Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group

We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in \cite{Zhong} of the H\"older regularity of $p-$harmonic functions in the Heisenberg group $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$


Introduction
In this paper we establish the C ∞ smoothness of solutions of a certain class of quasilinear parabolic equations in the Heisenberg group H n . In a cylinder Q = Ω × (0, T ), where Ω ⊂ H n is an open set and T > 0, we consider the equation modeled on the regularized parabolic p-Laplacian where p ≥ 2. The term regularized here refers to the fact that the non-linearity (1 + |∇ 0 u| 2 ) p− 2 2 affects the ellipticity of the right hand side only when the gradient blows up, and not when it vanishes, thus presenting a weaker version of the singularity in the p−Laplacian. Here, we indicate with x = (x 1 , ..., x 2n , x 2n+1 ) the variable point in H n . We alert the reader that, although it is customary to denote the variable x 2n+1 in the center of the group with the letter t, we will be using z instead, since we have reserved the letter t for the time variable. Consequently, we will indicate with ∂ i partial differentiation with respect to the variable x i , i = 1, ..., 2n, and use the notation Z = ∂ z for the partial derivative ∂ x 2n+1 . The notation ∇ 0 u ∼ = (X 1 u, ..., X 2n u) represents the so-called horizontal gradient of the function u, where As it is well-known, the 2n+1 vector fields X 1 , ..., X 2n , Z are connected by the commutation relation [X i , X j ] = δ ij Z, all other commutators being trivial. We now introduce the relevant structural assumptions on the vector-valued function (x, ξ) → A(x, ξ) = (A 1 (x, ξ), ..., A 2n (x, ξ)): there exist p ≥ 2, δ > 0 and 0 < λ ≤ Λ < ∞ such that for a.e.
It is worth mentioning here that the prototype for the class of equations (1.1) and for their parabolic approximation comes from considering the regularized p−Laplacian operator where M is the underlying differentiable manifold, ω is the contact form and g 0 is a Riemannian metric on the contact distribution. The measure µ 0 is the corresponding Popp measure. The approximants are constructed through Darboux coordinates, considering the p−Laplacians associated to a family of Riemannian metrics g ε that tame g 0 and such that the metric structure of the spaces (M, g ε ) converge in the Gromov-Hausdorff sense to the metric structure of (M, ω, g 0 ). For a more detailed description, see [8, Section 6.1]. As an immediate corollary of Theorem 1.1 one has the following. Theorem 1.3. Let (M, ω, g 0 ) be a contact, sub-Riemannian manifold and let Ω ⊂ M be an open set. For p ≥ 2, consider u ∈ L p ((0, T ), W 1,p 0 (Ω)) be a weak solution of For any open ball B ⊂⊂ Ω and T > t 2 ≥ t 1 ≥ 0, there exist constants C = C(n, p, d(B, ∂Ω), T − t 2 , δ) > 0 and α = α(n, p, d(B, ∂Ω), T − t 2 , δ) ∈ (0, 1) such that The C 1,α estimates in (1.10) in Theorem 1.2 allow us to apply the Schauder theory developed in [5,35], and finally deduce the following result.
The present paper is the first study of higher regularity of weak solutions for the non stationary p-Laplacian type in the sub-Riemannian setting, and it is based on the techniques introduced by Zhong in [36]. The stationary case has been developed so far essentially only in the Heisenberg group case thanks to the work of Domokos, [15], Manfredi, Mingione [26], Mingione, Zatorska-Goldstein and Zhong [27], Ricciotti [32], [31] and Zhong [36]. Regularity in more general contact sub-Riemannian manifolds, including the rototraslation group, has been recently established by the two of the authors and coauthors [8] and independently by Mukherjee [30] based on an extension of the techniques in [36]. Domokos and Manfredi [16] are rapidly making substantial progress in higher steps groups and in some special non-group structures, using the Riemannian approximation approach as in the work [8].
