On an evolution equation in sub-Finsler geometry

1 Introduction

Singular spaces occupy a prominent position in analysis and geometry. Examples of basic interest are Alexandrov spaces and Finsler manifolds. These latter ambients have received considerable attention over the past few decades as they often occur as scaling limits of crystalline or Riemannian structures, see [21] and especially the seminal work [3]. In this article, we introduce a nonlinear evolution equation that represents the gradient flow of an energy which is at the interface of Finsler and sub-Riemannian geometry. Our main objective is to understand in detail the relevant heat kernel. To keep the presentation self-contained, we confine ourselves to the model setting of a product of Euclidean spaces, i.e., R N = R m × R k , with coordinates z ∈ R m , σ ∈ R k . We assume that on each of the two layers, R m and R k , Minkowski norms Φ and Ψ have been assigned. By this, we mean that Φ 2 ∈ C 2 ( R m \ { 0 } ) , Ψ 2 ∈ C 2 ( R k \ { 0 } ) , and that these functions are strongly convex and 1-homogeneous. We, respectively denote by Φ 0 and Ψ 0 their dual norms, defined as in (2.2), and acting on the dual spaces of R m and R k , which are in turn identified with R m and R k themselves, with coordinates z and σ , respectively. The level sets of the functions Φ 0 and Ψ 0 are often referred to as Wulff shapes. This situation represents a simplified, yet significant, model for the more general one of a Riemannian manifold in which the tangent space is stratified in layers, each endowed with a different anisotropic structure, and having mixed homogeneities weighted according to the relative position in the stratification.

We are interested in the L 2 gradient flow

of the following energy:

Notable features of (1.2) are as follows:

1. The nonlinear dependence on the degenerate gradient

   (1.3) ∇ X f = ( X 1 f , … , X m f , X m + 1 f , … , X N f )

   associated with the N vector fields

   (1.4) X i = ∂ z i , i = 1 , … , m , X m + j = Φ 0 ( z ) 2 ∂ σ j , j = 1 , … , k

   (note from (1.4) that for i = 1 , … , N one has X i ⋆ = − X i in L 2 ( R N ) . We also note that, under the given assumptions on Φ , we have Φ 0 ∈ C 1 ( R m \ { 0 } ) , see [20, Corollary 1.7.3].

2. It degenerates along the manifold M = { 0 } z × R k , but because of the anisotropic nature of the dual norm Φ 0 ( z ) , it does so at different scales along regions of approach to M .

3. It is invariant with respect to the family of mixed dilations in R N

   (1.5) δ λ ( z , σ ) = ( λ z , λ 2 σ ) , λ > 0 .

where we have denoted by Δ Φ and Δ Ψ the Finsler Laplacians in the spaces R m and R k , see (2.6). Furthermore, if with δ λ as in (1.5), we define

then if f solves (1.6), so does also f ∘ D λ , for every λ > 0 .

We emphasise that (1.6) is a nonlinear evolution equation with quadratic growth in the degenerate “gradient” (1.3). By this, we mean that there exist constants γ , γ ⋆ > 0 such that

To explain (1.8), note that (1.3) and (1.4) give

Therefore, thanks to the 1-homogeneity of Φ and Ψ , if we let

then the PDE (1.6) can be alternatively written as

where the functions A i : R N → R are the components of the vector field

We now note that

where in the last equality, we have used the 1-homogeneity of Φ and Ψ , which is well-known to be equivalent to the Euler equations

By the condition (2.7), from (1.9) and (1.11), we finally infer the existence of γ , γ ⋆ > 0 such that (1.8) holds. Although the present work is not concerned with the local theory of the PDE (1.6), we remark that the growth (1.8) would allow us to apply the results in [4] once the relevant volume doubling condition and Poincaré inequality are available. For these aspects, see the seminal works [13,19], and also the more recent article [5] that develops the local theory in Finsler manifolds with Ricci lower bounds.

