Abstract
The purpose of this article, is to study the Dirichlet problems of the subLaplace equation Lu + f(ξ, u) = 0, where L is the subLaplacian on the Carnot group G and f is a smooth function. By extending the Perron method in the Euclidean space to the Carnot group and constructing barrier functions, we establish the existence and uniqueness of solutions for the linear Dirichlet problems under certain conditions on the domains. Furthermore, the solvability of semilinear Dirichlet problems is proved via the previous results and the monotone iteration scheme corresponding to the subLaplacian.
Mathematics Subject Classifications: 35J25, 35J70, 35J60.
Keywords:
Carnot group; subLaplace equation; Dirichlet problem; Perron method; monotone iteration scheme1 Introduction
In this article we consider Dirichlet problems of the type
where Ω is a bounded domain in a Carnot group G and L is the subLaplacian. Some knowledge on G and L see next section. Hörmander's theorem permits us to judge the hypoellipticity of the operator L, i.e., if Lu ∈ C^{∞ }then u ∈ C^{∞ }(see [1]).
The investigation of the boundary value problems, concerning the operators in the form of the sum of squares of vector fields fulfilling Hörmander condition, has turned into the subject of several works, see [24]. The precursory work of Bony [2] proved a maximum principle and the solvability of the Dirichlet problem in the sense of PerronWiener. The Wiener type regularity of boundary points for the Dirichlet problem was considered in [3]. Thanks to the previous results, Capogna et al. [4] established the solvability of the Dirichlet problem when the boundary datum belongs to L^{p}, 1 < p ≤ ∞, in the group of Heisenberg type.
The Perron method (see [5,6]) and the monotone iteration scheme (see [7,8]) are wellknown constructive methods for solving linear and semilinear Dirichlet problems, respectively. Brandolini et al. [9] applied these methods to the Dirichlet problems for subLaplace equations on the gauge balls in the Heisenberg group which is the simplest Carnot group of step two. Let us notice that the balls possess of legible properties. However, we do not see the reseach to the problems on other domains using these methods. Concerning the construction of barrier function, Brandolini et al. [9] used the result given in [10], which holds in the setting of Heisenberg group.
Our work is motivated by [9]. We try to extend the existence of solutions for semilinear Dirichlet problems on the Heisenberg balls in [9] to general Carnot domains. To do so, the Perron method in the Carnot group is used in this article. Based on the work in [3], we construct a barrier function in a domain of the Carnot group (see Lemma 3.10) under the hypothesis of the outer sphere condition to discuss the boundary behaviour of the Perron solutions. The method to obtain a barrier function is essentially similar to the one in [9]. Then we prove the existence of solutions for linear subLaplace Dirichlet problems. In the discussion of semilinear Dirichlet problems, we will use monotone iteration scheme. The main difficulty we meet is that the subLaplacian L does not have explicit expression. To overcome it, we use the regularity of L in [1].
The article is organized as follows. In the next section, we recall some basic definitions and collect some known results on the Carnot group which will play a role in the following sections. Section 3 is devoted to the study of the Perron method for linear equations. By finding a barrier function related to the subLaplacian L, we prove that the Perron solutions for linear Dirichlet problems are continuous up to the boundary. The main results are Theorems 3.8, 3.11, and 3.13. In Section 4, using the results in Section 3 and the monotone iteration scheme, we provide the solutions of the semilinear Dirichlet problems in Carnot groups with some available supersolutions and subsolutions. Finally, we give an existence of solution to the subLaplace equation on the whole group of Heisenberg type (a specific Carnot group of step two). The main results in this section are Theorems 4.2 and 4.3.
2 Carnot groups
We will consider G = (ℝ^{N}, ·) as a Carnot group with a group operation · and a family of dilations, compatible with the Lie structure.
