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Dirichlet problems for linear and semilinear sub-Laplace equations on Carnot groups
- Zixia Yuan^{1}Email author and
- Guanxiu Yuan^{2}
https://doi.org/10.1186/1029-242X-2012-136
© Yuan and Yuan; licensee Springer. 2012
- Received: 7 December 2011
- Accepted: 12 June 2012
- Published: 12 June 2012
Abstract
The purpose of this article, is to study the Dirichlet problems of the sub-Laplace equation Lu + f(ξ, u) = 0, where L is the sub-Laplacian 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 sub-Laplacian.
Mathematics Subject Classifications: 35J25, 35J70, 35J60.
Keywords
- Carnot group
- sub-Laplace equation
- Dirichlet problem
- Perron method
- monotone iteration scheme
1 Introduction
where Ω is a bounded domain in a Carnot group G and L is the sub-Laplacian. 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 [2–4]. The precursory work of Bony [2] proved a maximum principle and the solvability of the Dirichlet problem in the sense of Perron-Wiener. 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 well-known constructive methods for solving linear and semilinear Dirichlet problems, respectively. Brandolini et al. [9] applied these methods to the Dirichlet problems for sub-Laplace 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 sub-Laplace Dirichlet problems. In the discussion of semilinear Dirichlet problems, we will use monotone iteration scheme. The main difficulty we meet is that the sub-Laplacian 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 sub-Laplacian 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 sub-Laplace 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.
where dG(η) denotes a fixed Haar measure on G.
- (1)
ρ(ξ) ≥ 0; Moreover, ρ(ξ) = 0 if and only if ξ = e;
- (2)
ρ(ξ) = ρ(ξ^{-1}).
- (i)for 0 < β < 1,${\Gamma}^{\beta}:=\left\{f\in {L}^{\infty}\cap {C}^{0}:\underset{\xi ,\eta}{sup}\frac{|f\left(\eta \cdot \xi \right)-f\left(\eta \right)|}{{\left(\rho \left(\xi ,e\right)\right)}^{\beta}}<\infty \right\},$
- (ii)for β = 1,${\Gamma}^{1}:=\left\{f\in {L}^{\infty}\cap {C}^{0}:\underset{\xi ,\eta}{sup}\frac{|f\left(\eta \cdot \xi \right)+f\left(\eta \cdot {\xi}^{-1}\right)-2f\left(\eta \right)|}{{\left(\rho \left(\xi ,e\right)\right)}^{\beta}}<\infty \right\},$
- (iii)for β = k + β' where k = 1, 2, 3, . . . and, 0 < β' ≤ 1,${\Gamma}^{\beta}:=\left\{f\in {L}^{\infty}\cap {C}^{0}:{D}^{\alpha}f\in {\Gamma}^{{\beta}^{\prime}},\phantom{\rule{2.77695pt}{0ex}}\left|\alpha \right|\le k\right\}.$
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 $f,g\in {\mathcal{D}}^{\prime}\left(\Omega \right)$ 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 $\beta =k-\frac{Q}{p}>0$, 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
where $\lambda \left(\xi \right)\in C\left(\stackrel{\u0304}{\Omega}\right)$ satisfies λ(ξ) > 0.
The definition in the case of general degenerate elliptic operator can be seen in [3]. Notice that in the H-type group case, every bounded convex subset accords with the condition of the outer sphere. In particular, the gauge balls in H-type 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.
then u cannot achieve a nonnegative maximum at an interior point unless u ≡ constant in Ω.
has a unique distributional solution $u\in C\left(\stackrel{\u0304}{\omega}\right)$ for any $\omega \in \mathcal{F}$, $f\in C\left(\stackrel{\u0304}{\omega}\right)$ 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).
- (i)
u ≤ φ on ∂Ω;
- (ii)
for every $\omega \in \mathcal{F}$ and for every $h\in {C}^{2}\left(\omega \right)\cap C\left(\stackrel{\u0304}{\omega}\right)$ 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.
and hence all the equalities above hold. By Lemma 3.2 it follows that $\mathit{\u016b}-\stackrel{\u0304}{v}\equiv M$ 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. □
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 ${\omega}^{\prime}\in \mathcal{F}$ and $h\in {C}^{2}\left({\omega}^{\prime}\right)\cap C\left(\overline{{\omega}^{\prime}}\right)$ 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 $\mathit{\u016b}\le h$ on ∂(ω' ∩ ω). As $L\left(\mathit{\u016b}-h\right)-\lambda \left(\xi \right)\left(\mathit{\u016b}-h\right)=0$ in ω' ∩ ω and $\mathit{\u016b}-h\le 0$ on ∂(ω' ∩ ω), it yields by Lemma 3.2 that $\mathit{\u016b}\le h$ in ω' ∩ ω, and therefore U ≤ h in ω' ∩ ω. □
The following result is a trivial consequence of Definition 3.4.
