefficients b j are more appropriate. It is called the Kato condition, see [3] and [13].
It was proved in [8] that inequality (4) still holds if |b| 2 ∈ K n,2 . For Hölder continuous coefficients a j i (4) was proved in [15] under the assumption |b| ∈ K n,1 . We note that from the last assertion it follows (4) when |b| ∈ L p loc , p > 1, depends only on one variable and the leading coefficients are Hölder continuous.
For our applications we need the leading coefficients to be only continuous. So all above mentioned results are not enough for our purpose. Here we prove a theorem which deals with slightly discontinuous leading coefficients and allows b α ∈ K n,α with α close to 1 for lower order coefficients. In order to formulate this result we need some definitions.
Definition 2 Let f (x) be a measurable and locally integrable function. Define a quantity We say that f ∈ V MO(R n ) if ω f (ρ) is bounded and ω f (ρ) → 0 as ρ → 0. In this case the function ω f (ρ) is called VMO-modulus of f . For a bounded Lipschitz domain the space f ∈ V MO( ) is introduced in the same way but the integrals in the definition of f #r (x) are taken over B r (x) ∩ .

Definition 3
We say that a function σ : [0, 1] → R + belongs to the Dini class D if σ is increasing, σ (t)/t is summable and decreasing.
It should be noted that assumption about the decay of σ (t)/t is not restrictive (see Remark 1.2 in [1] for more details). We use the notation κ(ρ) = ρ 0 σ (t) t dt. Theorem 1.2 Let n ≥ 3. Assume that the leading coefficients a ij ∈ V MO( ). Suppose that |b| α ∈ K n,α and for some α > 1 and σ ∈ D.
If a function u ∈ W 1,p ( ), p > n, satisfies u ≥ 0 and Lu ≥ 0 in then either u > 0 in or u ≡ 0 in .
For n = 2 we need a stronger assumption.
If a function u ∈ W 1,p ( ), p > 2, satisfies u ≥ 0 and Lu ≥ 0 in then either u > 0 in or u ≡ 0 in .
For γ ∈ (0, 1) we define the annulus If the location of the center is not important we write simply B r and X r,γ .
As usual, for a bounded domain we denote by W 1,q 0 ( ), q > 1, the closure in W 1,q ( ) of the set of smooth compactly supported function, with the norm

Coercivity
Let ′ be a bounded subdomain in . Consider the problem We say that the operator L 0 is q-coercive in ′ for some q > 1, if for each f ∈ W −1,q ( ′ ) the problem (9) has a unique solution u ∈ W 1,q 0 ( ′ ) and this solution satisfies with C q independent on f and u.
It is well known that for arbitrary measurable and uniformly elliptic coefficients the operator L 0 is 2-coercive in arbitrary bounded ′ . Further, if the coefficients a ij ∈ V MO( ) then the operator L 0 is q-coercive for arbitrary q > 1 in arbitrary bounded ′ ⊂ with ∂ ′ ∈ C 1 , see [2]. The coercivity constant C q depends on ′ and VMO-moduli of a ij . Moreover, by dilation we can see that for ′ = B r , r ≤ 1, this constant does not depend on r. For ′ = X r,γ , r ≤ 1, C q depends on γ but not on r.
Let now the operator in Eq. 9 be q-coercive for certain q > 2. Put . . , f n ) ∈ (L q ( ′ )) n and f 0 ∈ L nq/(n+q) ( ′ ). Then by the imbedding theorem f ∈ W −1,q ( ′ ) and Eq. 10 takes the form We need the following local estimate.

Theorem 2.1 Let ′ be a bounded Lipschitz subdomain of and let the operator in
Then for a fixed λ ∈ (0, 1) where C may depend on the domain ′ , q, λ and the coercivity constant C q but it is independent of r.
Proof First, we claim that the problem (9) is s-coercive for any s ∈ [2, q]. Indeed, we have coercivity for s = 2 and s = q, and the claim follows by interpolation.