Following [11,12], a Carnot group G of step r ≥ 1 is a simple connected nilpotent Lie group whose Lie algebra admits a stratification. That is, there exist linear subspaces V_{1}, . . ., V_{r }of such that
Via the exponential map, it is possible to induce on G a family of nonisotropic dilations defined by
Here ξ = (x^{(1)}, x^{(2)}, . . ., x^{(r)}), for i = 1, . . ., r and N_{1 }+ · · · + N_{r }= N. We denote by the homogeneous dimension of G attached to the dilations {δ_{λ}}_{λ > 0}. Let m = N_{1 }and X = {X_{1}, . . ., X_{m}} be the dimension and a basis of V_{1}, respectively. Let Xu = {X_{1}u, . . ., X_{m}u} denote the horizontal gradient for a function u. The subLaplacian associated with X on G is given by
If u and v are two measurable functions on G, their convolution is defined by
where dG(η) denotes a fixed Haar measure on G.
Let e be the identity on G. For ξ ∈ G, we denote by ξ^{1 }the inverse of ξ with respect to the group operation. By [1], there exists a norm function satisfying
(1) ρ(ξ) ≥ 0; Moreover, ρ(ξ) = 0 if and only if ξ = e;
(2) ρ(ξ) = ρ(ξ^{1}).
The open ball of radius R centered at ξ is expressed as the set:
Let denote the space of distributions on G. The nonisotropic Sobolev space S^{k, p }is defined by
where α = (α_{1}, . . ., α_{l}) is a multiindex, , and . In the space S^{k, p}, we shall adopt the norm
For a domain Ω in G, we define S^{k, p}(Ω, loc) as the space of distributions f such that for every we have fψ ∈ S^{k, p}. Let 0 < β < ∞, we employ the following nonisotropic Lipschitz spaces:
(i) for 0 < β < 1,
(ii) for β = 1,
(iii) for β = k + β' where k = 1, 2, 3, . . . and, 0 < β' ≤ 1,
We refer the reader to [1] for more information on the above.
The following results are useful.
Proposition 2.1. (i) Suppose Ω ⊂ G is an open set, and suppose satisfy Lf = g in Ω. If g ∈ S^{k, p}(Ω, loc) (1 < p < ∞, k ≥ 0) then f ∈ S^{k+2,p}(Ω, loc).
(ii) Suppose 1 < p < ∞ and , then S^{k, p }⊂ Γ^{β}.
Part (i) and (ii) are contained, respectively, in Theorems 6.1 and 5.15 of [1].
3 The Perron method and barrier function for linear problem
In this section, we study the solvability of the following linear subLaplace Dirichlet problem
Definition 3.1. A bounded open set Ω ⊂ G is said to satisfy the outer sphere condition at ξ_{0 }∈ ∂Ω, if there exists a ball B_{G}(η, r) lying in G\Ω such that
The definition in the case of general degenerate elliptic operator can be seen in [3]. Notice that in the Htype group case, every bounded convex subset accords with the condition of the outer sphere. In particular, the gauge balls in Htype group are convex domains (see [4]). From Theorem 2.12 in [13] and Theorem 5.2 in [2] respectively, one has the following two lemmas.
Lemma 3.2. (Maximum principle) Let Ω be a connected open set in a Carnot group G. If u ∈ C^{2}(Ω) satisfies
then u cannot achieve a nonnegative maximum at an interior point unless u ≡ constant in Ω.
Lemma 3.3. Let Ω be a bounded domain in G. Then there exists a family of open subsets, denoted by , which is a base for the topology of Ω for which the Dirichlet problem
has a unique distributional solution for any , and φ ∈ C(∂ω). Furthermore, if f ∈ C^{∞}(ω), then u ∈ C^{∞}(ω).
We give notions of subsolution and supersolution for the Dirichlet problem (3.1).
Definition 3.4. Let φ ∈ C(∂Ω), . A function is called a subsolution of (3.1) if it fits the following properties:
(i) u ≤ φ on ∂Ω;
(ii) for every and for every such that Lh  λ(ξ)h = f and u ≤ h on ∂ω, we also have u ≤ h in ω.
