Uniqueness and comparison principles for semilinear equations and inequalities in Carnot groups

Abstract Variants of the Kato inequality are proved for distributional solutions of semilinear equations and inequalities on Carnot groups. Various applications to uniqueness, comparison of solutions and Liouville theorems are presented.


Introduction
It is well known that one of the fundamental tools for studying different questions related to coercive elliptic equations and inequalities on ℝ N is the so-called Kato inequality [14]. One of the earlier and main contributions in this direction has been proved by Brezis [3]. As a consequence of a modified Kato inequality he considered, among other things, distributional solutions of elliptic inequalities of the form ∆u ≥ |u| q− u on ℝ N , (1.1) where q > . The main conclusion of Brezis is that if u ∈ L q loc (ℝ N ) solves (1.1), then u(x) ≤ a.e. on ℝ N . results proved in [8]. In Section 4, we prove some uniqueness results for a general semilinear second-order inequality and give some concrete applications. In Section 5, we shall briefly discuss the ideas pointed out in the preceding section to systems of semilinear inequalities; see [9] for other applications of Kato inequalities to semilinear elliptic systems. Finally, in Section 6 we prove a modified version of Kato complex inequalities in the setting of Carnot groups and present some applications to the so-called reduction principles and to uniqueness of solutions of complex problems; see [6].

Preliminaries on Carnot groups
In this section, we recall some preliminary facts concerning Carnot groups (for more information and proofs we refer the interested reader to [2,12]). A Carnot group is a connected, simply connected, nilpotent Lie group of dimension N ≥ with graded Lie algebra G = V ⊕ ⋅ ⋅ ⋅ ⊕ V r such that [V , V i ] = V i+ for i = , . . . , r − and [V , V r ] = . A Carnot group of dimension N can be identified, up to an isomorphism, with the structure of a homogeneous Carnot group (ℝ N , ∘ , δ λ ) defined as follows: We identify with ℝ N endowed with a Lie group law ∘ . We consider ℝ N split into r subspaces ℝ N = ℝ n × ℝ n × ⋅ ⋅ ⋅ × ℝ n r with n + n + ⋅ ⋅ ⋅ + n r = N and ξ = (ξ ( ) , . . . , ξ (r) ) with ξ (i) ∈ ℝ n i . We shall assume that there exists a family of Lie group automorphisms, called dilation, δ λ with λ > of the form δ λ (ξ) = (λξ ( ) , λ ξ ( ) , . . . , λ r ξ (r) ). The Lie algebra of left-invariant vector fields on (ℝ N , ∘) is G. For i = , . . . , n = l, let X i be the unique vector field in G that coincides with ∂/∂ξ ( ) i at the origin. We require that the Lie algebra generated by X , . . . , X n is the whole G.
With the above hypotheses, we call = (ℝ N , ∘ , δ λ ) a homogeneous Carnot group. The canonical sub-Laplacian on is the second-order differential operator L = ∑ l i= X i . Now, let Y , . . . , Y l be a basis of span{X , . . . , X l }; the second-order differential operator is called a sub-Laplacian on . We denote by Q = ∑ r i= in i the homogeneous dimension of . In the sequel, we assume Q ≥ .
A nonnegative continuous function S : ℝ N → ℝ + is called a homogeneous norm on in the case that S(ξ) = if and only if ξ = and it is homogeneous of degree 1 with respect to δ λ (i.e., S(δ λ (ξ)) = λS(ξ)). We say that a homogeneous norm is symmetric if S(ξ − ) = S(ξ).
The Lebesgue measure is the bi-invariant Haar measure. For any measurable set E ⊂ ℝ N , we have |δ λ (E)| = λ Q |E|. Since Y , . . . , Y l generate the whole G, any sub-Laplacian ∆ G satisfies the Hörmander hypoellipticity condition. Moreover, the vector fields Y , . . . , Y l are homogeneous of degree 1 with respect to δ λ .
In what follows, we fix the vector fields Y , . . . , Y l . In this setting, we use the symbol ∇ to denote the vector field (Y , . . . , Y l ), and − div := ∇ * , where ∇ * is the formal adjoint of ∇ . Finally, we set W , loc := u ∈ L loc : |∇ u| ∈ L loc .