The plan of the paper is as follows. In Section 2 we collect some preparatory material that will be used in the main body of the paper. Section 3 is devoted to proving the first part of Theorem 1.2, which establishes the Lipschitz regularity of the approximating solutions u ε . In Section 4 we prove the Hölder regularity of derivatives of u ε in Theorem 1.2. Finally, in Section 5 we use the comparison principle and Theorem 1.2 to establish Theorem 1.1.
Some final comments are in order. The non-degeneracy hypothesis δ > 0 in (1.3) (see also (1.7)) is not needed in the Euclidean setting and, in the stationary regime, it is not needed in the Heisenberg group either. We suspect the C 1,α regularity of weak solutions for (1.1) still holds without this hypothesis, but at the moment we are unable to prove it. In this note we use δ > 0 notably in (3.17) and in Theorem 3.13.
In order to extend the parabolic regularity theory to the sub-Riemannian setting one has to find a way to implement, in this non-Euclidean framework, some of the techniques introduced by Di Benedetto [14] which rely on non-isotropic cylinders in space-time. The key idea is to work with cylinders whose dimensions are suitably rescaled to reflect the degeneracy exhibited by the partial differential equation. To give an example, if one sets x ∈ Ω, R, µ > 0, one can define the intrinsic In contrast with the usual parabolic cylinders of the linear theory, the shape of the Q R (µ) cylinders is stretched in the time dimension by a factor of the order |∇ 0 u| 2−p .
The use of such non-isotropic cylinders seems necessary in order to make-up for the different homogeneity of the time derivative and the space derivatives in the degenerate regime δ = 0. In a future study we plan to return to the problem of extending Di Benedetto's Caccioppoli inequalities on non-isotropic cylinders to the Heisenberg group and beyond.
Acknowledgements. We thank Vira A. Markasheva, who collaborated with us on an earlier version of this project.

Preliminaries
In this section we collect a few definitions and preliminary results that will be used throughout the rest of the paper. As indicated in the introduction, for each ε ∈ (0, 1) we define g ε to be the Riemannian metric in H n such that X 1 , ..., X 2n , εZ is an orthonormal frame, and denote such frame as X ε 1 , ..., X ε 2n+1 . The corresponding gradient operator will be denoted by ∇ ε .
⊂ Ω denotes the g ε -Riemannian ball of center x 0 and t 0 ∈ (0, T ). We call parabolic boundary of the cylinder First of all we recall the Hölder regularity, and local boundedness of weak solutions of (1.1) and (1.8) from [2].
When ε > 0 and δ > 0, classical regularity results (e.g., [25]) yield that weak solutions have bounded gradient, and hence (1.8) is strongly parabolic, thus leading to weak solutions being smooth. Clearly such smoothness may degenerate as ε → 0, and the main point of this paper is to show that this does not happen.
Let Ω ⊂ R n be a bounded open set and let Q = Ω × (0, T ). For a function u : Q → R, and 1 ≤ p, q we define the Lebesgue spaces L p,q (Q) = L q ([0, T ], L p (Ω)), endowed with the norms When p = q, we will refer to L p,p (Q) as L p (Q). One has the following useful reformulation of the Sobolev embedding theorem [18] in terms of L p,q spaces. In the next statement, we will indicate with N = 2n+2 the homogenous dimension of H n with respect to the non-isotropic group dilations, and we will denote by N 1 = N + 2 = 2n + 4 the corresponding parabolic dimension with respect to the dilations (x, t) → (δ λ x, λ 2 t).
Let v be Lipschitz function in Q, and assume that for all 0 < t < T , v(·, t) has compact support in Ω × {t}.
(i) There exists C = C(n) > 0 such that for any ε ∈ [0, 1] one has , and there exists C > 0, depending on n, such that for any ε ∈ [0, 1] one has We note that as ε decreases to zero, the background geometry shifts from Riemannian to sub-Riemannian. The stability with respect to ε of the constant C in the Lemma 2.3 is not trivial, see [10,7].
