Abstract
In this paper, when \(1<p<2\), we establish the \(C^{1,\alpha }_{\,\textrm{loc}\,}\)-regularity of weak solutions to the degenerate subelliptic p-Laplacian equation
on SU(3) endowed with the horizontal vector fields \(X_1,\dots ,X_6\). The result can be extended to a class of compact connected semi-simple Lie group.
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1 Introduction
In this research article, we consider the special unitary group of \(3\times 3\) complex matrices SU(3) endowed with a horizontal vector field \(X_1,X_2,\dots ,X_6\); see Sect. 2 for more geometries and properties of SU(3). Given a domain \(\Omega \subset \mathrm{SU(3)}\), we consider the quasilinear subelliptic equation
Here \({\nabla _{{{\mathcal {H}}}}u}=(X_1u,X_2u,\dots ,X_6u)\) is the horizontal gradient of a function \(u\in C^1(\Omega )\); \(X^*_i\) is the formal adjoint of \(X_i\); the vector function \(a:=(a_1,a_2,\dots ,a_6)\in C^2({{\mathbb {R}}}^6,{{\mathbb {R}}}^6)\) satisfies the following growth and ellipticity conditions:
for all \(\xi ,\eta \in {{\mathbb {R}}}^6\), where \(0\le \delta \le 1\), \(1<p<\infty \) and \(0<l_0<L\). Note that conditions (1.2) and (1.3) are the same as conditions [5, (2.3) and (2.4)], but the condition (1.4) is stronger than the condition [5, (2.5)]
We call a function \(u\in W^{1,p}_{{{\mathcal {H}}},{\,\textrm{loc}\,}}(\Omega )\) as a weak solution to (1.1) if
Here \(W^{1,p}_{{{\mathcal {H}}},{\,\textrm{loc}\,}}(\Omega )\) is the first order p-th integrable horizontal local Sobolev space, namely, all functions \(u\in L^p_{\,\textrm{loc}\,}(\Omega )\) with their distributional horizontal gradients \({\nabla _{{{\mathcal {H}}}}u}\in L^p_{\,\textrm{loc}\,}(\Omega )\). Given the typical example \(a(\xi )=(\delta +|\xi |^2)^{\frac{p-2}{2}}\xi \), equation (1.1) becomes the non-degenerate p-Laplacian equation
and the p-Laplacian equation
Particularly, we call weak solutions to (1.8) as p-harmonic functions in \(\Omega \subset {\mathrm{SU(3)}}\).
In the linear case \(p=2\), p-harmonic functions in SU(3) are harmonic functions and their \(C^\infty \)-regularity was established by Hörmander [11]. In the quasilinear case \(p\ne 2\), Domokos-Manfredi [5] obtained the local boundedness of horizontal gradient \({\nabla _{{{\mathcal {H}}}}u}\) of p-harmonic functions u in SU(3), that is, \({\nabla _{{{\mathcal {H}}}}u}\in L^\infty _{\,\textrm{loc}\,}(\Omega )\). Moreover, when \(2<p<\infty \), they obtain the Hölder regularity of \({\nabla _{{{\mathcal {H}}}}u}\), that is, \({\nabla _{{{\mathcal {H}}}}u}\in C^{0,\alpha }(\Omega )\) for some \(\alpha \in (0,1)\) independent of u. But when \(1<p<2\), the Hölder regularity of \({\nabla _{{{\mathcal {H}}}}u}\) was unknown.
For the general quasi-linear equation (1.1) in SU(3), Domokos-Manfredi [5] also built up analogue regularity. To be precise, if a satisfies conditions (1.2), (1.3) and (1.5) for some \(1<p<\infty \), weak solutions u are proved to satisfy \({\nabla _{{{\mathcal {H}}}}u}\in L^\infty _{\,\textrm{loc}\,}(\Omega )\). If \(2\le p<\infty \), they further proved \({\nabla _{{{\mathcal {H}}}}u}\in C^{\alpha }_{\,\textrm{loc}\,}(\Omega )\) for some \(\alpha \in (0,1)\). But when \(1<p<2\), the Hölder regularity of \({\nabla _{{{\mathcal {H}}}}u}\) was also unavailable.
In this paper, we focus on the case \(1<p<2\). Moreover, instead of the condition (1.5) assumed by [5] when \(2\le p<\infty \), we work with the stronger condition (1.4). Indeed, if a satisfies (1.2), (1.3) and (1.4) for some \(1<p<2\), we obtain an Hölder regularity of horizontal gredient of weak solutions u to (1.1). As a consequence, when \(1<p<2\), the horizontal gradient of p-harmonic functions in SU(3) has an Hölder regularity. See Theorem 1.1 below for details. We always denote by \(B_r(x)\) the Carnot–Carathéodory ball centered at \(x\in {\mathrm{SU(3)}}\) with radius r with respect to the Carnot–Carathéodory distance \(d_{CC}\) which is defined in Sect. 2. For convenience we write \(B_r\) as \(B_r(x)\) and denote by \(C(a,b,\dots )\) a positive constant depending on parameters \(a,b,\dots \) whose value may change line to line.
Theorem 1.1
Suppose that \(a\in C^2({{\mathbb {R}}}^6,{{\mathbb {R}}}^6)\) satisfies the conditions (1.2), (1.3) and (1.4) for some \(l_0\) , L and such \(1<p<2\) , \(\delta \ge 0\). If \(u\in W^{1,p}_{{{\mathcal {H}}},{\textrm{loc}}}(\Omega )\) is a weak solution to (1.1), then \({\nabla _{{{\mathcal {H}}}}u}\in C^\alpha _{{\textrm{loc}}}(\Omega )\) for some \(\alpha \in (0,1)\) depending on \(l_0\) , L and such p , \(\delta \). Moreover, for all \(B_{r_0}\subset \Omega \) and any \(0<r\le r_0\) , we have
where \(0<\alpha <1\) depends on p , \(l_0\) and L , and the constant \(C>0\) depends on p , \(l_0\) , L and \(r_0\).
Consequently, when \(1<p<2\) , the horizontal gradients of p-harmonic functions on SU(3) have the Hölder regularity and satisfy (1.9).
Recall that in the Euclidean space \({{\mathbb {R}}}^n\) (corresponding to the vector field \(\{\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\dots ,\frac{\partial }{\partial x_n}\}\)), the \(C^{0,1}\) and \(C^{1,\alpha }\)-regualrity of solutions to Eq. (1.1) under conditions (1.2), (1.3) and (1.4) have been established by [8, 14, 21,22,23]. In the Heisenberg group \({{\mathbb {H}}}^n\), the \(C^{0,1}\) and \(C^{1,\alpha }\)-regularity of solutions to Eq. (1.1) under conditions (1.2), (1.3) and (1.4) have been established by [4, 7, 16,17,18, 20, 25].
We will prove Theorem 1.1 in Sect. 5 by borrowing some ideas from [18] to use De Giorgi’s method in [3], and also adapting some apriori estimates in [5] under conditions (1.2), (1.3) and (1.4). Indeed, given any weak solution u to (1.1), to get the Hölder regularity for \({\nabla _{{{\mathcal {H}}}}u}\), the central is to show that \({\nabla _{{{\mathcal {H}}}}u}\) belongs to De Giorgi’s class in SU(3), which will be recalled in Sect. 3. To this end, we consider the double truncation of \(X_lu\), that is,
where \(\mu (r)=\mathop {\max }\nolimits _{1\le {k}\le 6}\mathop {\sup }\nolimits _{B_r}|X_ku|\) and \(l\in \{1,\dots ,6\}\). It then suffices to buil up the following crucial Caccioppoli inequality
for all \(\beta \ge 0\), where \(\gamma >1\) is a constant and \(\eta \in C^\infty _0(B_r)\) is a standard cut-off function in \(B_r\). Indeed, if (1.10) holds true, following [18] line by line and using an interation argument as in [9, Lemma 7.3], we are able to conclude (1.9) and hence Theorem 1.1 holds true. For details see Sect. 5.