Kato's inequality for a sub-Laplacian operator on
In this section, we shall prove that a modified version of the Kato inequality for distributional solutions holds for a sub-Laplacian operator on a Carnot group . Similar inequalities can be proved for more general classes of linear differential operators. For instance, one can handle second-order operators generated by a system of smooth vector fields in ℝ N satisfying the Hörmander condition, and left invariant differential operators on homogeneous groups; see [12]. However, we shall not discuss these kinds of generalizations here.
Brought to you by | University of Sussex Library Authenticated Download Date | 9/20/17 9:29 AM As usual, we denote by sign, sign + and u + the functions defined by Throughout this paper, Ω ⊂ ℝ N denotes an open subset.
is a function u ∈ L loc (Ω) such that for any nonnegative ϕ ∈ C (Ω) we have that Ω u∆ G ϕ ≥ Ω fϕ.

Theorem 3.2 (Kato inequality).
Let u, f ∈ L loc (Ω) be such that Then The proof is a consequence of the following lemma; see [1] for a related result. Lemma 3.3. Let γ ∈ C (ℝ) be a convex function with bounded first derivative. Let u, f ∈ L loc (Ω) be such that Then γ(u) ∈ L loc (Ω) and Proof. We need to prove that for any nonnegative ϕ ∈ C (Ω) we have and that the following inequality holds: Fix ϕ ∈ C (Ω). Let (m η ) η be a family of symmetric mollifiers associated to a fixed homogeneous norm S. Set u η := u ⋆ m η in Ω η := {x ∈ Ω : dist(x, ∂Ω) > η}, that is, For η small enough, it follows that supp(ϕ) ⊂ Ω η .
Brought to you by | University of Sussex Library Authenticated Download Date | 9/20/17 9:29 AM Let x ∈ Ω η . Since m η (x ∘ ⋅ − ) is a nonnegative test function in Ω, by using the Fubini-Tonelli theorem we obtain On the other hand, by the convexity of γ it follows that The convergence of γ(u η ) → γ(u) in L loc (Ω) is assured by the convergence of u η → u in L loc (Ω) and the fact that γ is a Lipschitz function (since γ ὔ is bounded). By observing that To this end, we first claim that Indeed, since γ ὔ is continuous and u η → u a.e. in Ω (if necessary by passing to a subsequence), it follows that γ ὔ (u η )ϕ → γ ὔ (u)ϕ a.e. in Ω. Now, by γ ὔ being bounded, an application of the Lebesgue dominated conver- Next, if necessary by passing to a subsequence, we may suppose that the convergence in (3.3) is a.e. on Ω.
Finally, by the Lebesgue theorem we have This completes the proof. Proof of Theorem 3.2. The idea is first to approximate the function sign + with a family of convex functions γ ϵ having bounded derivatives, and then apply Lemma 3.3 above.
Let m ∈ C (ℝ) be nonnegative with supt(m) ⊂ [− , ] and ∫ m = . For ϵ > , set m ϵ := ϵ m( t−ϵ ϵ ) and consider γ ϵ as the solution of the problem and by the Lebesgue theorem we obtain The proof of (3.2) follows from a similar argument as above, so we shall omit it. Remark 3.4. Theorem 3.2 holds if we replace the functions sign + and u + respectively with To this end, we can argue as in the proof of Theorem 3.2, replacing γ ϵ (t) by γ ϵ (t − h).
Clearly, in order to prove (3.1) we need to know that at least a.e. This is not always possible. Indeed, we can construct a function u (even continuous) such that each mollification u η has sign + (u η ) ≡ , while sign + (u) ̸ ≡ . We shall prove this when Ω = ] , [. Since the above series is uniformly convergent, the function u is continuous. Moreover, u(x) > if and only if there exists n ≥ such that ϕ n (x) > . This is obviously equivalent to the fact that x ∈ I n . In other words, u vanishes on S and it is positive on I.
Let η > and let u η be a mollification of u, that is, u ⋆ m η , where (m η ) η is a standard family of mollifiers. We claim that u η (x ) > for any