To state our first result, we need to introduce some notations. Henceforth, points of R m will be denoted by z , ζ , etc., points of R k by σ , τ , λ , etc. We will use the notation X = ( z , σ ) , Y = ( ζ , τ ) , etc., for points in the product space R N . Since the notation ∇ X will no longer appear in this work, there will be no risk of confusion of this notation with the subscript in (1.8), (1.9), and (1.11). Also, we will, respectively, indicate with σ Φ and σ Ψ the intrinsic measures of the Wulff spheres defined by

where H m − 1 and H k − 1 denote ( m − 1 ) -dimensional and ( k − 1 ) -dimensional Hausdorff measure in R m and R k , respectively. Recall the classical formulas σ m − 1 = H m − 1 ( S m − 1 ) = 2 π m 2 Γ m 2 , σ k − 1 = H k − 1 ( S k − 1 ) = 2 π k 2 Γ k 2 . It is clear from (1.12) that, in the linear isotropic case Φ ( z ) = ∣ z ∣ , Ψ ( σ ) = ∣ σ ∣ , one has σ Φ = σ m − 1 , σ Ψ = σ k − 1 . For a number ν ∈ C , we will denote with J ν the Bessel function of the first kind and order ν and indicate by

We have the following.

To state our second result, we mention that in recent paper [7], the authors have studied in the product space R N the following degenerate energy with mixed homogeneity:

It is clear that (1.2) corresponds to the special case α = 1 and p = 2 of (1.16). One of their main results is an explicit fundamental solution for the Euler-Lagrange equation of (1.16). To formulate the relevant result, consider the following anisotropic Minkowski gauge:

In [7], a new Legendre transformation Θ 0 was introduced, namely

The remarkable feature of (1.18) is underscored by the following result, established in [7, Proposition 3.3]:

where Φ 0 and Ψ 0 are the classical dual Minkowski norms defined as in (2.2). The reader should observe the perfect symmetry between (1.17) and (1.19). Furthermore, in [7, Theorem 1.2], the authors proved that the function

is a fundamental solution, with pole in ( 0 , 0 ) , of the Euler-Lagrange equation of (1.16). The number Q in (1.20) is given by

It is clear from (1.20) that such number plays the role of a dimension. In (1.20), the positive constants C α , p and C α are implicitly given, and they involve the Wulff shapes of the gauge Θ 0 . However, in order to fully understand the gradient flow of the energy (1.2), it is important to have an explicit knowledge of the constants. For this reason, in Section 3, we turn to this problem, and in Lemma 3.4, we show that, with σ Φ and σ Ψ as in (1.12), then

where we have denoted by B ( x , y ) Euler beta function (3.7). Since in [7, Theorem 1.2] it was proved that

it is clear that (1.22) provides an expression of C α , p . We explicitly note from (1.19), (1.20), and (1.21) that, when α = 1 and p = 2 , we have

and

We also mention that the above theorem (1.20) generalised a result first established in [10] for the linear case Φ ( z ) = ∣ z ∣ , Ψ ( σ ) = ∣ σ ∣ , when p = 2 .

To put our second result in some historical context, consider the standard Gauss-Weierstrass kernel in R n (here, we are assuming n ≠ 2 )

A well-known manifestation of the fact that G ( x , t ) is a fundamental solution of the heat equation, and of the Bochner subordination principle, is that the L loc 1 ( R N ) function defined by

provides a fundamental solution of − Δ with pole in 0 ∈ R n . Furthermore, an elementary, yet beautiful computation shows that

The next theorem shows that, surprisingly, despite its strongly nonlinear character, the evolution equation (1.6) displays the same linear phenomenon as the standard heat equation. Furthermore, Theorem 1.3 shows the remarkable fact that, had we not known the magic Minkowski gauge Θ 0 in (1.24), by running the nonlinear heat flow (1.1), we are forced to discover it!

We close this introduction with a short description of the organisation of the article. In Section 2, we collect some basic properties of Minkowski norms and of the Finsler Laplacian and heat equation. Section 3 is devoted to proving Proposition 3.5. In Section 4, we prove Theorem 1.1. Finally, in Section 5, we establish Theorem 1.3.

