Abstract
Let \(n \in \mathbb {N}\) and \(\mathbb {H}^n\) be the Heisenberg group of dimension \(2n+1\). Let \(\Omega \) be a bounded open subset of \(\mathbb {H}^n\) and \(p \in (1, Q)\), where Q is the homogeneous dimension of \(\mathbb {H}^n\). Within an appropriate framework, we prove an interior pointwise gradient estimate for a weak solution to the problem
where \(\mathfrak {X}\) represents the horizontal gradient on the Heisenberg group.
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1 Introduction
This paper concerns a regularity estimate for an elliptic equation in the setting of Heisenberg group. While much is known for the regularity theory in the Euclidean spaces, its Heisenberg counterpart is less developed. Recently, there is a drastic movement toward the latter. Namely, we refer the readers to [5, 8,9,10, 14, 15, 17] and the references therein. Here, we wish to add a result in this direction to the existing literature. The result is in the spirit of the Euclidean pointwise estimates in [2,3,4, 11]. Also see [1, 6, 7, 12,13,14] and the references therein for more new trends in elliptic equations and systems.
Before delivering the main content, we quickly review the Heisenberg group. Let \(n \in \mathbb {N}\) and \(\mathbb {H}^n\) be the Heisenberg group of dimension \(2n+1\). That is, \(\mathbb {H}^n\) is a two-step nilpotent Lie group with underlying manifold \(\mathbb {C}^n \times \mathbb {R}\). The group operation is given by
for all \((z,t), (w,s) \in \mathbb {H}^n\), where \(z = (z_1,\ldots ,z_n)\), \(w = (w_1,\ldots ,w_n)\) and
For each \(u = (z,t) \in \mathbb {H}^n\), the inverse element is
and the identity is \(0 = (0,0)\).
By writing
the corresponding Lie algebra of the left-invariant vector fields on \(\mathbb {H}^n\) is spanned by
where \(j \in \{1,\ldots ,n\}\).
All non-trivial commutation relations are given by
with \([\cdot ,\cdot ]\) being the usual Lie bracket.
The horizontal gradient \(\mathfrak {X}\) is defined by
Let \(\Omega \subset \mathbb {H}^n\). We obtain the horizontal Sobolev spaces \(HW^{1,p}(\Omega )\) and \(HW^{1,p}_0(\Omega )\) by replacing the usual gradient \(\nabla \) with \(\mathfrak {X}\) in the definitions of \(W^{1,p}\) and \(W^{1,p}_0\) spaces. The horizontal divergence operator is then given by
For each \(a > 0\) and \((z,t) \in \mathbb {H}^n\), a dilation on \(\mathbb {H}^n\) is defined by
which is also an automorphism of \(\mathbb {H}^n\).
The homogeneous norm of \(u = (z,t) \in \mathbb {H}^n\) is given by
With this in mind,
where \(a > 0\). The homogeneous norm enjoys the triangle inequality and hence gives rise to a left-invariant distance
for each \(u,v \in \mathbb {H}^n\). Then, we may define the open ball with center \(u \in \mathbb {H}^n\) and radius \(r \in (0,\infty )\) by
Both left and right Haar measures on \(\mathbb {H}^n\) coincide with the Lebesgue measure \(dz \, dt\) on \(\mathbb {C}^n \times \mathbb {R}\). We denote the Lebesgue measure of a measurable set \(E \subset \mathbb {H}^n\) by |E|. Then,
for all \(u \in \mathbb {H}^n\) and \(r \in (0,\infty )\), where
is called the homogeneous dimension of \(\mathbb {H}^n\). Moreover,
and \(\Gamma (\cdot )\) is the usual Gamma function.