Applications to uniqueness of solutions
In this section, we consider weakly elliptic linear differential operators of the form and the associated uniqueness problem for the semilinear equation Notice that since Lu = div(B(x)∇u), where B is a positive semidefinite matrix, by writing B as B = μ T ⋅ μ and defining div L = div(μ T ⋅) and ∇ L = μ∇, it follows that This means that a Kato inequality holds for L; see [8]. If L = ∆ G is a sub-Laplacian on a Carnot group, then a distributional solution of (4.1) is a function u ∈ L loc (Ω) such that f(u) ∈ L loc (Ω), and for any nonnegative ϕ ∈ C (Ω) we have has no nontrivial weak [distributional] solution belonging to X. Let h ∈ L loc (Ω) and let f ∈ C (ℝ) be such that Then the equation

.3) has at most one weak [distributional] solution belonging to X.
Proof. Let h ∈ L loc (Ω) and let u, v ∈ X be solutions of (4.3). The function u − v ∈ X is a weak solution of An application of the appropriate Kato inequality (3.1) or [8, Theorem 2.1] yields which in turn implies that the function w := (u − v) + is a weak (or distributional) solution of In other words, w solves (4.2). Hence w ≡ a.e. on Ω, that is, u ≤ v a.e. on Ω. Inverting the role of u and v, the claim follows.
A concrete application of Theorem 4.2 is contained in the following result.
[ is a continuous function satisfying the following assumptions: Brought to you by | University of Sussex Library Authenticated Download Date | 9/20/17 9:29 AM Let h ∈ L loc (ℝ N ). Then the problem has at most one distributional solution u ∈ L loc (ℝ N ). Moreover, if h ≥ , then u ≤ a.e. on ℝ N . Proof. The obvious idea is to apply Theorem 4.2. To this end, it is enough to check that the inequality has only the trivial solution. Indeed, let us assume that v ∈ L loc (ℝ N ) is a solution of (4.6). By a mollification argument (as in the proof of Lemma 3.3) we have Next, by the convexity of b and the Jensen inequality, it follows that Now v η is smooth and solves (4.7) with the function b nondecreasing (indeed, it satisfies (i) and it is convex) and satisfying (4.5), thus we are in the position to apply [7, Theorem 3.10] (by changing u := −v η ), so we deduce that v η ≡ . Thus, by letting η → we obtain v ≡ . Remark 4.4. When dealing with C solutions, hypothesis (iii) can be relaxed by assuming that b is nonincreasing; see [7].
where q , q > . Theorem 4.3 applies to such f . Indeed, for t ≥ define g(t) := min{t q , t q }. The function b that we need is the convexification of cg for a small constant c > .
We claim that there exists a constant c > such that for any t > s we have Assume that q ≤ q . By the well-known inequality we have the following three cases: (ii) Let > t > s. Then Brought to you by | University of Sussex Library Authenticated Download Date | 9/20/17 9:29 AM (iii) Let t > > −s. The proof of the claim will follow if we prove that t q + s q ≥ cg(t + s) for any t, s > .

By using the inequality
for a, b > and p > and distinguishing three different cases, we have the following: (a) Let s ≥ and t > . Then Then (c) Let t ≥ and > s > . Then Next, by choosing c := min c q , c q , −q , −q , −q , we get the claim. By defining b := conv(cg), it follows that assumptions (4.4) and Theorem 4.3 (i) and (iii) are fulfilled. Notice that Theorem 4.3 (ii) is satisfied since at infinity the function b behaves like t q with q > . We point out that f does not satisfy the Brezis condition f ὔ (t) ≥ |t| q− for any t ∈ ℝ unless q = q . The interested reader may compare this with [3].