To prove (1.10), firstly, under the stronger condition (1.4), by choosing different test functions, we are able to adapt or modify the arguments in [5] so to get several a priori estimates as in Lemma 2.3, which are stronger than the corresponding estimates in [5]. See Sect. 2 for details.
Next, from these apriori estimates in Lemma 2.3, in Sect. 4 we deduce two auxiliary Caccioppoli inequalities for \({\nabla _{{{\mathcal {H}}}}}u\) and \({\nabla _{{{\mathcal {T}}}}}u\) involving v as in Lemmas 4.5 and 4.6, whose proofs are postponed to Sect. 6. Since there is no nilpotent structure in SU(3), we have to deal with all integrals involving \([X_i,X_j]u\) for all \(i,j\in \{1,\dots ,8\}\) according to Table 1 and use a priori estimates in Lemma 2.3 to bound them.
Finally, we choose the function \(\eta ^{\beta +4}v^{\beta +3}\) to test (4.7) in Lemma 4.4 and obtain (4.14), where \(\beta \ge 0\). Then using conditions (1.2), (1.3), (1.4), a priori estimates in Lemma 2.3 and Lemmas 4.5, 4.6, we conclude (1.10) in Lemma 4.1. The proof is postponed to Sect. 4.
To prove Lemma 4.5, first, we choose the function \(\eta ^{\beta +2}v^{\beta +2}|{\nabla _{{{\mathcal {H}}}}}u|^2X_lu\) to test (4.7) in Lemma 4.4 and obtain (6.1), where \(\beta \ge 0\) and \(l\in \{1,\dots ,6\}\). Then using conditions (1.2), (1.3), (1.4), we conclude (4.8) in Lemma 4.5. The proof is postponed to Sect. 6.
The proof of Lemma 4.6 is based on Lemma 4.5. To prove Lemma 4.6, first, we choose the function \(\eta ^{\tau (\beta +2)+4}v^{\tau (\beta +4)}|\nabla _{{\mathcal {H}}}u|^4X_lu\) to test (4.7) in Lemma 4.4 and obtain (6.6), where \(\beta \ge 0\), \(\tau \in (1/2,1)\) and \(l\in \{7,8\}\). Then using conditions (1.2), (1.3), (1.4), a priori estimates in Lemmas 2.3 and 4.5, we conclude (4.12) in Lemma 4.6. The proof is postponed to Sect. 6.
2 Preliminaries
We identify the group SU(3) with the unitary group of \(3\times 3\) complex matrices
Its Lie algebra is given by
with the inner product \( \langle {X,Y}\rangle :=-{\frac{1}{2}}{{\textrm{tr}}}(XY). \)
Noting that the two-dimensional maximal torus
we choose its Lie algebra
as the Cartan subalgebra.
Recalling the definition of su(3), we use a set of Gell Mann matrices \({{\mathcal {G}}}\) to form its orthonormal basis, that is,
Consider the following two vector fields:
Since \(T_1=\frac{1}{2} X_7\) and \(T_2=\frac{1}{2\sqrt{3}}X_7-\frac{1}{\sqrt{3}}X_8\), we choose the vertical vector field \({\nabla _{{{\mathcal {T}}}}}=\{X_7,X_8\}\) as an orthonormal basis of the Cartan subalgebra \({{\mathcal {T}}}\).
The following table is [5, Table 2.1], which gives all \([X_i,X_j]\) for any two vector fields \(X_i,X_j\in \{X_1,X_2,\dots ,X_8\}\).
Table 1 shows that
and that
where \(C^k_{i,j}\in {{\mathbb {R}}}\) are constants and are completely determined by Table 1. Note that the orthonormal basis \( {\nabla _{{{\mathcal {H}}}}}=\{X_1,X_2,\dots ,X_6\} \) generates the horizontal subspace \({{\mathcal {H}}}\) in SU(3). Since the matries \({{\mathcal {G}}}\) are left-invariant vector fields, the basis \({\nabla _{{{\mathcal {H}}}}}\) is left-invariant. From Table 1, at every point of SU(3) the basis \({\nabla _{{{\mathcal {H}}}}}\) satisfies the Hörmander condition. From this, the horizontal distribution of a sub-Riemannian manifold is generated by \({\nabla _{{{\mathcal {H}}}}}\).
In the following, we define the Carnot–Carathéodory distance \(d_{CC}\). An absolutely continuous function curve \(\gamma :[0,T]\rightarrow {\mathrm{SU(3)}}\) is subunitary associated to \({\nabla _{{{\mathcal {H}}}}}\) if there are measurable \(\{\alpha _i\in {L^\infty [0,T]}\}_{1\le i\le 6}\) such that
According to [2], since \({\nabla _{{{\mathcal {H}}}}}\) satisfies the Hörmander condition, there exist curves \(\gamma \) subunitary associated to \({\nabla _{{{\mathcal {H}}}}}\) connecting any two given points \(x,y\in {\mathrm{SU(3)}} \). The Carnot–Carathéodory distance \(d_{CC}\) is then defined as
We denote by dx the bi-invariant Harr-measure, by |E| the Lebesgue measure of a measurable set \(E\subset {\mathrm{SU(3)}}\) and by 
the average of an integrable function f over set E.
In the rest of this section, we recall some Caccioppoli inequalities established by Domokos-Manfredi [5] and use similar methods to get stronger estimates; see [5, Theorem 2.1, Lemmas 3.1, 3.2, 3.3 and 3.4, Remark 3.2 and Corollary 3.1].
Domokos-Manfredi [5] established the following uniform gradient estimate; see [5, Theorem 2.1].
Proposition 2.1
[5, Theorem 2.1] Let \(1<p<\infty \) and \(0\le \delta \le 1\). Assume that \(a\in C^2({{\mathbb {R}}}^6,{{\mathbb {R}}}^6)\) satisfies the conditions (1.2), (1.3) and (1.4). If \(u\in W^{1,p}_{{{\mathcal {H}}},{\textrm{loc}}}(\Omega )\) is a weak solution to (1.1), then \({\nabla _{{{\mathcal {H}}}}u}\in L^\infty _{\,\textrm{loc}\,}(\Omega )\). Moreover, for all ball \(B_r\subset \Omega \) , we have
Combining Proposition 2.1 and [6, Theorem 1.1], we have the following corollary.
Corollary 2.2
Let \(1<p<\infty \) and \(\delta >0\). If \(u\in W^{1,p}_{{{\mathcal {H}}},{{\textrm{loc}}}}(\Omega )\) is a weak solution to (1.1), then \(u\in C^\infty (\Omega )\).
The following lemma gives some Caccioppoli estimates which will be used to prove Lemmas 4.5, 4.6 and 4.1; see Sect. 2.1 for its proof. For simplicity, we write \(w=(\delta +|{\nabla _{{{\mathcal {H}}}}u}|^2)\). Note the fact that in all the integral terms in the following lemma, only \(w^{\frac{p-2}{2}}\) includes \(\delta \). This fact is necessary for us to establish the Caccioppoli inequality for v in Sect. 4; see Lemma 4.1.