2 Minkowski norms

Let M : R n → [ 0 , ∞ ) be a Minkowski norm in R n . By this, we mean that M 2 ∈ C 2 ( R n \ { 0 } ) is a strongly convex function such that M ( λ x ) = ∣ λ ∣ M ( x ) for every x ∈ R n and λ ∈ R . By strong convexity, we mean that the Hessian matrix 1 2 ∇ 2 ( M 2 ) is positive definite in R n \ { 0 } . Since all norms in R n are equivalent, there exist constants β ≥ α > 0 such that

We denote by

its Legendre transform, also known as the dual norm of M . The Cauchy-Schwarz inequality trivially holds

A basic property of the norms M and M 0 is the following, see [3, Lemma 2.1] and also (3.12) in [6],

Given a function u ∈ C 1 ( R n ) , an elementary, yet useful, consequence of the homogeneity of M is

A less obvious basic fact is the following formula, which can be found in [3, Lemma 2.2].

The Euler-Lagrange equation of the energy

is the so-called Finsler Laplacian

It is worth emphasising here that the operator in (2.6) is quasilinear, but not linear, unless of course M ( x ) = ∣ x ∣ . However, the operator Δ M is elliptic. In fact, since M 2 is homogeneous of degree 2, the Euler formula gives

and from (2.1), we thus have for every ξ ∈ R n

An important property of the dual norm is the following result that follows from (2.3) and from Lemma 2.1, see [6] and [9].

A remarkable consequence of Proposition 2.2 is that the nonlinear operator Δ M acts linearly on functions of the dual norm M 0 . The Finsler heat equation arises from the gradient flow of the energy (2.5). It is the quasilinear PDE in R n × R

In the framework of Finsler manifolds, an in-depth study of (2.8) has been carried out in the previous studies [17,18], see also [1] for the case of R n . Using Proposition 2.2, it is possible to construct the following notable explicit solution of the Finsler heat equation, see [17, Example 4.3] and also [1].

We leave it as an exercise for the reader to verify that the function G ( x , t ) satisfies Proposition 2.4, the Finsler counterpart of the extremal case of the famous result in [15]. A Li-Yau theory in Finsler manifolds was developed in the cited work [18]. In connection with the present work, a version of the theory in [2] will be presented in a forthcoming article.

Using the coarea formula, one can easily show that

where we have let

Note that, on the one hand, the coarea formula gives

and therefore,

On the other hand, a rescaling gives

and therefore,

Equating (2.12) and (2.14), we infer

In particular, when r = 1 , we obtain from (2.10) and (2.15)

We will need the following useful observation.

3 Equalities matter

This section is devoted to the explicit computation of constants C α , p and C α in the above cited (1.20) from [7, Theorem 1.2]. The main result of the section is Proposition 3.5. This result plays a critical role in recognising that the function G ( X , t ) introduced in (1.14) of Theorem 1.1 is, in fact, a fundamental solution of the PDE (1.6). Throughout this section, R N = R m × R k , and Φ and Ψ will be strongly convex Minkowski norms in R m and R k , respectively. We recall from [7] that, with Θ 0 ( z , σ ) given by (1.19), the constants in (1.20) are prescribed by the formulas (1.23), where, with Q as in (1.21), we have let

If we now consider the Wulff ball of the anisotropic gauge

and set

then it should be clear to the reader that, if we define

where we have denoted by 1 K ⋆ the indicator function of the set K ⋆ , then we have

We now state a generalisation of Lemma 2.6, whose proof we leave to the reader.

If we now apply (3.6), keeping (3.1), (3.2), and (3.3) in mind, we reach the following conclusion.

In the next lemma, we compute in closed form the integral on the right-hand side of the equation in Lemma 3.2. Such a result will be crucial in the proof of Proposition 3.5. In its statement, we denote by

Euler beta function. For the reader’s convenience, we recall its well-known property

Combining Lemmas 3.2 and 3.3, we obtain the following result.

In connection with Theorem 1.3, we are particularly interested in the case α = 1 and p = 2 of Lemma 3.4. Our objective is to establish the following important fact.