In the sequel, we write
Then, it follows from (1) that
Next we formulate our problem precisely. Let \(n \in \mathbb {N}\) and \(\Omega \) be a bounded open subset of \(\mathbb {H}^n\). Let \(p \in (1, Q)\). Consider
where
We assume that
This means that \(\textbf{A}(x, z)\) is measurable in x for every z and is continuous in z for a.e. x. Moreover, \(\textbf{A}(x, z)\) is differentiable in \(z \ne 0\) for a.e. x. The following structural conditions are also imposed on \(\textbf{A}\):
for some constant \(\Lambda \ge 1\) as well as for all \(x, x_0 \in \mathbb {H}^n\) and \((z,\eta )\in \mathbb {R}^{2n}\times \mathbb {R}^{2n}{\setminus }\{(0,0)\}\), where \(D_2\textbf{A}(x,z)\) denotes the Jacobian matrix of \(\textbf{A}\) with respect to the second variable \(z\in \mathbb {R}^{2n}{\setminus } \{0\}\). In (8), the function \(\omega : [0,\infty ) \longrightarrow [0,1]\) is required to be non-decreasing and satisfies
together with Dini’s condition
with \(\tau _0:= \frac{2}{p} \wedge 1\).
Note that (2) encapsulates well the p-Laplace equation with mixed data
Our aim here is to derive an interior pointwise estimate for the gradient of a weak solution to (2). To state our main result, we first need some definitions.
Definition 1.1
A function \(u\in HW^{1,p}_0(\Omega )\) is a weak solution to (2) if
for all \( \varphi \in HW^{1,p}_0(\Omega ).\)
In what follows, for each measurable function \(h: \Omega \longrightarrow \mathbb {R}\) denote
where \(B \subset \Omega \) is a ball. The oscillation of h on a set \(A \subset \Omega \) is defined by
Also set
for each \(q \in (1,n)\) and \(R > 0\), where \(q'\) is the conjugate index of q.
Our main result is as follows.
Theorem 1.2
Let \(n \in \mathbb {N}\) and \(\Omega \) be a bounded open subset of \(\mathbb {H}^n\). Let \(p \in (1, Q)\). Assume (3), (4), (5), (6), (7) and (8). Suppose that \(u\in C^1(\Omega )\) is a weak solution to (2). Then, there exists a constant \(C = C(n, p,\Lambda ,W_0) > 0\) such that
for all ball \(B_R(x)\subset \Omega \).
A technical remark is worth mentioning.
Remark 1.3
Working with Heisenberg group has its own intrinsic difficulties. See Lemma 3.1 below. Therein the appearance of the term \(M_r\) is new, which does not happen in the Euclidean setting. This results in weaker estimates compared to the corresponding Euclidean versions. The phenomenon was also observed in [10, Remark 3.1].
Despite all these, we still manage to achieve the pointwise estimate by carefully handling the new term \(M_r\).
The procedure for proving Theorem 1.2 involves the construction of certain comparison estimates in Sect. 2, an iteration argument in Sect. 3. With these in place, we prove Theorem 1.2 in Sect. 4.
Throughout assumptions. In the whole paper, let \(n \in \mathbb {N}\) and \(\Omega \) be a bounded open subset of \(\mathbb {H}^n\). Let \(p \in (1, Q)\). We always assume the set of conditions (3), (4), (5), (6), (7) and (8). If further assumptions are required, they will be explicitly stated in the corresponding statements.
2 Comparison Estimates
In this section, we construct two comparison estimates that serve to prove Theorem 1.2. The second one in Lemma 2.4 is new as it adapts our general setting proposed in this paper.
For later use, we draw some nice consequences from the aforementioned structural conditions. The first inequality in (5) and the Caratheodory property together yield
Meanwhile (6) implies the strict monotonicity condition
for all \((z,\eta ) \in \mathbb {R}^{2n} \times \mathbb {R}^{2n} \setminus \{(0,0)\}\) and for a.e. \(x \in \mathbb {H}^n\), where \(\Phi : \mathbb {R}^{2n}\times \mathbb {R}^{2n}\rightarrow \mathbb {R}\) is defined by
due to [16, Lemma 1].
Let \(u \in HW_0^{1,p}(\Omega )\) be a weak solution to (2). Suppose \(B_{2r}(x_0) \Subset \Omega \). Hereafter we write
for each \(\rho > 0\). Let \(w\in u+ HW_{0}^{1,p}(B_{2r})\) be the unique weak solution to the problem
and let \(v\in w+ HW_0^{1,p}(B_r)\) be the unique solution of
Recall the following uniform estimate from [10, (2.12)].