Lemma 2.3
Let \(0<\delta <1\). Assume that \(a\in C^2({{\mathbb {R}}}^6,{{\mathbb {R}}}^6)\) satisfies the conditions (1.2), (1.3) and (1.4). If \(u\in W^{1,p}_{{{\mathcal {H}}},{\textrm{loc}}}(\Omega )\) is a weak solution to (1.1), then for any \(\eta \in {C^\infty _0(\Omega )}\) with \(0\le \eta \le 1\) , the followings hold:
-
(i)
If \(\beta \ge 0\) , then
$$\begin{aligned} \int _\Omega \eta ^2w^{\frac{p-2}{2}}|\nabla _{{\mathcal {T}}}u|^{2\beta }| \nabla _{{\mathcal {H}}}\nabla _{{\mathcal {T}}}u|^2dx&\le {C(p,L,l_0)}\int _\Omega |\nabla _{{\mathcal {H}}}\eta |^2w^{\frac{p -2}{2}}|\nabla _{{\mathcal {T}}}u|^{2\beta +2}dx\nonumber \\&\quad {+}C(p,L,l_0)(\beta {+}1)^2\!\int _\Omega \!\eta ^2w^{\frac{p-2}{2}}| \nabla _{{\mathcal {H}}}u|^2|\nabla _{{\mathcal {T}}}u|^{2\beta }dx. \end{aligned}$$(2.3) -
(ii)
If \(\beta \ge 0\) , then
$$\begin{aligned} \int _\Omega \eta ^2w^{\frac{p-2}{2}}|\nabla _{{\mathcal {H}}}u|^{2\beta }| \nabla _{{\mathcal {H}}}\nabla _{{\mathcal {H}}}u|^2dx&\le {C(p,L,l_0)}(\beta +1)^4\int _\Omega \eta ^2w^{\frac{p-2}{2}}| \nabla _{{\mathcal {H}}}u|^{2\beta }|\nabla _{{\mathcal {T}}}u|^2dx\nonumber \\&\quad +C(p,L,l_0)(\beta {+}1)^2K_\eta \int _{supp(\eta )}w^{\frac{p-2}{2}} |\nabla _{{\mathcal {H}}}u|^{2\beta +2}dx, \end{aligned}$$(2.4) -
(iii)
If \(\beta \ge 1\) , then
$$\begin{aligned}&\int _\Omega \eta ^{2\beta +2}w^{\frac{p-2}{2}}|\nabla _{{\mathcal {T}}}u |^{2\beta }|\nabla _{{\mathcal {H}}}\nabla _{{\mathcal {H}}}u|^2dx\nonumber \\&\quad \le {C(p,L,l_0)^\beta }(\beta +1)^{4\beta }{\Vert \nabla _{{\mathcal {H}}} \eta \Vert }^{2\beta }_{L^\infty } \int _\Omega \eta ^2w^{{\frac{p-2}{2}}}|\nabla _{{\mathcal {H}}}u|^{2 \beta }|\nabla _{{\mathcal {H}}}\nabla _{{\mathcal {H}}}u|^2dx. \end{aligned}$$(2.5) -
(iv)
If \(\beta \ge 1\) , then
$$\begin{aligned}{} & {} \int _\Omega \eta ^2w^{{\frac{p-2}{2}}}|{\nabla _{{{\mathcal {H}}}}u}|^{2\beta }|\nabla _{{\mathcal {H}}} \nabla _{{\mathcal {H}}}u|^2dx \nonumber \\{} & {} \quad \le {C(p,L,l_0)}(\beta +1)^{12}K_\eta \int _{supp(\eta )}w^{{\frac{p-2}{2}}}| \nabla _{{\mathcal {H}}}u|^{2\beta +2}dx. \end{aligned}$$(2.6) -
(v)
If \(\beta \ge 1\) , then
$$\begin{aligned}{} & {} \int _\Omega \eta ^{2\beta +2}w^{\frac{p-2}{2}}|\nabla _{{\mathcal {T}}} u|^{2\beta }|\nabla _{{\mathcal {H}}}\nabla _{{\mathcal {H}}}u|^2dx \nonumber \\{} & {} \quad \le {C(p,L,l_0,\beta )}K^{\beta +1}_\eta \int _{supp(\eta )}w^{{\frac{p-2}{2}}}|\nabla _{{\mathcal {H}}}u|^{2\beta +2}dx. \end{aligned}$$(2.7) -
(vi)
If \(\beta \ge 1\) , then
$$\begin{aligned}{} & {} \int _\Omega \eta ^{2\beta +4}w^{\frac{p-2}{2}}|\nabla _{{\mathcal {T}}}u |^{2\beta }|\nabla _{{\mathcal {H}}}\nabla _{{\mathcal {T}}}u|^2dx\nonumber \\{} & {} \quad \le {C(p,L,l_0,\beta )}K^{\beta +2}_\eta \int _{supp(\eta )}w^{{\frac{p-2}{2}}}|\nabla _{{\mathcal {H}}}u|^{2\beta +2}dx. \end{aligned}$$(2.8)
Above
2.1 Proof of Lemma 2.3
In this subsection, we prove Lemma 2.3. To this end, we consider the Riemannian approximation to (1.1) as in [5]. For any fixed constant \(\varepsilon \in (0,1)\), we write \(\nabla ^\varepsilon =(\nabla ^\varepsilon _{{\mathcal {H}}},\nabla ^\varepsilon _{{\mathcal {T}}})\), where \(\nabla ^\varepsilon _{{\mathcal {H}}}={\nabla _{{{\mathcal {H}}}}}\) and \(\nabla ^\varepsilon _{{\mathcal {T}}}=\varepsilon {\nabla _{{{\mathcal {T}}}}}\). Consider a Riemannian approximation to (1.1), that is,
see [5, Section 3] for details. Let \(u_\varepsilon \) be a weak solution to (2.10) with conditions (1.2), (1.3) and (1.4). By [5, Remark 3.1], we obtain that \(u_\varepsilon \in C^\infty (\Omega )\) when \(\delta >0\) and \(\varepsilon >0\).
We have the following series of a priori estimates for \(u_\varepsilon \), that is, Lemmas 2.4, 2.5, 2.6 and 2.7. They correspond to Lemmas 3.1, 3.2, 3.3 and 3.4 in [5], where they assume (1.2), (1.3) and (1.5). Here we work with (1.2), (1.3) and the stronger one (1.4). For simplicity, we write \(w_\varepsilon =(\delta +|\nabla ^\varepsilon {u_\varepsilon }|^2)\).
Lemma 2.4
Let \(0<\delta <1\). For any \(\beta \ge 0\) and any \(\eta \in {C^\infty _0(\Omega )}\) with \(0\le \eta \le 1\) , we have
Lemma 2.4 is a stronger version of [5, Lemma 3.1], since \({w_\varepsilon }^{\frac{p-2}{2}}|\nabla ^\varepsilon {u_\varepsilon }|^2\) in the second term in the right hand side of (2.11) is more accurate than \(w_\varepsilon ^{\frac{p}{2}}\) in the second term in the right hand side of [5, (3.3)]. To prove Lemma 2.4, we follows the proof of [5, Lemma 3.1] line by line. The only difference is that we use the condition (1.4) instead of the condition (1.5) they used in the proof of [5, Lemma 3.1]. Here we omit the details.
Lemma 2.5
Let \(0<\delta <1\). For any \(\beta \ge 0\) and any \(\eta \in {C^\infty _0(\Omega )}\) with \(0\le \eta \le 1\) , we have
Lemma 2.5 is similar to [5, Lemma 3.2]. The only difference between the two lemmas is that \(w_\varepsilon ^{{\frac{p-2}{2}}+\beta }\) and \(w_\varepsilon ^{{\frac{p}{2}}+\beta }\) in [5, (3.8)] are replaced by \({w_\varepsilon }^{\frac{p-2}{2}}|\nabla ^\varepsilon {u}_\varepsilon |^{2\beta }\) and \({w_\varepsilon }^{\frac{p-2}{2}}|\nabla ^\varepsilon {u_\varepsilon }|^{2\beta +2}\) separately. The proof of Lemma 2.5 follows the same line as that of [5, Lemma 3.2]. To prove [5, Lemma 3.2], they used \(\phi =\eta ^2w_\varepsilon ^\beta X^\varepsilon _iu_\varepsilon \) to test [5, (3.10)]. Now, to prove Lemma 2.5, we use \(\phi =\eta ^2|\nabla ^\varepsilon u_\varepsilon |^{2\beta } X^\varepsilon _iu_\varepsilon \) instead of \(\phi =\eta ^2w_\varepsilon ^\beta X^\varepsilon _iu_\varepsilon \) as the new test function in [5, (3.10)] in the proof of [5, Lemma 3.2]. Then we use the condition (1.4) whenever the condition (1.5) is used in the rest of the proof of [5, Lemma 3.2]. Here we omit the details.
Lemma 2.6 follows from Lemma 2.4.