Estimates of the Green Functions
Let L be an operator of the form Eq. 3, and the assumptions of Theorem 1.2 are fulfilled. We establish the existence and some estimates of the Green function G = G(x, y) for the problem in sufficiently small ball B r ⊂ .
where the constants C 1 and C 2 depend on the same quantities as R.
Proof We use the idea from [15]. Denote by G 0 (x, y) the Green function of the problem (9) in the ball B r . The estimates (13) for G 0 were proved in [9] (see also [5]): where the constants C 10 and C 20 depend only on n and ν.
By Remark 1, we can assume without loss of generality α < n/(n − 1). Put q = α ′ > n and denote by C q the coercivity constant for L 0 in the ball. We begin with the estimate for any B ρ (y) ⊂ By Eqs. 11 and 14, we have A j 1 ≤ C(n, ν, C q )(2 −j −1 ρ) 1− n q ′ , and Eq. 6 gives Therefore, Next, we write down the equation for G and obtain provided this series converges. We claim that for a proper constant C. Indeed, Denote 2ρ = |x − y|. Then We have by Eq. 15 Br \Bρ (x) |b(z)| |D z G 0 (z, y)| dz ≤ 2 n−1 AC 10 C k κ k+1 (r) |x − y| n−2 (here we used an evident inequality κ(2r) ≤ 2κ(r)). Further, By Eqs. 11 and 14, Using the assumption q > n we get and the claim follows if we put C = 2 n−2 (A + 2C 10 ). Thus, the series in Eq. 16 converges if κ(r) is sufficiently small. Moreover, if 2 n−2 (A + 2C 10 )κ(r) ≤ C 20 C 20 +2C 10 then Eqs. 14 implies 13 with C 1 = C 10 + C 20 2 , C 2 = C 20 2 . To prove the continuity of G we take x such that |x − x| ≤ ρ/2 = |x − y|/4. Since q > n, the estimate (11) and the Morrey embedding theorem give We write down the relation and deduce, similarly to Eq. 17, that Therefore, if κ(r) is sufficiently small, Remark 2 In fact, since q can be chosen arbitrarily large, G is locally Hölder continuous w.r.t.

Lemma 2.2
The statement of Lemma 2.1 holds for the problem (18). The constants R, C 1 and C 2 may depend on the same quantities as in Lemma 2.1 and also on γ .
The proof of Lemma 2.1 runs without changes.

Approximation Lemma and Weak Maximum Principle
Proof It is easy to see that the difference v m = u m − u solves the problem Using the Green function G m of the operator L m in B r with the Dirichlet boundary conditions we get By Lemma 2.1, |G m (x, y)| ≤ C|x − y| 2−n with constant independent of m. Thus, the supremum of the last integral is bounded. The first integral in brackets tends to zero by the Lebesgue Dominated convergence Theorem, and the Lemma follows.
This statement follows from standard weak maximum principle and Lemma 2.3.  Proof First suppose that G(x * , y) < 0 for certain x * ∈ B r and for a positive measure set of y. By continuity of G in x we have G(x, y) < 0 for a (maybe smaller) positive measure set of y and an open set of x. Therefore, we can choose a bounded nonnegative function f such that u(x) = B r G(x, y)f (y) < 0 on an open set. But this would contradict to the weak maximum principle, see Corollary 2.1. Thus, we can change G on a null measure set and assume it nonnegative.
Proof of Theorem 1.2. We repeat in essential the proof of Lemma 2.5. Denote the set S = {x ∈ : u(x) = 0} and suppose that S = . Then we can choose x 0 ∈ S and y 0 ∈ \ S such that ρ := |x 0 − y 0 | = dist(y 0 , S), and ρ can be chosen arbitrarily small. Repeating the proof of Lemma 2.5 we introduce the same Dirichlet Green function G(·, y 0 ) of L in B 2ρ (y 0 ) and show that δ G(·, y 0 ) with sufficiently small δ > 0 is a lower barrier for u in the annulus X 2ρ,1/4 (y 0 ). This ends the proof.

The Case n = 2
The case n = 2 is treated basically in the same way as the case n ≥ 3, but some changes must be done mostly due to the fact that the estimate of the Green function contains logarithm.
Let us explain what changes must be done in the argument in compare with n ≥ 3. Denote by G 0 (x, y) the Green function of the problem (9) in the disc B r . Then for x = y ∈ B r , 0 < G 0 (x, y) ≤ C ′ 10 log r |x − y| + 2 ; if |x − y| ≤ dist(x, ∂B r )/2 then G 0 (x, y) ≥ C ′ 20 log where the constants C ′ 10 and C ′ 20 depend only on the ellipticity constants of the operator L 0 . Indeed by [9] these estimates can be reduced to similar estimates for the Laplacian, when they can be verified directly (in this case the Green function can be written explicitly). material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.