In the sequel we will use an interpolation inequality that will take the place of the Sobolev inequality in a Moser type iteration, just as, for example, in [11,Proposition 4.2]. Although the result does not use the equation at all, we state it in terms that will make it immediately applicable later on. Henceforth, to simplify the notation, we will routinely omit the indication of dx, dxdt, etc. in all integrals involved, unless there is risk of confusion.
Lemma 2.4. Let u ε be a weak solution of (1.8) in Q. If β ≥ 0, and η ∈ C 1 ([0, T ], C ∞ 0 (Ω)) vanishes on the parabolic boundary of Q, then there is a constant C > 0, depending only on ||u ε || L ∞ (Q) , such that To conclude the argument, it suffices to apply Young's inequality.
In this section we establish Lipschitz regularity for the derivatives of the solutions u ε . The main results of this section are summarized in the following estimates, which are unform in ε > 0.
The proof of Theorem 3.1 will follow from combining the results in Theorem 3.11, Lemma 3.12, Proposition 3.13 and Proposition 4.1, that are all established later in the section. The Caccioppoli inequalities needed to prove Theorem 3.1 will take up most of the section, and they all apply to a solution u ε of the approximating equation (1.8) in a cylinder Q = Ω × (0, T ). We begin with two lemmas in which we explicitly detail the pde's satisfied by the smooth approximants X ε ℓ u ε and Zu ε .
.., 2n, and s ℓ = (−1) [ℓ/n] , then the function v ε ℓ solves the equation Proof. Differentiating (1.8) with respect to X ε ℓ , when ℓ ≤ n, we find Taking the derivative with respect to X ε ℓ when ℓ ≥ n + 1, we obtain Lemma 3.3. Let u ε be a solution of (1.8) in Q. Then, the function Zu ε is a solution of the equation Proof. It follows by a straightforward differentiation of (1.8) with respect to Z, and we omit the details.
Proof. We use φ = η 2 |Zu ε | β Zu ε as a test function in the equation satisfied by Zu ε , see Lemma 3.3, to obtainˆt The left-hand side of the latter equation can be expressed as follows: Considering the first term in the right-hand side, we obtain As for the second term in the right-hand side, we havê Combining the latter three equations, we find The structure conditions (1.3) yield thus concluding the proof.
. If in the first term in the right-hand side of (3.2) we use the fact that Fix η ∈ C ∞ 0 (Ω) and let φ = η 2 (δ + |∇ ε u ε | 2 ) β/2 X ε ℓ u ε . Taking such φ as the test-function in the weak form of (3.10), and integrating by parts the terms in divergence form, one has 1 2ˆt The latter equation implies that for every ℓ = 1, ..., 2n one has Summing over ℓ = 1, ..., 2n, by a simple application of the chain rule, and using the structural assumption (1.7), we see that the left-hand side can be bounded from below by Since the last term in the right-hand side of this estimate is nonnegative, we obtain from this bound Next, we estimate each of the terms in the right-hand side separately. Recalling that from (1.7) one has |A iξ j (x, η)| = |∂ ξ j A ε i (x, η)| ≤ C(δ + |η| 2 ) p−2 2 , one has that for any α > 0 there exists C α > 0 depending only on α, p, n and the structure constants, such that 2n ℓ=1 In a similar fashion, we obtain Finally, integrating by parts twice, and using the structural assumptions, one has 2n ℓ=1 Combining (3.6)-(3.9) with (3.5), we reach the desired conclusion (3.3).
In the case β = 0 we obtain the following stronger estimate, which we will need in the sequel. We denote by || · || the L ∞ norm of a function on the parabolic cylinder Q.