Lemma 2.6
Let \(0<\delta <1\). For any \(\beta \ge 1\) and any \(\eta \in {C^\infty _0(\Omega )}\) with \(0\le \eta \le 1\) , we have
Lemma 2.6 is stronger than [5, Lemma 3.3], since \({w_\varepsilon }^{\frac{p-2}{2}}|\nabla ^\varepsilon {u_\varepsilon }|^2\) in the right hand side of (2.13) is more accurate than \(w_\varepsilon ^{\frac{p}{2}}\) in the right hand side of [5, (3.11)]. To prove Lemma 2.6, we follows the proof of [5, Lemma 3.3] line by line. The only difference is that we use (2.11) with \(\eta \rightarrow \eta ^{\beta +2}\) and the condition (1.4) to replace [5, (3.7)] and the condition (1.5) they used in the proof of [5, Lemma 3.3] separately. Here we omit the details.
By Lemma 2.6, we obtain Lemma 2.7.
Lemma 2.7
Let \(0<\delta <1\). For any \(\beta \ge 1\) and \(\eta \in {C^\infty _0(\Omega )}\) with \(0\le \eta \le 1\) , we have
Lemma 2.7 is a stronger version of [5, Lemma 3.4], since \({w_\varepsilon }^{\frac{p-2}{2}}|\nabla ^\varepsilon {u_\varepsilon }|^{2\beta }\) in the right hand side of (2.14) is more accurate than \(w_\varepsilon ^{\frac{p-2}{2}+\beta }\) in the right hand side of [5, (3.13)]. To prove Lemma 2.7, we follows the proof of [5, Lemma 3.4] line by line. The only difference is that we use Lemma 2.6 and the condition (1.4) to replace [5, Lemma 3.3] and the condition (1.5) they used in the proof of [5, Lemma 3.4] separately. Here we omit the details.
Lemma 2.3 follows from Lemmas 2.4, 2.5 and 2.7. Specifically, by Theorem 2.2, letting \(\varepsilon \rightarrow 0\) in the above estimates, we get some intrinsic Cacciopoli inequalities for weak solutions u to (1.1) from Lemmas 2.4, 2.5 and 2.7, which are stronger than that in [5, Corollary 4.1]; see [5, Section 4] for more details.
Proof of Lemma 2.3
The estimates (2.3), (2.4) and (2.5) follow from Lemmas 2.4, 2.5 and 2.7 respectively in a direct way.
Next we prove (2.6). By Hölder’s inequality, we have
Noting that \(|{\nabla _{{{\mathcal {T}}}}u}|^2\le 2|{\nabla _{{{\mathcal {H}}}}\nabla _{{{\mathcal {H}}}}u}|^2\), we use (2.5) to bound the first term in the right hand side of (2.15). Then by Young’s inequality, we have
which, together with (2.4), yields (2.6).
Combining (2.5) and (2.6), we conclude (2.7).
Next we prove (2.8). By Hölder’s inequality, we have
By changing \(\eta \) to \(\eta ^{\beta +2}\) in (2.3), we have
We combine (2.16) and (2.17). Then by Young’s inequality therein, we have
Since \(|{\nabla _{{{\mathcal {T}}}}u}|^2\le 2|{\nabla _{{{\mathcal {H}}}}\nabla _{{{\mathcal {H}}}}u}|^2\), applying (2.7) to the first term in the right hand side of (2.18), we conclude (2.8). \(\square \)
2.2 Sobolev inequalities
In this subsection, we recall local Sobolev inequalities and Poincaré inequalities on SU(3). Denote by \(Q=10\) the homogeneous dimension of SU(3). The following lemma follows from [10, Theorem 11.20 and Corollary 9.8] and [1, Proposition 2.1].
Lemma 2.8
Let \(1\le p_1<Q\). For any ball \(B_\rho \subset \Omega \) , we have
for any \(f\in C^\infty _0(B_\rho )\) , and
for any \(f\in C^\infty (B_\rho )\) , where 
.
3 De Giorgi’s class of functions
In this section, we recall De Giorgi’s class of functions defined in SU(3). Let \(\Omega \) be a domain in SU(3).
Definition 3.1
Let \(B_{\rho _0}\subset \Omega \) be a ball. We call a function \(u\in {W^{2,2}_{{{\mathcal {H}}}}(B_{\rho _0})}\cap {L^\infty (B_{\rho _0})}\) belongs to \({DG^+(B_{\rho _0})}\) if there exists non-negative constants \(\gamma \) and \(\chi \) such that for any balls \(B_{\rho '},B_\rho \) with the same center as \(B_{\rho _0}\) and \(0<\rho '<\rho <\rho _0\), for any \(k\in {{\mathbb {R}}}\), and for any \(q>Q\), the inequality
holds true, where \(A^+_{k,\rho }(u)=\{x\in {B_\rho }:(u(x)-k)^+=\max (u(x)-k,0)>0\}\).
Remark 3.2
By replacing \(A^+_{{k,\rho }}(u)\) with \(A^-_{k,\rho }(u)\) in (3.1), we define that \(u\in {DG^-(B_{\rho _0})}\) in a similar way, where \(A^-_{k,\rho }(u)=\{x\in {B_\rho }:(u(x)-k)^-=\min (u(x)-k,0)<0\}. \)
It is apparent that \(u\in {DG^-(B_{\rho _0})}\) if \(-u\in {DG^+(B_{\rho _0})}\). Denote \({DG(B_{\rho _0})}={DG^+(B_{\rho _0})}\cap {DG^-(B_{\rho _0})}\).
The following lemma is almost the same as [25, Lemma 4.1]; see the “Appendix” for its proof.
Lemma 3.3
For any \(b\in (0,1)\) , there exists a constant \(\theta _1=\theta _1(\gamma ,q,b)\in (0,1)\) such that for any function \({u}\in {DG^+(B_{\rho })}\) and for any constant k , the following holds:
If
for any \(q>Q\) , then
implies that
The following lemma is almost the same as [25, Lemma 4.2]; see the “Appendix” for the details of its proof.
Lemma 3.4
For any \(\tau \in (0,1]\) , there exists a constant \(s=s(\gamma ,q,\tau ,R)>0\) such that for any function \({u}\in {DG^+(B_{\rho })}\) and for any \(k\in {{\mathbb {R}}}\) , the following holds:
If
then
where \(H=\mathop {\sup }\nolimits _{B_{\rho }}u(x)-k\).
Remark 3.5
Changing k to \(-k\) and u(x) to \(-u(x)\) in Lemma 3.4, for any function \({u}\in {DG^-(B_{\rho })}\) the following holds:
implies that
where \(H=-\mathop {\inf }\nolimits _{B_{\rho }}u(x)+k\).
Applying Lemma 3.4 and Remark 3.5 with \(k={\frac{1}{2}}\left( \mathop {\sup }\limits _{B_{\rho /4}}u(x)+\mathop {\inf }\limits _{B_{\rho /4}}u(x)\right) \), we have the following theorem. We omit its proof.
Theorem 3.6
For any function \({u}\in {DG(B_{\rho })}\) , there exists \({s_0}=s_0(\gamma ,q)>0\) such that
4 Crucial Caccioppli inequality
In this section, we give the crucial Caccioppli inequality and its proof. Let \(1<p<\infty \) and \(u\in W^{1,p}_{{{\mathcal {H}}},{\textrm{loc}}}(\Omega )\) be a weak solution to (1.1) with \(0<\delta \le 1\). Fix a ball \(B_{r_0}\subset \Omega \). For any ball \(B_r\) with the same center as \(B_{r_0}\) and \(0<r\le r_0/16\), we denote \(\varpi (r)=\mathop {\max }\limits _{1\le {k}\le 6}{{\textrm{osc}}_{B_r}}X_ku\) and
where \(\mu (r)=\mathop {\max }\nolimits _{1\le {k}\le 6}\mathop {\sup }\nolimits _{B_r}|X_ku|\) and \(l\in \{1,\dots ,6\}\). For any fixed \(l\in \{1,\dots ,6\}\), we consider the set
Note that for any \(i\in \{1,\dots ,6,7,8\}\),
We have the following lemma, which includes the crucial Caccioppli inequality for v. In this paper, for convenience we write \(a_{i,j}(\xi )=\frac{\partial a_i(\xi )}{\partial \xi _j}\) for any \(\xi \in {{\mathbb {R}}}^6\) and \(i,j\in \{1,2,\dots ,6\}\).