Lemma 2.1
There exists a constant \(C = C(n,p,\Lambda ) > 0\) such that
for all \(1< t < \infty \), \(r > 0\) and \(0< \delta < 1\).
As a direct consequence of Lemma 2.1, we obtain the following useful estimate.
Corollary 2.2
There exists a constant \(C = C(n,p,\Lambda ) > 0\) such that
for all \(1< t < \infty \) and \(0 < \rho \le \frac{r}{2}\).
The first comparison estimate relates v to w.
Lemma 2.3
There exists a constant \(C = C(n,p,\Lambda ) > 0\) such that
Proof
Using \(v-w\) as a test function in (12), we obtain
or equivalently
Then, (10), (5) and Holder’s inequality together imply
Next we use \(v-w\) as a test function in (11) and then combine with (13) to see that
or equivalently
Now we apply (10) and (8) to derive
Consequently,
due (14) and the fact that \(0 \le \omega \le 1\). \(\square \)
The second comparison estimate is between w and u.
Lemma 2.4
There exists a constant \(C = C(n,p,\Lambda ) > 0\) such that
for \(p \in (1,2)\) and
for \(p\in [2,Q)\).
Proof
Let \(m \in \mathbb {R}^n\). Then,
for all \(\varphi \in HW^{1,p}_0(B_{2r})\). Choosing \(\varphi =u-w\) as a test function in (15) gives
Using (10) and Holder’s inequality, this leads to
Next we apply Sobolev’s inequality (cf. [17, Theorem 2.1]) to obtain
Consequently,
Now we consider two cases.
Case 1: Suppose \(p \in (1,2)\). Then rewrite
Using Young’s inequality in the form
with an appropriate \(\epsilon > 0\), we arrive at
whence it follows from (17) that
By integrating both sides of the above estimate on \(B_{2r}\) and the apply Holder’s inequality with exponents \(\frac{2}{p}\) and \(\frac{2}{2-p}>1\), we obtain
Combining (16) with (18) yields
Another application of Young’s inequality yields
as required.
Case 2: Suppose \(p \in [2,Q)\). Starting from (16), we have
as required, where we used Young’s inequality in the second step.
This completes the proof. \(\square \)
3 Iteration Argument
Besides the comparison estimates, the proof of Theorem 1.2 requires an iteration argument given by Proposition 3.2 below. Proposition 3.2 in turn requires an auxiliary result stated in Lemma 3.1.
In what follows, let \(u\in HW_{0}^{1,p}(\Omega )\) be a weak solution of (2). Also w and v are given by (11) and (12), respectively, with \(B_{2r}(x_0) \Subset \Omega \).
Lemma 3.1
There exist constants \(C \ge 1\), \(\kappa \in (0,1)\) and \(\sigma _0 \in (0,1/2]\), all of which depend on n, p and \(\Lambda \) only, such that
for all \(q \in [1,\infty )\) and \(0<\rho <\sigma _0 r\), where
Proof
Let \(q \in [1,\infty )\). In view of [10, (3.5) and (3.6)], there exist constants \(C\ge 1\), \(\kappa \in (0,1)\) and \(\sigma _0 \in (0,1/2]\), all of which depend on n, p and \(\Lambda \) only, such that
for all \(0< \rho < \sigma _0 r\). Also observe that
for all \(\rho > 0\).
The claim is now justified by combining the above estimates together. \(\square \)
Next define
for a ball \(B_\rho =B_\rho (x_0)\subset \Omega \).
Proposition 3.2
There exist constants \(\kappa = \kappa (n,p,\Lambda ) \in (0,1)\) and \(\sigma _0 = \sigma _0(n,p,\Lambda ) \in (0,1/2]\) such that
for all \(\delta \in (0,\sigma _0]\) and \(B_r(x_0)\Subset \Omega \), where \(C_0 = C_0(n,p,\Lambda )\), \(C' = C'(n,p,\Lambda )\) and \(M_r\) is given by (19).
Remark 3.3
We may choose \(C' = 0\) in Proposition 3.2 if \(p \in [2,Q)\).
Proof
By virtue of Lemma 3.1, there exist constants \(C \ge 1\), \(\kappa \in (0,1)\) and \(\sigma _0 \in (0,1/2]\), all of which depend on n, p and \(\Lambda \) only, such that
for all \(\delta \in (0,\sigma _0]\).