Lemma 3.6. Let u ε be a weak solution of (1.8) in Q, let t 2 ≥ t 1 ≥ 0, and η ∈ C 1 ([0, T ], C ∞ 0 (Ω)) be such that 0 ≤ η ≤ 1, and for which ||∂ t η|| ≤ C||∇ ε η|| 2 , where C > 0 is a universal constant. For every α > 0 there exists C α > 0 such that With η as in the statement of the lemma, we take φ = η 2 X ε ℓ u ε as a test function in the weak form of (3.10). Integrating by parts the terms in divergence form, one has 1 2ˆt The gives Summing over ℓ = 1, ..., 2n, in view of the structural hypothesis (1.7), after an integration by parts in the first term in the left-hand side we obtain the following bound Next, we estimate each of the terms in the right-hand side separately. Recalling that |A ε iξ j (x, η)| ≤ C(δ + |η| 2 ) p−2 2 , we find that for any α 1 , α 2 > 0 there exist C α 1 , C α 2 > 0, depending only on α 1 , α 2 , p, n and the structure constants, such that 2n ℓ=1 Now, we apply Lemma 3.4 to find, for any α > 0, Analogously, Using the structure conditions, one has thus concluding the proof.
Next, we need to establish mixed type Caccioppoli inequalities, where the left-hand side includes terms with both horizontal derivatives and derivatives along the second layer of the stratified Lie algebra of H n . Lemma 3.7. Set T > t 2 > t 1 > 0. Let u ε be a weak solution of (1.8) in Q = Ω × (0, T ). Let β ≥ 2 and let η ∈ C 1 ((0, T ), C ∞ 0 (Ω)), with 0 ≤ η ≤ 1. For all α ≤ 1 there exist constants C Λ , Proof. Let η ∈ C ∞ 0 (Ω × (0, T )) be a nonnegative cutoff function. Fix β ≥ 2 and ℓ ∈ {1, ..., 2n}. Note that , which suggests to use 2X ε ℓ u ε |Zu ε | β as a test function in the equation (3.2) satisfied by X ε ℓ u ε and to choose β|X ε ℓ u ε | 2 |Zu ε | β−2 Zu ε as a test function in the equation (3.3) satisfied by Zu ε . Equation (3.2) becomes in weak form Consequently, if we substitute the test function φ = 2η β+2 |Zu| β X ε ℓ u, we obtain We will show that these terms satisfy the following estimate We first note that The last term can be estimated, as follows, using Lemma 3.4: From here estimate (3.13) holds. Integrating by parts we have From here, using inequality (3.14), we deduce that I 2 ℓ satisfies inequality (3.13). The estimate of I 3 ℓ can be made as follows: From here and (3.14) the inequality (3.13) follows. The estimate of I 4 ℓ is analogous: We now recall the following pde from Lemma 3.3 Substituting in this equation the test function φ = β|Zu| β−2 Zuη β+2 |∇ ε u ε | 2 , one obtains We observe that the ellipticity condition yields Let us now consider I 5 : The estimate of I 6 is identical to that I 1 ℓ and we thus omit it. Let us consider I 7 . One has Similar consideration holds for I 8 Finally, we estimate I 9 .
It follows that Summing up equations (3.12) and (3.15), we obtain Applying (3.16), the proof is completed.
Corollary 3.10. In the hypotheses of the previous corollary we have where c = c(n, p, L) > 0.
Proof. In order to handle the first term in the right-hand side of the sought for conclusion, it suffices to observe that The conclusion then follows from Hölder's inequality. We can handle the second term in the same way The key step in the proof of the Lipschitz regularity of solutions is the following Caccioppoli type inequality which is a parabolic analogue of [36, Theorem 3.1].
Proof. In view of Lemma 3.6, the conclusion will follow once we provide an appropriate estimate of the termˆt The first step is to apply Hölder's inequality to obtain (the first integral in the right-hand side can be bounded by applying Corollary 3.10, resulting in the estimate) (by Young' s inequality, recalling C 0 from the statement of Lemma 3.6) Now we note that β β + 2 4C 0 (β + 1) 4 (β + 2) In the next result we establish Lipschitz bounds that are uniform in ε. The argument consists in implementing Moser iterations, and rests on the observation that the quantity δ + |∇ ε u ε | 2 is bounded from below by δ > 0, and that for every β ≥ 0 it is bounded in L p+β in a parabolic cylinder, uniformly in ε.