Lemma 4.1
Let \(B_r\subset \Omega \) be a ball. Assume that the cut-off function \(\eta \in C^\infty _0(B_r)\) satisfies (4.10) and (4.11). Then for any \(\gamma >1\) and any \(\beta \ge 0\) , we have
Remark 4.2
For any \(l\in \{1,\dots ,6\}\), we use the same method as that of Lemma 4.1 to get (4.2) with
From Lemma 4.1 and Remark 4.2, we get the following lemma; see Sect. 6 for its proof.
Lemma 4.3
For any ball \(B_r\subset \Omega \) with the same center \(B_R\) and \(0<r\le {R}\) , there is \(\theta =\theta (p,L,l_0,R)>0\) such that the followings hold:
-
(i)
If
$$\begin{aligned} |\{x\in {B}_r:X_ku<\mu (r)/4\}|\le \theta |B_r| \end{aligned}$$(4.3)holds true for an index \(k\in \{1,\dots ,6\}\) , then
$$\begin{aligned} \mathop {\inf }\limits _{B_{r/2}}X_ku\ge 3\mu (r)/16. \end{aligned}$$(4.4) -
(ii)
If
$$\begin{aligned} |\{x\in {B}_r:X_ku>-\mu (r)/4\}|\le \theta |B_r| \end{aligned}$$(4.5)holds true for an index \(k\in \{1,\dots ,6\}\) , then
$$\begin{aligned} \mathop {\sup }\limits _{B_{r/2}}X_ku\le -3\mu (r)/16. \end{aligned}$$(4.6)
4.1 Proof of Lemma 4.1
Before proving Lemma 4.1, we need the following lemmas. See Sect. 6 for their proofs.
Lemma 4.4
For any \(l\in \{1,\dots ,6,7,8\}\) , \(X_lu\) is a weak solution to the equation
Lemma 4.5
For any \(\beta \ge 0\) and any non-negative \(\eta \in C^\infty _0(\Omega )\) , we have
Before starting the following lemma, we note that if \(\delta \ge 0\) and \(q\ge 0\) such that \(p+q-2\ge 0\), the following holds:
where \(C(p,q)=6^{(q+p-2)/2}\) when \(p\ge 2\) and \(C(p,q)=6^{q/2}\) when \(1<p<2\). In the rest of this section, for any fixed ball \(B_r\subset \Omega \) we consider a cut-off function \(\eta \in {C^\infty _0(B_r)}\) such that
and
The following lemma gives a Caccioppoli inequality for \({\nabla _{{{\mathcal {T}}}}}u\) weighted with \(|{\nabla _{{{\mathcal {H}}}}}u|^4\) involving v; see Sect. 6 for its proof.
Lemma 4.6
Let \(B_r\subset \Omega \) be a ball. Assume that the cut-off function \(\eta \in C^\infty _0(B_r)\) satisfies (4.10) and (4.11). Then for any \(\tau \in (1/2,1)\) , any \(\gamma \in (1,2)\) and any \(\beta \ge 0\) , we have
where
Now we use Lemmas 4.4, 4.5 and 4.6 to prove Lemma 4.1.
Proof of Lemma 4.1
We consider two cases: \(1<\gamma <3/2\) and \(\gamma \ge 3/2\). In the case that \(\gamma \ge 3/2\), we use Hölder’s inequality for the integral in the right hand side of (4.2) when (4.2) holds for some \(1<\gamma _0<3/2\). Below we assume \(1<\gamma <3/2\). Recall that
where \(\mu (r)=\mathop {\max }\nolimits _{1\le {k}\le 6}\mathop {\sup }\nolimits _{B_r}|X_ku|\) and \(l\in \{1,\dots ,6\}\). Let \(B_r\subset \Omega \) be a ball. Consider the cut-off function \(\eta \in C^\infty _0(\Omega )\) with (4.10) and (4.11). For any \(\beta \ge 0\) and any \(l\in \{1,\dots ,6\}\), letting \(\varphi =\eta ^{\beta +4}v^{\beta +3}\) be a test function in (4.7), we have
which, together with \(X_lX_ju=X_jX_lu+[X_l,X_j]u\), yields
Recalling that
and that
by the condition (1.2), we have
Here the integration domain in the right hand side of the above inequality is the set E.
Next, we bound each item in the right hand of (4.14) in turn. By integration by parts, we have
By the condition (1.4) and (4.9), we have
Applying Young’s inequality to the final item in the right hand side of (4.15), we have
For \(I_2\), by (2.1) and the condition (1.3), we have
For \(I_{21}\), by (4.9) and Young’s inequality, we have
For \(I_{22}\), by Hölder’s inequality with \(q={\frac{2\gamma }{\gamma -1}}\), we have
Here the integration domains in (4.16) and (4.18) are essentially the set E. Applying (2.7) with \(\beta =\frac{q-2}{2}\) in Lemma 2.3 to the final integral in the right hand side of (4.18), we have
Combining (4.18) and (4.19), from (4.9), we have
Combining (4.16), (4.17) and (4.20), we have
For \(I_3\), by integration by parts and (2.1), we have
Since
holds for any \(k\in \{7,8\}\), by (2.2) and the condition (1.4), we have
For \(I_{31}\), \(I_{33}\) and \(I_{35}\), by (4.9), we have
For \(I_{34}\), by (4.9) and Young’s inequality, we have
For \(I_{32}\), Hölder’s inequality yields
Noting that the integration domains in the above inequality are the set E, we have
where
We use Lemma 4.6 with \(\tau =2-\gamma \) to estimate M. Thus
where J is as in (4.12). Combining (4.21) and (4.22), we have
where J is as in (4.12). From this, by Young’s inequality again, we have
where J is as in (4.12).
Combining all the above estimates together, we use Hölder’s inequality to conclude (4.2). \(\square \)
5 Proof of Theorem 1.1
In this section, we prove Theorem 1.1 by considering two cases: \(\delta >0\) and \(\delta =0\). First, we prove Theorem 1.1 for the case \(\delta >0\).
The following lemma is similar to [5, Lemma 4.1]. It is a preparation for proving Theorem 1.1. We omit its proof.
Lemma 5.1
Let \(1<p<2\) and \(u\in C^\infty (\Omega )\) be a solution to (1.1) with \(\delta >0\). Let \(B_{3r_0/2}\) be a ball in \(\Omega \). Assume that for any ball \(B_r\) with the same center as \(B_ {3r_0/2}\) and \(0<r<r_0/2\) , there is \(\tau >0\) such that
holds for an index \(l\in \{1,2,\dots ,6\}\) and for a constant \(k\in {{\mathbb {R}}}\). Then for any \(q\ge 4\) and any \(0<r''<r'\le r\) , the following holds:
Here the range of p in Lemma 5.1 is \(1<p<2\). When \(1<p<2\), we use the extra assumption (5.1) to replace the condition \(2\le p<\infty \) in the proof of [5, Lemma 4.1]. Since the assumption (5.1) implies that
we can prove Lemma 5.1 by following the proof of [5, Lemma 4.1] line by line. We omit the details.
Remark 5.2
By changing \((X_lu-k)^+\) to \((X_lu-k)^-\) and \(A^+_{{k,r}}(X_lu)\) to \(A^-_{k,r}(X_lu)\), we get the same conclusion as (5.2).
By an interation argument (see [9, Lemma 7.3]), Theorem 1.1 follows from the following theorem. For simplicity we write \(\varpi (r)=\mathop {\max }\limits _{1\le {k}\le 6}{{\textrm{osc}}_{B_r}}X_ku\); see Sect. 4 for details.
