1. Introduction
The study of the regularity for partial differential equations involving the
p-Laplacian operator has always been a hot topic. In the Euclidean space, the
,
,
-regularities and other second-order Sobolev regularities for the
p-Laplacian equation have been proved in [
1,
2,
3,
4,
5,
6,
7]. In recent years, there has been significant progress in the study of the regularity for the
p-Laplacian equation in sub-Riemannian manifolds. Many scholars have made outstanding contributions. In the Heisenberg group,
, Domokos-Manfredi [
8,
9], Manfredi-Mingione [
10], Migione et al. [
11], Ricciotti [
12], and Zhong-Mukherjee [
13,
14] established the
and
-regularities for the
p-Laplacian equation in the full range
; Domokos [
15] and Lie et al. [
16] proved the
-regularity for the
p-Laplacian equation in the range of
with
. In the group SU(3), the
,
, and
-regularities of the
p-Laplacian equation were established by [
17,
18]. The method in [
13,
14] is extended by Citti-Mukherjee [
19] to include Hörmander vector fields of step two, and the
and
-regularities for the
p-Laplacian equation have been successfully established. The
-regularity for inhomogeneous quasi-linear equations on the Heisenberg group
were established by [
20,
21] when
. New ideas and perspectives behind the development of research on regularity include certain hybrid-type Caccioppoli-type inequalities, as first proposed and introduced by Zhong [
13]. In comparison, for the degenerate parabolic
p-Laplacian equation, such inequalities are not applicable due to the differences in homogeneity between the time and spatial derivatives. Therefore, we need to find and create new methods and techniques to establish more suitable Caccioppoli-type inequalities.
In this study paper, we propose a new method to construct a crucial Caccioppoli-type inequality. Based on the inequality, when
, we establish the
-regularity for the parabolic
p-Laplacian equation on the group SU(3). To be specific, we focus on a special type of unitary group composed of
complex matrices. We denote by SU(3) this unitary group and endow it with horizontal vector fields
. More exhaustive geometries and properties of SU(3) are shown in
Section 2. We select an open domain
in the group SU(3). For
, we define a cylinder
, as first proposed in [
22]. In
, we consider the following equation:
Here,
is the formal adjoint of
;
is the horizontal gradient; the vector function
meets the following condition:
for every
, where
,
and
. If, for every function
, the equation
holds true, then we name the function
as a weak solution to Equation (
1). Here,
is the first-order
p-th integrable horizontal local Sobolev space, which is composed of total functions
, whose distributional horizontal gradients are
. In the classic case,
, Equation (
1) becomes the parabolic
p-Laplacian equation:
The study of the parabolic
p-Laplacian equation originated from DiBenedetto-Friedman [
22]. They established the
-regularity of the weak solution in the Euclidean space; Wiegner [
23] also proved the same result. For more exhaustive results on the parabolic
p-Laplacian equation and more general cases in the Euclidean space, we refer to the book by DiBenedetto [
24]. For the study of the parabolic
p-Laplacian equation in the sub-Riemannian manifold, Capogna et al. [
25] established, when
, the
-regularity of the weak solution to the non-degenerate parabolic
p-Laplacian equation in the Heisenberg group
, as follows:
Recently, for the degenerate parabolic
p-Laplacian equation in the Heisenberg group,
, when
, Capogna et al. [
26] established the
-regularity of the weak solution.
In this study paper, we focus on the
-regularity of the weak solution
u to (
3) on SU(3). As a consequence, when
, we establish the
-regularity of
u; that is,
. See Theorem 1 below for details.
Theorem 1. Suppose is a weak solution to (1), satisfying condition (2), in . Then, for . Moreover, when , for every , we have the following:where , and is the homogeneous dimension of SU(3).
Consequently, when , the weak solution to the parabolic p-Laplacian equation on SU(3)
has the -regularity and satisfies (4). To prove Theorem 1, it requires us to contemplate the following regularized equation:
where
u is a weak solution to (
1), and
is the parabolic boundary of the cylinder
, with the following condition:
for every
, where
,
,
and
. Here, from [
17], since
are the left-invariant vector fields, we have
. Simultaneously, we also need to consider the Riemannian approximation equation (see
Section 2 for details):
where
is a weak solution to (
5), with the following condition:
for every
, where
,
and
. Above,
depend only on
. Let
be a weak solution to (
7). When
, we write
and
; see ([
26], Section 2) for details. The Riemannian approximation technique has become a mature technique widely used in studying equations; see [
17,
19,
25,
26] for the definition and more details of the technique. It is proven in [
25,
26] that
and
as
, and that
and
as
; also see [
13,
14,
17,
19] for an example.