The definition of supersolution is analogous.
Lemma 3.5. Assume that u is a subsolution of (3.1) and v is a supersolution of (3.1), then either u < v in Ω or u ≡ v.
Proof. Suppose that at some point η ∈ Ω we have u(η) ≥ v(η). Set . Take ξ_{0 }∈ Ω such that (u  v)(ξ_{0}) = M, and we can know that u  v ≡ M in a neighborhood of ξ_{0}. Otherwise there exists such that ξ_{0 }∈ ω but u  v is not identically equal to M on ∂ω. Letting and denote the solutions of Lw  λ(ξ)w = f in ω, equal to u and v on ∂ω respectively. Since u and v are the subsolution and the supersolution respectively, we deduce from Definition 3.4 that and in ω. One sees that
and hence all the equalities above hold. By Lemma 3.2 it follows that in ω and hence u  v ≡ M on ∂ω, which contradicts the choice of ω.
The previous argument implies u  v ≡ M in Ω. Combining this with Definition 3.4(i) we obtain u ≡ v in Ω. □
Let be a subsolution of (3.1) and . Denote by the solution of the Dirichlet problem (see Lemma 3.3)
and define in Ω the lifting of u (in ω) by
Lemma 3.6. U(ξ) is a subsolution of (3.1).
Proof. Since u(ξ) is a subsolution of (3.1), it follows that U(ξ) = u(ξ) ≤ φ(ξ) on ∂Ω. Let and such that Lh  λ(ξ)h = f and U ≤ h on ∂ω'. If ω ∩ ω' = ϕ, then u = U ≤ h on ∂ω'. It leads to U = u ≤ h in ω';
Suppose now ω ∩ ω' = ϕ. Since u ≤ U, we have u ≤ h on ∂ω' and then u ≤ h in ω'. In particular, u ≤ h in ω'\ω, i.e. U ≤ h in ω'\ω. Thus, we have on ∂(ω' ∩ ω). As in ω' ∩ ω and on ∂(ω' ∩ ω), it yields by Lemma 3.2 that in ω' ∩ ω, and therefore U ≤ h in ω' ∩ ω. □
The following result is a trivial consequence of Definition 3.4.
Lemma 3.7. Let u_{1}, u_{2}, . . ., u_{l }be subsolutions of (3.1). Then the function
is also a subsolution of (3.1).
Let S denote the set of all subsolutions of (3.1). Notice that S is not empty, since k^{2 }∈ S for k large enough. The basic result via the Perron method is contained in the following theorem.
Theorem 3.8. The function satisfies Lu  λ(ξ)u = f in Ω.
Proof. Notice that k^{2}, for k large enough, is a supersolution of (3.1). By Lemma 3.5, we deduce v ≤ k^{2 }for any v ∈ S, so u is well defined. Let η be an arbitrary fixed point of Ω. By the definition of u, there exists a sequence {v_{n}}_{n∈ℕ }such that v_{n}(η) → u(η). By replacing v_{n }with max {v_{1}, . . ., v_{n}}, we may assume that v_{1 }≤ v_{2 }≤ · · · ≤ v_{n }≤ · · ·. Let be such that η ∈ ω and define V_{n}(η) to be the lifting of v_{n }in ω according to (3.3). From Lemma 3.2, V_{n }is also increasing and, since V_{n }∈ S (see Lemma 3.6) and V_{n }≥ v_{n}, it gets V_{n}(η) → u(η). Set . Obviously, we have that V ≤ u in Ω and V (η) = u(η). Noting that every V_{n }satisfies LV_{n } λ(ξ)V_{n }= f in ω, we have, by the dominated convergence theorem that the function V satisfies LV  λ(ξ)V = f in the distributional sense in ω. Since f ∈ C^{∞}(ω), we have V(ξ) ∈ C^{∞}(ω) in view of the hypoellipticity of the operator L  λ(ξ).