Theorem 5.3
Let \(1<p<2\) and \(B_{r_0}\) be a ball in \(\Omega \). There is \(s=s(p,L,l_0,r_0)\ge 1\) such that for any \(0<r\le {r_0}/16\) the following holds:
Proof
Let \(B_{r_0}\) be a ball in \(\Omega \). For any ball \(B_r\) with the same center as \(B_{r_0}\) and \(0<r\le {r_0}/16\), we may assume that
otherwise, the inequality (5.3) holds true for \(s=1\). Below we assume that (5.4) holds true. To prove Theorem 5.3, we consider the following two cases.
\(Case\ 1\). There exists a constant \(\theta =\theta (p,L,l_0,r_0)>0\) as in Corollary 4.3 such that either
or
holds true for at least one index \(l\in \{1,\dots ,6\}\). Below we assume that (5.5) holds true. We use the same method to deal with the case (5.6). By (i) in Corollary 4.3, we have
Thus
From this, Lemma 5.1 with \(q=2Q=20\) implies that
holds true for any \(0<r''<r'\le 2r\), any \(i\in \{1,\dots ,6\}\) and any constant k. Hence for each \(i\in \{1,\dots ,6\}\), we have \(X_iu \in DG^+(B_{2r})\); see Definition 3.1 for details. On the other hand, Remark 5.2 implies that \(X_iu \in DG^-(B_{2r})\) holds true for each \(i\in \{1,\dots ,6\}\). According to Remark 3.2, for each \(i\in \{1,\dots ,6\}\) we have \(X_iu \in DG(B_{2r})\). By Theorem 3.6, there exists \(s_0=s_0(p,L,l_0,r_0)>0\) such that for each \(i\in \{1,\dots ,6\}\), we have
Since \(1<p<2\), one has
which, together with (5.7), yields
Choosing \(s_0=s_0(p,L,l_0,r_0)\ge 1\) large enough such that \(2^{s_0}\ge C(p,L,l_0,r_0)\), we conclude (5.3) in this case.
\(Case\ 2\). If Case 1 does not happen, then
and
hold true for every \(i\in \{1,\dots ,6\}\), where the constant \(\theta =\theta (p,L,l_0,r_0)>0\) is as in Corollary 4.3. Consider the set \(\{x\in {B_{8r}}:X_iu>\mu (8r)/4\}\). Since
holds true for all \(k\ge \mu (8r)/4\), by Lemma 5.1 with \(q=2Q=20\), for any \(0<r''<r'\le 8r\), any \(i\in \{1,\dots ,6\}\) and any \(k\ge k_0=\mu (8r)/4\), we have
Since (5.8) implies that
by Lemma 3.4, there exists a constant \(s_1=s_1(p,L,l_0,r_0)>0\) such that
On the other hand, by (5.9) and Remark 3.5, we have
Combining (5.10) and (5.11), we have
Since \(1<p<2\), one has
Thus (5.12) becomes
Note that the conditions (5.8) and (5.9) implies that
which, together with (5.13), yields
Choosing \(s_0=s_0(p,L,l_0,r_0)\ge 1\) large enough such that \(2^{s_0}\ge C(p,L,l_0,r_0)\), we conclude (5.3) in this case.
Finally, we choose the constant \(s=\max \{1,s_0,s_1+2,\log _2C(p,L,l_0,r_0)\}\) to complete the proof of Theorem 5.3. \(\square \)
In the rest of this section, we prove Theorem 1.1 for the case \(\delta =0\).
Proof of Theorem 1.1for the case \(\delta =0\). When \(1<p<2\) and \(\delta =0\), the vector function \(a:=(a_1,a_2,\dots ,a_6)\in C^2({{\mathbb {R}}}^6,{{\mathbb {R}}}^6)\) satisfies conditions (1.2), (1.3) and (1.4) with \(\delta =0\), that is,
for all \(\xi ,\eta \in {{\mathbb {R}}}^6\), where \(0<l_0<L\). For any \(\delta >0\) and all \(\xi \in {{\mathbb {R}}}^6\), we define the new vector function \(a^\delta :=(a^\delta _1,a^\delta _2,\dots ,a^\delta _6)\in C^2({{\mathbb {R}}}^6,{{\mathbb {R}}}^6)\) as
Here by [15, P343], we choose \(\eta _\delta \in C^{0,1}([0,\infty ))\) such that \(a^\delta \) converges to a uniformly on compact subsets of \({{\mathbb {R}}}^6\) as \(\delta \rightarrow 0\) and satisfies the conditions:
for all \(\xi ,\eta \in {{\mathbb {R}}}^6\), where \({{{\tilde{L}}}}={{{\tilde{L}}}}(p,L,l_0)\ge 1\). Let \(\Omega \) be a domain in SU(3). Given an any domain \(\Omega '\Subset \Omega \) and a weak solution \(u\in W^{1,p}_{{{\mathcal {H}}}}(\Omega )\) to (1.1) with the conditions (5.14), (5.15) and (5.16), we let \(u^\delta \in W^{1,p}_{{{\mathcal {H}}}}(\Omega ')\) be the unique weak solution to the following Dirichlet problem
By Theorem 1.1 for the case \(\delta \in (0,1]\), we get the uniform estimate (1.9) for \({\nabla _{{{\mathcal {H}}}}}u^\delta \) (see [19, Section 3.4]). Since the constant C in (1.9) is independent of \(\delta \), letting \(\delta \rightarrow 0\), we conclude (1.9) for the case \(\delta =0\). \(\square \)
6 Proofs of Lemmas 4.4, 4.5, 4.6 and 4.3
In this section, we prove Lemmas 4.4, 4.5, 4.6 and 4.3. Firstly, we prove Lemma 4.4.
Proof of Lemma 4.4
For any function \(\phi \in {C^\infty _0(\Omega )}\) and any \(l\in \{1,\dots ,6,7,8\}\), letting \(X_l\phi \) be a test function in (1.1), we have
Since \(X_iX_l=X_lX_i+[X_i,X_l]\), integration by parts implies that
Since \(X_lX_j=X_jX_l+[X_l,X_j]\) again, we have
By integration by parts again, we conclude (4.7). \(\square \)
Secondly, we prove Lemma 4.5.
Proof of Lemma 4.5
Let \(\eta \in {C^\infty _0(\Omega )}\) be a non-negative cut-off function. For any \(\beta \ge 0\) and any \(l\in \{1,\dots ,6\}\), taking \(\varphi =\eta ^{\beta +2}v^{\beta +2}|\nabla _{{\mathcal {H}}}u|^2X_lu\) as a test function in (4.7), we have
Noting that
by the condition (1.2), we have
From this, summing \(L^l\) with respect to l from 1 to 6, we have
Next, we estimate each item in the right hand side of (6.1) in turn. By the condition (1.3), we have
Since
one gets
Note that (2.1) shows
From this, then by (6.2) and the condition (1.3), we have
By (2.1) again, we have
where
By the condition (1.3) again, we have
Now we estimate \(I^l_{42}\). We only consider the case that \(k=7\) below. In the case that \(k=8\), we use the similar method. Since \(X_7=-[X_1,X_2]\), integration by parts yields
Denote \(\phi :=\eta ^{\beta +2}|\nabla _{{\mathcal {H}}}u|^2X_lu\). Rewriting \(\varphi =v^{\beta +2}\phi \) in (6.4), we have
which yields
We estimate \(J^l\) as below. By the condition (1.3) again, we have
We estimate \(K^l\) as below. By the integration by parts again, we have
For \(K^l_1\), the condition (1.4) yields
For \(K^l_2\), since
we have
Next we use (2.2) and the condition (1.4) to bound the first, third and fifth terms in the right hand side of (6.5), and use integration by parts and the condition (1.3) to bound the second and fourth terms. Thus
Finally, combining all the above estimates together, then by Young’s inequality, we conclude (4.8). \(\square \)
Thirdly, based on Lemma 4.5, we prove Lemma 4.6.