Some applications to a class of semilinear systems
In this section, as in the previous Section 4, we consider weakly elliptic linear differential operators of the form Lu = div L (∇ L u). We refer to Definition 4.1 for the appropriate notion of solutions. Theorem 5.1. Let X be a subspace of L loc (Ω) such that if u ∈ X, then u + ∈ X. Let b : [ , ∞[→ [ , +∞[ be a continuous function such that b( ) = and the problem has no nontrivial weak [distributional] solutions belonging to X. Let f ∈ C (ℝ) be such that

be a weak [distributional] solution of the system of inequalities
Then the following assertions hold: on Ω.
(ii) Let C ≥ and assume that the function f (t) := −Cf(−t) satisfies (5.2). Let (u, v) ∈ X × X be a weak [distributional] solution of the system on Ω. (5.4) Brought to you by | University of Sussex Library Authenticated Download Date | 9/20/17 9:29 AM Then u = −v a.e. on Ω. Therefore, u satisfies on Ω, (5.5) and the function f must be odd on the range of u, that is, for any t ∈ u(Ω) the condition f(t) = −f(−t) holds.
Proof. Let (u, v) ∈ X × X be a solution of (5.3). The function u + v ∈ X solves An application of the Kato inequality yields which in turn implies that the function w := (u + v) + is a weak solution of that is, w solves (5.1). Hence w ≡ a.e. on Ω, that is, u + v ≤ a.e. on Ω. This proves case (i).
From the first inequality in (5.4) it follows that u solves (5.5). Adding (5.5) and the second inequality of (5.4) (and taking into account that v = −u), we obtain This last chain of inequalities implies that f(u) = −f(−u), completing the proof.
Then the conclusions of Theorem 5.1 hold.
Proof. It is enough to check that the inequality has only the trivial solution. This follows from the proof of Theorem 4.3. Remark 5.4. Dealing with C solutions, hypothesis (iii) can be weakened, assuming that b is nonincreasing.
Brought to you by | University of Sussex Library Authenticated Download Date | 9/20/17 9:29 AM Corollary 5.5. Let q > . Let (u, v) be a distributional solution of the problem Then u = −v a.e. on ℝ N and An immediate consequence is the following corollary. Corollary 5.6. Let q > . Let (u, v) be a distributional solution of the problem Then u = v a.e. on ℝ N .
The above results improve some theorems obtained in [4].

A note on the complex case
In this section, we shall prove a complex version of some results stated in Section 3 and [8] in the framework of Carnot groups. For the Euclidean case, see [13,14]. Theorem 6.1 (Kato's inequality: The complex case). Let u, f ∈ L loc (Ω; ℂ) be such that Then The proof is based on the following lemma. Lemma 6.2. Let γ ∈ C (ℝ ) be a convex function with bounded first derivatives. Let u, f ∈ L loc (Ω; ℂ) be such that where ∂γ ∂z is the Wirtinger operator defined by Proof. We shall use the same notations as in the proof of Lemma 3.3. Without loss of generality, we assume that u and f are smooth (if this is not the case we can use a mollification process as in the proof of Lemma 3.3). Let u := s + it. By computation it follows We claim that ∆ G γ(u) ≥ γ x ∆ G s + γ y ∆ G t.
As an application of Theorem 6.1 we have the following result.
has no nontrivial distributional solutions. If u ∈ L loc (Ω; ℂ) is a complex distributional solution of such that |u| ∈ X, then u ≡ a.e. on Ω.
By assumption it follows that |u| ≡ a.e. on Ω.
We end this section with easy consequences that follow from the proof of Theorem 4.3.
Proof. Let u and v be distributional solutions of (6.3) and set w := u − v. The function w satisfies ∆ G w = |u| q− u − |v| q− v.
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