We conclude that V ≡ u in ω. In fact, suppose V(ζ) < u(ζ) for some ζ ∈ ω, then there exists a function such that . Define the increasing sequence and then the corresponding liftings W_{n}. Set . Analogously to V, W satisfies LW  λ(ξ)W = f. Since V_{n }≤ w_{n }≤ W_{n}, we obtain V ≤ W. The equalities V(η) = u(η) = W(η) and Lemma 3.2 imply that V ≡ W in Ω. This is in contradiction with . Consequently, V ≡ u in ω and u satisfies Lu  λ(ξ)u = f in the classical sense. The arbitrariness of η leads to the desired result. □
Definition 3.9. Let ζ ∈ ∂Ω. Then a function is called a barrier function related to the subLaplacian L at ζ if the following two conditions hold:
(i) Lw(ξ) ≤ 1 in Ω;
Lemma 3.10. Let Ω ⊂ G be a bounded open domain which satisfies the outer sphere condition at every point of the boundary ∂Ω. Then for every ζ ∈ ∂Ω, the Dirichlet problem
has a unique solution fulfilling w(ξ) > 0 on and w(ζ) = 0.
Proof. From [1], let Γ(ξ) = C_{Q}ρ(ξ, e)^{(Q2) }be the fundamental solution of the subLaplacian L. Define the convolution
where χ_{Ω }denotes the indicator function. Since for , it yields .
According to Corollary 10 in [3], the problem
has a unique solution . Since (see Corollary 2.8 in [1]), it follows that is the desired solution of (3.4). □
Theorem 3.11. Let Ω be as in Lemma 3.10. Suppose φ ∈ C(∂Ω) and . Then the Dirichlet problem (3.1) possesses a unique solution .
Proof. Uniqueness is a direct consequence of Lemma 3.2. Theorem 3.8 provides the existence of the solution u ∈ C^{∞}(Ω). To complete the proof of the theorem, it needs only to examine that u is continuous up to the boundary of Ω.
Let ζ ∈ ∂Ω. Since φ ∈ C(∂Ω), it follows that for any ε > 0 there exists some δ > 0 such that for every ξ ∈ ∂Ω with ρ(ξ, ζ) < δ, we have
Let w(ξ) be the barrier function related to L at ζ constructed in Lemma 3.10. Set and choose k_{1 }> 0 such that k_{1}w(ξ) ≥ 2M if ρ(ξ, ζ) ≥ δ. Set , and k = max{k_{1}, k_{2}}. Define that w_{1}(ξ): = φ(ζ) + ε + kw(ξ) and w_{2}(ξ): = φ(ζ)  ε  kw(ξ). Then we see in view of Lemma 3.10,
On the one hand, w_{1}(ξ) = φ(ζ) + ε + kw(ξ) ≥ φ(ζ) + ε > φ(ξ) when ρ(ξ, ζ) < δ; On the other hand, w_{1}(ξ) ≥ φ(ζ) + ε + 2M > φ(ξ) when ρ(ξ, ζ) ≥ δ. Combining these with Lemma 3.2 we can conclude that w_{1}(ξ) is a supersolution of (3.1). Analogously, w_{2}(ξ) is a subsolution of (3.1). Hence from the choice of u and the fact that every supersolution dominates every subsolution, we have in Ω that
and then
Since w(ξ) → 0 as ξ → ζ, we obtain u(ξ) → φ(ζ) as ξ → ζ. □
Remark 3.12. Let and u be the solution of
Elementary calculations show that and are a subsolution and a supersolution of (3.5) respectively. Thus, . It provides a L^{∞ }estimate for the solution of (3.5).
Theorem 3.13. Set φ ∈ C(∂Ω) and . Then there exists a unique solution to (3.1) in the sense of distribution.