Proof of Lemma 4.6
For simplicity we write the left hand side of (4.12) as
where \(\tau \in (1/2,1)\). Let \(B_r\subset \Omega \) be a ball. Consider the cut-off function \(\eta \in C^\infty _0(\Omega )\) with (4.10) and (4.11). For any \(\beta \ge 0\) and any \(l\in \{7,8\}\), letting \(\varphi =\eta ^{\tau (\beta +2)+4}v^{\tau (\beta +4)}|\nabla _{{\mathcal {H}}}u|^4X_lu\) be a test function in (4.7), we have
By the condition (1.2), we have
Next we estimate each item in the right hand side of (6.6) in turn. For simplicity we write
By the condition (1.3), then by Hölder’s inequality, we have
By the condition (1.3), Hölder’s inequality and Lemma 4.5, we have
where I is denoted as the right hand side of (4.8) in Lemma 4.5, that is,
Combining (6.7), (6.8) and (6.9), we have
Now we bound \({\tilde{K}}\). By Hölder’s inequality, we have
where
Let \(q=2/(1-\tau )\). By (2.8) with \(\beta =(q-2)/2\) in Lemma 2.3 and (4.9), we have
Here we apply Lemma 2.3 to estimate the second inequality in (6.13), and apply (4.9) to estimate the last inequality.
Next we bound I as in (6.10). Fix \(1<\gamma <2\). For the second item of I, by Hölder’s inequality and (4.9), we have
For the second item of I, by (4.9), we have
For the finally item of I, by Hölder’s inequality, (2.7) with \(\beta =\frac{1}{\gamma -1}\) in Lemma 2.3 and (4.9), we have
Here we apply Hölder’s inequality to estimate the first inequality in (6.16) and apply Lemma 2.3 and (4.9) to estimate the finally inequality. Combining (6.14), (6.15) and (6.16), we have
where J is as in (4.13). Combining (6.11), (6.12), (6.13) and (6.17), by Young’s inequality therein, we have
Next, we bound \(K^l_4\). Noting that
by (2.2) and the condition (1.3), we have
Before bounding each item in the right hand side of (6.18), we bound \(K_5\). By (2.2) and the condition (1.3), we have
From this, to bound \(K_5\), we only need to bound \(K_{43}\). Now we bound each item in the right hand side of (6.18). By Hölder’s inequality, we have
Applying the fact \(|{\nabla _{{{\mathcal {T}}}}u}|^2\le 2|{\nabla _{{{\mathcal {H}}}}\nabla _{{{\mathcal {H}}}}u}|^2\), (2.7) with \(\beta =\frac{\tau }{1-\tau }\) in Lemma 2.3 and (4.9) to estimate the final item in the right hand side of (6.19), we have
Combining (6.19) and (6.20), we have
where \(K_\eta \) is as in (2.9). Since \(\eta \in C^\infty _0(B_r)\) satisfies (4.10) and (4.11), one gets
For \(K_{42}\), by Hölder’s inequality, Young’s inequality and (4.9), we have
Applying Hölder’s inequality, (2.7) with \(\beta =\frac{1}{1-\tau }\) in Lemma 2.3 and (4.9) to estimate the final item in the right hand side of (6.21), we have
Combining (6.21) and (6.22), we have
where \(K_\eta \) is as in (2.9). For \(K_{43}\), Hölder’s inequality yields
where I is as in (6.10). Applying Hölder’s inequality to estimate the final item in the right hand side of (6.23), we have
Applying Hölder’s inequality, (2.7) with \(\beta =\frac{3\tau -1}{1-\tau }\) in Lemma 2.3 and (4.9) to estimate the first item in the right hand side of (6.24), we have
Applying (4.9) to estimate the final item in the right hand side of (6.24), we have
Combining (6.23), (6.24), (6.25) and (6.26), we have
where \(K_\eta \) is as in (2.9). For \(K_{44}\), by Young’s inequality, Hölder’s inequality and (4.9), we have
Finally, combining all the above estimates together, we conclude (4.12). \(\square \)
Finally, we use Lemma 4.1 and Remark 4.2 to prove Lemma 4.3.
Proof of Lemma 4.3
We consider two cases: (i) that (4.3) holds true for an index \(k\in \{1,\dots ,6\}\) and (ii) that (4.5) holds true for an index \(k\in \{1,\dots ,6\}\). In the case (i), we use Lemma 4.1 with v to conclude (4.4). Similarly, in the case (ii), we use Remark 4.2 with \(v'\) to conclude (4.6). Below we assume that (4.3) holds true for an index \(k\in \{1,\dots ,6\}\).
Let \(B_R\) be a ball in \(\Omega \). For any ball \(B_r\subset \Omega \) with the same center \(B_R\) and \(0<r\le {R}\), we consider the cut-off function \(\eta \in {C^\infty _0(B_r)}\) with (4.10) and (4.11). Recall that
For any \(\beta \ge 0\), we write \(\psi =\eta ^{\beta /2+2}v^{\beta /2+2}\). Note that
For \(\gamma >1\), by Hölder’s inequality and Lemma 4.1, we have
Here we use Hölder’s inequality and Lemma 4.1 to get the second inequality in (6.27). By Lemma 2.8 with \(p_1=2\), we have
where \(Q=10\) is the homogeneous dimension of SU(3). Here for any ball \(B_r\) we use [24, Theorem V.4.1 on P69] to control its volume, that is, \(C_1 r^Q \le |B_r| \le C_2 r^Q\). Combining (6.27) and (6.28), we have
Now we choose \(\gamma =\frac{Q-1}{Q-2}=\frac{9}{8}\) such that \(1<\gamma <\frac{Q}{Q-2}=\frac{5}{4}\). For simplicity we write
Since \(\gamma \beta _{i+1}=\frac{Q}{Q-2}(\beta _i+4)\), by (6.29) with \(\beta =\beta _i\), we have
Denote
Thus (6.30) becomes
where
For any natural number \(N\ge 1\), we iterate (6.31) with respect to i from 0 to \(N-1\). Thus
Letting \(N \rightarrow \infty \), one gets
Recall that the cut-off function \(\eta \in C^\infty _0(B_r)\) satisfies (4.10) and (4.11). Then the assumption (4.3) implies that
We choose \(\theta =\theta (p,L,l_0,R)>0\) small enough such that
Then (6.32) becomes
which yields that \(X_ku\ge 3\mu (r)/16\) in \(B_{r/2}\). We conclude (4.4). \(\square \)
7 A class of compact connected semi-simple Lie group
In this section, we consider a class of compact connected semi-simple Lie group \({{\mathbb {L}}}{{\mathbb {G}}}\), which was first proposed by Domokos-Manfredi [5]. The semi-simple Lie group \({{\mathbb {L}}}{{\mathbb {G}}}\) is connected and compact. We notate \({{\mathcal {L}}}{{\mathcal {G}}}\) as its Lie algebra. The inner product on \({{\mathcal {L}}}{{\mathcal {G}}}\) satisfies properties
and
Let \({{\mathbb {L}}}{{\mathbb {S}}}\) be the maximal torus of \({{\mathbb {L}}}{{\mathbb {G}}}\). We notate \({{\mathcal {L}}}{{\mathcal {S}}}\) as its Lie algebra. Owing to that \({{\mathcal {L}}}{{\mathcal {S}}}\) is a maximal commutative subalgebra of \({{\mathcal {L}}}{{\mathcal {G}}}\), we call it as the Cartan subalgebra. Denote by \({\mathcal {R}}\) the set of all roots, where we say that \(R\in {{\mathcal {L}}}{{\mathcal {S}}}\) is a root if \(R\ne 0\) with the root space \({{\mathcal {L}}}{{\mathcal {G}}}_R \ne \{0\}\). Here \({{\mathcal {L}}}{{\mathcal {G}}}_R=\{Z\in {{\mathcal {L}}}{{\mathcal {G}}}_{{\mathbb {C}}}:[S,Z]=i\langle R,S \rangle Z,\quad \forall S\in {{\mathcal {L}}}{{\mathcal {S}}}\}\).