Hence, to obtain Theorem 1, we only need to prove that
have the following
-regularity uniformly in
. Finally, letting
, from the following theorem, we can apply the standard method as [
25,
26] to derive Theorem 1.
Theorem 2. Assume that is a weak solution to (7) with condition (8), in . If , then . Moreover, when , for any , we have the following:where and . The proof of Theorem 2 relies on Moser’s iteration; see
Section 4 for details. The key point, by the approach in [
25,
26], is to establish a crucial Caccioppoli-type estimate for
involving
(see Lemma 6). To obtain the crucial Caccioppoli-type estimate, when
, we establish two Caccioppoli-type inequalities for
and
in Lemmas 4 and 5, proven in
Section 3. Applying Lemma 5 to re-estimate the integral terms on the right hand of (
26) in Lemma 4, we prove the crucial Caccioppoli-type estimate in
Section 3.
Consequently, we construct a crucial Caccioppoli-type inequality (
38). Based on the inequality we establish, when
, the
-regularity for the parabolic
p-Laplacian equation on the group SU(3). Compared to the Heisenberg group
, our new result achieves the same range of
p as [
26]. Unfortunately, the
-regularity for the range
cannot be achieved with our current technology because our argument rests in a crucial way on Lemma 5 with the condition
. The difficulties in the proof arise from handling and estimating integral terms involving
. In the Heisenberg group
, there exists the property that
; however, it does not hold true on SU(3). For example,
(see
Table 1). This means that we need to handle more integral terms when estimating integral terms involving
. Our approach can also be applied to more general sub-Riemannian manifolds. For instance, it can be used with a special class of semi-simple Lie groups as proposed in [
17], and Hörmander vector fields of step two as discussed in [
19], to establish the regularity for the parabolic
p-Laplacian equation. Technically speaking, our method can also be extended to other types of partial differential equations, for example, the non-homogeneous equation
. The establishment of the regularity for the range of
will be the focus and difficulty of our next work.
2. Preliminaries
The group SU(3) is a special type of unitary group composed of
complex matrices; that is,
where
E is the identity matrix. The Lie algebra of SU(3) is defined by the following:
granted with the inner product
The two-dimensional maximal torus on the group SU(3) is provided by the following:
whose Lie algebra is as follows:
is selected as the Cartan subalgebra. The following Gell–Mann matrices form a set of the orthogonal basis of su(3), namely the following:
The following two vector fields are generated from
and
, respectively; that is,
Since
and
, the vertical vector fields
form a set of orthogonal basis of
. Hence, the vertical gradient is defined by
.
We recall the Riemannian approximation technique. Given
, we define the Riemannian approximation to the vector fields
, as
From which, we denote
as the gradient,
The following table ([
17], Table 2.1), shows the total Lie bracket for any two vector fields belonging to
.
Table 1 shows that
and that
where
are constants determined entirely by
Table 1. From
Table 1, it is not difficult for us to discover that the horizontal subspace
in SU(3) is generated by the set of orthogonal bases
satisfying the Hörmander condition. Hence, the horizontal gradient is defined by
. Here, the basis
is left-invariant due to the left-invariance of the Gell–Mann matrices. To summarize, the basis
generates the horizontal distribution of a sub-Riemannian manifold.
6. Conclusions
In this article, we construct a crucial Caccioppoli-type inequality (
38). Based on the inequality, when
, we built up the
-regularity of weak solutions to the degenerate parabolic
p-Laplacian equation on the group SU(3) granted with the horizontal vector fields
. Compared to the Heisenberg group
, our new result achieves the same range of
p as [
26]. Unfortunately, the
-regularity for the range
cannot be achieved with our current technology because our argument rests in a crucial way on Lemma 5 with the condition
. Our approach can also be used for more general sub-Riemannian manifolds, for instance, a special class of the semi-simple Lie group proposed in [
17] and Hörmander vector fields of step two in [
19], to establish regularity for the parabolic
p-Laplacian equation. Technically speaking, our method can also be extended to other types of partial differential equations, for example, the non-homogeneous equation
. The establishment of the regularity for the range
will be the focus and difficulty of our next work.
In conclusion, the results shown in this article are original. We believe that our results will be widely applied in the study of regularities for equations involving the p-Laplacian operator and other areas of applied science.