In the iteration itself, we will consider metric balls B ε defined through the Carnot-Caratheodory metric associated to the Riemannian structure g ε defined by the orthonormal frame X ε 1 , ..., X ε 2n+1 . We recall here that g ε converges to the sub-Riemannian structure of the Heisenberg group in the Gromov-Hausdorff sense [24], and in particular B ε → B 0 in terms of Hausdorff distance. These considerations should make it clear that the estimates in the following theorem are stable as ε → 0.
Remark 4.2. Actually, a stronger result holds. Let α ∈ (0, 1) denote the Hölder exponent of Zu ε (which is uniform in ε > 0). By observing that w ε − w ε (x 0 , t 0 ) is also a solution of (4.1), then a standard Caccioppoli type argument yields This shows, in particular, that |∇ ε Zu| belongs to the parabolic Morrey class M 2,α/2 (Q), where for λ ∈ (0, 1) and q ≥ 1 we have indicated with M q,λ (Q) the space of all functions f ∈ L q (Q) such that for all B ⊂ Ω, and 0 < t 0 < T , one has We also recall that the parabolic Campanato spaces L q,λ (Q) is the collection of all f ∈ L q (Q) such that for all B = B(x 0 , r) ⊂ Ω, and 0 < t 0 < T , one has A standard argument, see for instance [13], shows that the Campanato space is isomorphic to the space of Hölder continuous functions. In particular, we rely on the following instance of this general result.
Next, we return to the study of the regularity of horizontal derivatives of solutions. By virtue of Lemma 3.2 we recall that if u ε is a solution of (1.8) in Q, if for a fixed ℓ = 1, ..., 2n we set v ε = X ε ℓ u ε , and s ℓ = (−1) [ℓ/n] , then the function v ε is a solution in Q of (4.5) and a ε (x, t) = s ℓ Z(A ε ℓ+s ℓ n (x, ∇ ε u ε )), with |a ε (x, t)| ≤ C|∇ ε Zu ε | ∈ L 2 loc (Q) ∩ M 2,α (Q) for some constant C > 0 depending only on the structure constants and on ||u|| L p,p (Q) . We need to invoke a standard result from the theory of Morrey-Campanato which adapts immediately to the Heisenberg group setting, see [28], [6]. In the statement of the next lemma we assume that Q = Ω × (0, T ) is a given cylinder, and that [b ij ] 2n+1 i,j=1 is a uniformly elliptic matrix-valued function in Q, with coefficients in L ∞ (Q). We also suppose that for some λ ∈ (0, 1) we are given functions b i ∈ M 2,λ (Q), and a function b such that for each 2B ⊂ Ω and r > 0 sufficiently small, (Q), where we recall that N 1 = N + 2 = 2n + 4 is the parabolic dimension with respect to the dilations (x, t) → (δ λ x, λ 2 t).
We can now conclude the proof of the second part of Theorem 1.2. To begin, as we need to apply Lemma 4.4 to the linear equation (4.5), we observe that (4.2) and Hölder inequality yield the needed hypothesis (4.6). At this point one can invoke Lemma 4.4 to conclude that for every ℓ = 1, ..., 2n the function ∇ ε X ε l u ε belongs locally to M 2,λ . In view of the Poincaré inequality, one then has that ∇ ε u ε belongs to the Campanato spaces L 2,λ and hence by virtue of Lemma 4.3 it is Hölder continuous, concluding the proof.

Proof of Theorem 1.1
We will need a simple form of the comparison principle, see [3] and [4].
We now show how Theorem 1.1 follows from the comparison principle and from Theorem 1.2.
The latter implies that u 0 is a weak solution of (1.1), in B(x 0 , r) × (t 0 − r 2 , t 0 ), which agrees with the function u on the parabolic boundary of B(x 0 , r) × (t 0 − r 2 , t 0 ). By the comparison principle, the solution to this boundary values problem is unique, and hence we conclude that u ∈ C 1,α loc (B(x 0 , r) × (t 0 − r 2 , t 0 )).