According to [5, Section 5], we can define the orthogonal complement of \({{\mathcal {L}}}{{\mathcal {S}}}\) denoted by \({{\mathcal {H}}}\), and we can choose its orthonormal basis satisfying Property 7.1. We notate \({\mathcal {B}}_{{\mathcal {H}}}=\{X_1,X_2,\dots ,X_{2n}\}\) as the orthonormal basis of \({{\mathcal {H}}}\).
Proposition 7.1
-
(i)
\(\forall 1\le k\le n,\ \exists R_k\in {\mathcal {R}}^+ \ s.t.\ {\textrm{span}}\{X_{2k-1},X_{2k}\}={{\mathcal {H}}}_{R_k}\).
-
(ii)
\([X_{2k-1},X_{2k}]=-R_k\), \([X_{2k},R_k]=-\Vert R_k\Vert ^2X_{2k-1}\), \([R_k,X_{2k-1}]=\Vert R_k\Vert ^2X_{2k}\).
-
(iii)
\([X_l,X_m]\in {{\mathcal {H}}}\) when \((l,m)\ne (2k-1,2k)\).
-
(iv)
\(\{[X_{2k-1},S],[X_{2k},S]\}\subset {{\mathcal {H}}}_{R_k}\) when \(S\in {{\mathcal {L}}}{{\mathcal {S}}}\).
Based on properties of \({\mathcal {B}}_{{\mathcal {H}}}\), a basis of \({{\mathcal {L}}}{{\mathcal {S}}}\) can be selected, that is, \(\{R_1,R_2,\dots ,R_\upsilon \}\). For any function v, we denote by
the horizontal and vertical gradients. Moreover, from Property 7.1, we draw the conclusion
Here \(\lambda _{i,j}^{(k)}\), \(\theta _{i,j}^{(l)}\) and \(\vartheta _{i,j}^{(k)}\) are constants.
Given a domain \(\Omega \subset {\mathrm{{{\mathbb {L}}}}{{\mathbb {G}}} }\), we consider the equation
Here \({\nabla _{{{\mathcal {H}}}}u}=(X_1u,\dots ,X_{2n}u)\) is the horizontal gradient of u, and the vector function \(a=(a_1,\dots ,a_{2n})\) satisfies the conditions:
for all \(\xi ,\eta \in {{\mathbb {R}}}^{2n}\), where \(0\le \delta \le 1\), \(1<p<\infty \) and \(0<l_0<L\). Based on the conclusion (7.1) and above conditions, our method can be expended to the semi-simple Lie group \({{\mathbb {L}}}{{\mathbb {G}}}\). We list our main results for \({{\mathbb {L}}}{{\mathbb {G}}}\) and omit the proof.
Theorem 7.2
Suppose that the conditions (7.3), (7.4) and (7.5) hold for some \(l_0\) , L and such \(1<p<2\) , \(\delta \ge 0\). If \(u\in W^{1,p}_{{{\mathcal {H}}},{\textrm{loc}}}(\Omega )\) is a weak solution to (7.2), then \({\nabla _{{{\mathcal {H}}}}u}\in C^\alpha _{{\textrm{loc}}}(\Omega )\) for some \(\alpha \in (0,1)\) depending on \(l_0\) , L and such p , \(\delta \). Moreover, for all \(B_{r_0}\subset \Omega \) and any \(0<r\le r_0\) , we have
where \(0<\alpha <1\) depends on p , \(l_0\) and L , and the constant \(C>0\) depends on p , \(l_0\) , L and \(r_0\).
Consequently, when \(1<p<2\) , the horizontal gradients of p-harmonic functions on \({{\mathbb {L}}}{{\mathbb {G}}}\) have the Hölder regularity and satisfy (7.6).
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Acknowledgements
The author would like to express his gratitude to Yuan Zhou and Fa Peng for their fruitful discussions. This work is supported by the National Natural Science Foundation of China (No. 12025102, No. 11871088).
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Appendix
Appendix
The following lemma is [13, Lemma 4.7], which will be used to prove Lemma 3.3.
Lemma 8.3
For a non-negative sequence \(\{y_l\}_{l=0,1,2,\dots }\) , \( y_{l+1}\le c_0 b^l_0 y^{1+\varepsilon }_l \) implies that
where \(c_0>0\) , \(\varepsilon >0\) and \(b_0>1\).
Moreover, if \( y_0\le \theta =c^{-\frac{1}{\varepsilon }}_0b^{-\frac{1}{\varepsilon ^2}}_0, \) then \( y_l\le \theta b^{-\frac{l}{\varepsilon }}_0. \) Consequently, \(y_l\rightarrow 0\) as \(l\rightarrow \infty \).
Proof of Lemma 3.3
For any \(b\in (0,1)\), letting \(\rho _h={\frac{\rho }{2}}+{\frac{\rho }{2^{h+1}}},\ \rho =\rho _h,\ \rho '=\rho _{h+1}\) and \(k_h=k+bH-b^{h+1}H\) in (3.1), we have
where \(h=0,1,2,\dots \) and \(q>Q=10\).
The following inequality comes from [12, Lemma 2.3]
Letting \(l=k_{h+1}\), \(k=k_{h}\) and \(\rho =\rho _{h+1}\) in (8.8), we have
The condition that \(|{A^+_{k,\rho }}|\le \theta _1|B_\rho |\) shows that
From this, we choose \(\theta _1\) small enough such that \({C}2^Q\theta _1\le 1/2\). Then (8.9) becomes
Applying Hölder’s inequality to (8.10), then combining (8.7), from the assumption that \(H=\mathop {\sup }\limits _{B_\rho }u(x)-k\ge \chi \rho ^{1-Q/q}\), we have
Here for any ball \(B_\rho \) we use [24, Theorem V.4.1 on P69] to control its volume, that is, \(C_1 \rho ^Q \le |B_\rho | \le C_2 \rho ^Q\). Thus
Denote
From (8.11), by Lemma 8.3 with \(y_l=\mu _h\), there exists \(\theta _1=\theta _1(\gamma ,q,b)\in (0,1)\) such that \(\mathop {\lim }\limits _{h\rightarrow \infty } \mu _h=0\), that is, \( |A^+_{k+bH,\rho /2}|=0. \) \(\square \)
Proof of Lemma 3.4
The following inequality comes from [12, Lemma 2.3]
Denote
Letting \(l=\mu _1-\frac{w_1}{2^{t+1}}\), \(k=\mu _1-\frac{w_1}{2^{t}}\) and \(\rho \rightarrow \rho /2\) in (8.12), we have
The assumption that \(|A^-_{k,\rho /2}|\ge \tau |B_{\rho /2}|\) implies that
which, together with Hölder’s inequality to (8.13), yields
Letting \(k=\mu _1-\frac{w_1}{2^t}\), \(\rho '\rightarrow \rho /2\) and \(\rho \rightarrow \rho \) in (3.1), we have
Below we may assume that
otherwise we have
that is,
Thus (3.2) holds. Combining (8.16), (8.15) and (8.14), we have
Summing (8.17) with respect to t from 1 to \(s-3\), we have
Here for any ball \(B_\rho \) we use [24, Theorem V.4.1 on P69] to control its volume, that is, \(C_1 \rho ^Q \le |B_\rho | \le C_2 \rho ^Q\). Thus
Denote
By (8.16) and Lemma 3.3 with \(b=1/2\) and \(k=\mu _1-\frac{w_1}{2^{s-2}}\), there exists \(s=s(\gamma ,q,\tau )>0\) such that
Thus (3.2) holds true. \(\square \)
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Yu, C. \(C^{1,\alpha }\)-regularity for p-harmonic functions on SU(3) and semi-simple Lie groups. Abh. Math. Semin. Univ. Hambg. 94, 57–94 (2024). https://doi.org/10.1007/s12188-024-00274-4
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DOI: https://doi.org/10.1007/s12188-024-00274-4
Keywords
- p-Laplacian equation
- \(C^{1,\alpha }\)-regularity
- SU(3)
- Caccioppoli inequality
- De Giorgi
- p-harmonic function
- Semi-simple Lie group