Proof. Take a sequence , n = 1, 2, . . ., so that {f_{n}(ξ)} converges uniformly to f in Ω. Denote by u_{n }the corresponding solution of the Dirichlet problem
We obtain, in view of Remark 3.12,
In conclusion, {u_{n}} converges uniformly to a continuous function u which is the required solution. □
4 The monotone iteration scheme for semilinear equation
Let Ω be a bounded open domain in a Carnot group G. Consider Dirichlet problem (1.1), where f(ξ, u) is a smooth function of ξ and u, φ ∈ C(∂Ω). A function is called a supersolution of (1.1) if it satisfies
Analogously, a function is called a subsolution of (1.1) if it satisfies
The above inequalities are both in the sense of distribution. Here, a function T ≥ 0 means that for any positive test function ψ, we have Tψ ≥ 0. In the following we are ready to construct a smooth solution of (1.1) commencing with a subsolution and a supersolution in S^{1,2}(Ω, loc) by the monotone iteration scheme. We first prove a maximum principle.
Lemma 4.1. Assume that satisfies
where and λ(ξ) > 0. If u ≤ 0 on ∂Ω, then .
Proof. Suppose that the conclusion fails. Since u is continuous on , there exists a point ξ_{0 }∈ Ω such that u(ξ_{0}) > 0. Fix ε > 0 so small that u(ξ_{0})  ε > 0. Consequently, the function u_{ε }: = max{u  ε, 0} is nonnegative and has compact support in Ω as u ≤ 0 on ∂Ω. By the distribution meaning of solutions, we get
When u_{ε }> 0, it follows Xu_{ε }= Xu and Xu is not identically zero. In fact, if Xu ≡ 0, then u ≡ u(ξ_{0}) > 0 in Ω which contradicts the assumption that u ≤ 0 on ∂Ω. Consequently the left hand side of (4.1) is positive, a contradiction. This completes the proof of the lemma. □
Theorem 4.2. Let Ω be as in Lemma 3.10. Let f ∈ C^{∞}(G × (a, b)) and φ ∈ C(∂Ω). Suppose that μ and ν are, respectively, a supersolution and a subsolution of (1.1) with μ, , ν ≤ μ, and a < min ν < max μ < b. Then there exists a solution of (1.1) satisfying ν ≤ u ≤ μ.
Proof. Take K > 0 such that
on . Let v = Tu denote the unique solution in of the Dirichlet problem (see Theorem 3.11)
We claim that the nonlinear transformation T is monotone. To establish this we set u_{1 }< u_{2 }and notice that
and Tu_{1 }= Tu_{2 }= φ on ∂Ω. Letting w = Tu_{1 } Tu_{2}, we can obtain
and w = 0 on ∂Ω. As f(ξ, u) + K^{2}u is increasing in u by (4.2), it yields (L  K^{2}) w ≥ 0. From
we get w S^{2,2}(Ω, loc) by Lw ∈ L^{2}(Ω) and Proposition 2.1(i). It follows that w ≤ 0 in Ω by applying Lemma 4.1, therefore, Tu_{1 }≤ Tu_{2 }and T is monotone. We now begin the iteration scheme.
Let u_{1 }= Tμ. As
and u_{1 }= φ on ∂Ω, we get by a trivial calculation that
and u_{1 } μ ≤ 0 on ∂Ω. Arguing as in the previous gives u_{1 }≤ μ in Ω.
Define u_{n+1 }= Tu_{n}. The monotoneity of T yields
Analogously, starting from ν, we obtain a nondecreasing sequence
where v_{1 }= Tν, v_{n+1 }= Tv_{n}. Moreover, ν ≤ μ implies v_{1 }= Tν ≤ Tμ = u_{1 }and, therefore, v_{n }≤ u_{n }for each n ∈ ℕ. Thus
so that the limit is well defined in . Recall that
The dominated convergence theorem shows that
in the distributional sense. According to Proposition 2.1(i) and the fact that f(ξ, u) ∈ L^{p}(Ω) for 1 < p < +∞ one has u ∈ S^{2,p}(Ω, loc). Iterating the process, we get u ∈ S^{k, p}(Ω, loc) for k ≥ 0. Let . The definition in Section 2 gives ψu ∈ S^{k, p}. Furthermore, we obtain u ∈ C^{∞}(Ω) in view of Proposition 2.1(ii). Combining this with (4.3) we have which is the desired solution. □
We assume henceforth that G is of Heisenberg type. Such group was introduced by Kaplan [14] and has been subsequently studied by several authors, see [4,11,13] and the references therein.