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 $u\left(\xi \right):=\underset{v\in S}{sup}v\left(\xi \right)$ 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 $\omega \in \mathcal{F}$ 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 $V\left(\xi \right):=\underset{n\to \infty}{lim}{V}_{n}\left(\xi \right)$. 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 $\mathit{\u016b}\in S$ such that $V\left(\zeta \right)<\mathit{\u016b}\left(\zeta \right)$. Define the increasing sequence ${w}_{n}=max\left\{\mathit{\u016b},{V}_{n}\right\}$ and then the corresponding liftings W_{ n }. Set $W\left(\xi \right):=\phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty}{lim}\phantom{\rule{2.77695pt}{0ex}}{W}_{n}\left(\xi \right)$. 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 $V\left(\zeta \right)<\mathit{\u016b}\left(\zeta \right)\le W\left(\zeta \right)$. 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. □
- (i)
Lw(ξ) ≤ -1 in Ω;
- (ii)
w(ξ) > 0 on $\stackrel{\u0304}{\Omega}\backslash \left\{\zeta \right\}$, w(ζ) = 0.
has a unique solution $w\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ fulfilling w(ξ) > 0 on $\stackrel{\u0304}{\Omega}\backslash \left\{\zeta \right\}$ and w(ζ) = 0.
where χ_{Ω} denotes the indicator function. Since $\Gamma \left(\xi \right)\in {L}_{loc}^{p}$ for $1\le p<\frac{Q}{Q-2}$, it yields $\mathit{\u0169}\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$.
has a unique solution $v\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$. Since $L\mathit{\u0169}=-{\chi}_{\Omega}$ (see Corollary 2.8 in [1]), it follows that $w:=v+\mathit{\u0169}$ is the desired solution of (3.4). □
Theorem 3.11. Let Ω be as in Lemma 3.10. Suppose φ ∈ C(∂Ω) and $f\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$. Then the Dirichlet problem (3.1) possesses a unique solution $u\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$.
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.
Since w(ξ) → 0 as ξ → ζ, we obtain u(ξ) → φ(ζ) as ξ → ζ. □
Elementary calculations show that $-\frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}\left|\right|f|{|}_{{L}^{\infty}\left(\Omega \right)}$ and $\frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}\left|\right|f|{|}_{{L}^{\infty}\left(\Omega \right)}$ are a subsolution and a supersolution of (3.5) respectively. Thus, $\left|\right|u|{|}_{{L}^{\infty}\left(\Omega \right)}\le \frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}|\left|f\right|{|}_{{L}^{\infty}\left(\Omega \right)}$. It provides a L^{ ∞ } estimate for the solution of (3.5).
Theorem 3.13. Set φ ∈ C(∂Ω) and $f\in C\left(\stackrel{\u0304}{\Omega}\right)$. Then there exists a unique solution $u\in C\left(\stackrel{\u0304}{\Omega}\right)$ to (3.1) in the sense of distribution.
In conclusion, {u_{ n }} converges uniformly to a continuous function u which is the required solution. □
4 The monotone iteration scheme for semilinear equation
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.
where $\lambda \left(\xi \right)\in C\left(\stackrel{\u0304}{\Omega}\right)$ and λ(ξ) > 0. If u ≤ 0 on ∂Ω, then $\underset{\xi \in \Omega}{sup}u\left(\xi \right)\le 0$.
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, $\nu \in {S}^{1,2}\left(\Omega ,loc\right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$, ν ≤ μ, and a < min ν < max μ < b. Then there exists a solution $u\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ of (1.1) satisfying ν ≤ u ≤ μ.
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.
and u_{1} - μ ≤ 0 on ∂Ω. Arguing as in the previous gives u_{1} ≤ μ in Ω.
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 $\psi \in {C}_{0}^{\infty}\left(\Omega \right)$. 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 $u\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ 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.
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 H-type group satisfy the outer sphere condition.
in G.
obtained by means of Theorem 4.2 using u_{ - } and u_{m-1}, respectively, as a subsolution and a supersolution.
This implies that u_{ m } is a supersolution, and we can restart the monotone iteration scheme on B_{ G }(e, m+1).
- (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 $u\left(\xi \right)=\underset{m\to \infty}{lim}{u}_{m}\left(\xi \right)$. 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). □
Declarations
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.
Authors’ Affiliations
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