Next we aim to bound the second term on the right-hand side of the above estimate. Observe that
where we used Lemma 2.3 in the second step and the fact that \(0 \le \omega \le 1\) in the third step. Here \(C = C(n,p,\Lambda )\).
It remains to bound the term
which can be done using Lemma 2.4.
This completes our proof. \(\square \)
4 Interior Pointwise Gradient Estimates
Now we have enough preparation to prove Theorem 1.2.
Proof of Theorem 1.2
Let \(B_R(x_0) \subset \Omega \). For short, we will write \(B_\rho = B_\rho (x_0)\) for each \(\rho > 0\) in the sequel. Set \(\epsilon \in (0,\sigma _0/2) \subset (0,1/4)\) be sufficiently small so that \( C_0\epsilon ^{\kappa }\le \frac{1}{4}\), where \(C_0\), \(\kappa \) and \(\sigma _0\) are the constants given by Proposition 3.2.
Let w and v be given by (11) and (12), respectively, with \(r = \frac{R}{4}\). Recall from Lemma 2.1 that
Also recall from (14) that
Likewise,
These three estimates together yield that there exists a constant \(C_1 = C_1(n,p,\Lambda ) > 0\) satisfying
For all \(j \in \mathbb {N}\) set
It suffices to show that
where \(C = C(n, p,\Lambda ,W_0)\).
Let \(\beta \in \mathbb {R}^n\). Then,
where we applied Proposition 3.2 with \(C_j = C_j(n,p,\Lambda ,\epsilon )\) for \(j \in \{2,3\}\) in the first step and (21) in the second step. Moreover, we may choose \(C_3 = 0\) if \(p \in [2,Q)\).
Let \(j_0, m \in \mathbb {N}\) be such that \(j_0 \ge 2\) and \(m \ge j_0+1\), whose appropriate values will be chosen later. Summing the above estimate up over \(j\in \{j_0, j_0+1, \ldots ,m-1\}\), we obtain
Observe that
Therefore, (23) implies
Estimating between an integral and its partial sum reveals that
and
In what follows, choose a \(j_0 = j_0(\epsilon ,C_2,W_0,\Omega )\) such that
where \(C_2\) is given in (23) and (24). Note that this choice is possible due to (9).
Now we consider three cases.
Case 1: Suppose \(|\mathfrak {X}u(x_0)|\le T_{j_0}\). Then, (22) trivially follows.
Case 2: Suppose there exists a \(j_1 \in \mathbb {N}\) such that \(j_1 \ge j_0\) and
for all \(j \in \{j_0, j_0+1,\ldots ,j_1\}\).
Then,
Now applying (23) and (24) with \(m=j_1\), we derive
where we used the fact that \(0< \epsilon < 1\) in the first step as well as (25), (26), (27) and (28) to estimate the last three terms in the second step.
It remains to estimate the fifth term on the right-hand side of the above inequality. Since \(C_3 = 0\) when \(p \in [2,n)\), we need only focus on \(p \in (1,2)\). In this case, it follows from Young’s inequality that
Either way we always have
where we used Corollary 2.2 in the second step. This is (22) as desired.
Case 3: Suppose \(T_j\le |\mathfrak {X}u(x_0)|\) for all \(j \in \{2,3,4,\ldots \}\). Then, we deduce from (24), (25) and (26) that
for all \(k \in \{j_0, j_0+1, j_0+2,\ldots \}\).
Simplifying the above estimate further and then letting \(k\longrightarrow \infty \), we arrive at
Now (22) follows after an application of Young’s inequality as we did in the last part of Case 2.
Thus, the proof is complete. \(\square \)
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This research is funded by University of Economics Ho Chi Minh City, Vietnam.
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Trong, N.N., Do, T.D. & Truong, L.X. Interior Pointwise Gradient Estimates for Quasilinear Elliptic Equations in Heisenberg Group. Bull. Malays. Math. Sci. Soc. 47, 22 (2024). https://doi.org/10.1007/s40840-023-01624-w
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DOI: https://doi.org/10.1007/s40840-023-01624-w