Let G be a Carnot group of step two whose Lie algebra . Consider the map J : V_{2 }→ End(V_{1}) defined by
G is said of Heisenberg type if for every ξ_{2 }∈ V_{2}, with ξ_{2} = 1, the map J (ξ_{2}): V_{1 }→ V_{1 }is orthogonal.
In the case of the Heisenberg type groups, the gauge balls coincide with the level sets of the fundamental solution (that is a radial function in this class of groups, see [14]), and the balls B_{G}(e, R) invade G as R tends to +∞ since the vector fields on G satisfy the Hörmander rank condition. Thus, we get the following existence theorem in the whole space G by making use of Theorem 4.2 and the result in [4] that the gauge balls in Htype group satisfy the outer sphere condition.
Theorem 4.3. Let G be a group of Heisenberg type. Let u_{}(ξ), u_{+}(ξ) S^{1,2}(G, loc) ∩ C(G) be respectively a subsolution and a supersolution of the problem
where f ∈ C^{∞}(G × (a, b)) and a < u_{}(ξ) ≤ u_{+}(ξ) < b. Then there exists a solution u ∈ C^{∞}(G) of (4.4) satisfying
in G.
Proof. Let u_{0 }= u_{+}, set B_{G}(e, m) be the gauge ball of radius m centered at identity e. We construct u_{m }inductively in the following manner. Let v_{m }be the solution of the Dirichlet problem
obtained by means of Theorem 4.2 using u_{ }and u_{m1}, respectively, as a subsolution and a supersolution.
Define
Obviously, u_{ }≤ u_{m }≤ u_{m1}. We need to prove that u_{m }is a supersolution of (4.4). To see this, take a positive test function . From the divergence theorem, we obtain
and
The above two identities give
where denotes the outerward normal to ∂B_{G}(e, m), and A is a fixed positive semidefinite matrix (see [4,13]). Therefore, we may restrict ourselves to the case in which represents the derivative of u_{+ } v_{m }in an outward direction with respect to ∂B_{G}(e, m). Moreover, since u_{+ } v_{m }≥ 0 in B_{G}(e, m) and u_{+ } v_{m }= 0 on ∂B_{G}(e, m), it follows
Substitution in (4.5) gives
This implies that u_{m }is a supersolution, and we can restart the monotone iteration scheme on B_{G}(e, m+1).
In this way we obtain iteratively a sequence of supersolutions {u_{m}} satisfying the following properties:
(i) {u_{m}} is nonincreasing, and u_{ }≤ u_{m }≤ u_{+};
(ii) Every u_{m }satisfies Lu_{m }+ f(ξ, u_{m}) = 0 in B_{G}(e, m).
Set . We observe that {u_{m}} is a sequence of solutions of (4.4) on any B_{G}(e, k) for m ≥ k. It follows that u is a solution on B_{G}(e, k). Arguing as in Theorem 4.2 we know u ∈ C^{∞ }(B_{G}(e, k)). The arbitrariness of k implies u ∈ C^{∞}(G). Therefore, it holds that u is the required solution of (4.4). □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors contributed equally in this article. They read and approved the final manuscript.
Acknowledgements
We would like to thank Pengcheng Niu for research assistance and the two anonymous referees for very constructive comments. Zixia Yuan thanks the Mathematical Tianyuan Youth Foundation of China (No. 11026082) for financial support.
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