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BY 4.0 license Open Access Published by De Gruyter February 11, 2020

Rigidity and trace properties of divergence-measure vector fields

Gian Paolo Leonardi ORCID logo and Giorgio Saracco ORCID logo EMAIL logo

Abstract

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ ( t ) is a non-negative convex function vanishing only at t = 0 . We show that this property is always satisfied in dimension n = 2 , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ ( t ) = c t 2 ) in dimension n 4 . The validity of the quadratic rigidity, which we prove in dimension n = 2 , implies the existence of the trace of a divergence-measure vector field ξ on an 1 -rectifiable set S, as soon as its weak normal trace [ ξ ν S ] is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

MSC 2010: 26B20; 28A75; 35L65

1 Introduction

The structure and the properties of vector fields, whose distributional divergence either vanishes or is represented by a locally finite measure, are of great interest in Mathematics and in Physics. Such vector fields arise, for instance, in fluid mechanics, in electromagnetism, and in conservation laws. We shall not give a detailed account, however the interested reader is referred to the monographs [20, 21], as well as to the papers [6, 18, 23, 24, 25], and to the references found therein.

In this paper we shall consider a rigidity property à la Liouville for divergence-free vector fields in n defined hereafter.

Definition 1.

Let φ : [ 0 , + ) [ 0 , + ) be a convex function such that φ ( t ) = 0 if and only if t = 0 . We say that the φ- rigidity property holds in n if, for any vector field η = ( η 1 , , η n ) L ( n ; n ) such that

  1. η = 0 on { x n : x n < 0 } ,

  2. div η = 0 on n in distributional sense,

  3. η n φ ( | η | ) almost everywhere on n ,

one has that η 0 almost everywhere on n .

We state here two results that establish this rigidity property.

Theorem 2 (Linear rigidity).

Let n 2 . Then the φ-rigidity property holds in R n when φ ( t ) = c t for some constant c > 0 .

Theorem 3 (φ-rigidity in the plane).

Let n = 2 . Then the φ-rigidity holds in R 2 for any choice of φ as in Definition 1.

Ideally, one would like to prove Theorem 3 in any dimension. Yet, proving φ-rigidity in dimension n > 2 when φ is not linear is a rather delicate issue and it would require some further hypotheses. Indeed, we have found counterexamples for the choice φ ( t ) = c t 2 in any dimension n 4 , as proved in Theorem 2. At the current stage it is unclear to us if Theorem 3 may hold in dimension n = 3 . Notice that the counterexamples found in dimension n 4 are cylindrically symmetric; however, we prove in Proposition 3 that no such vector field can be a counterexample for n = 3 .

The specific choice φ ( t ) = t 2 2 is quite interesting as it is closely related with a trace property of certain divergence-measure vector fields. More precisely, it allows us to deduce the existence of the trace of a divergence-measure vector field ξ on an oriented, 1 -rectifiable set S 2 , under a maximality assumption of the weak normal trace of ξ at S. In general, see Section 2.3, given a divergence-measure vector field ξ defined on a bounded open set Ω n , one is able to define its normal trace only in a distributional sense. More specifically, assuming that Ω is weakly-regular (that is, the perimeter of Ω is finite and coincides with the ( n - 1 ) -dimensional Hausdorff measure of Ω , see Definition 5) one can show [34, Section 3] that there exists a function [ ξ ν Ω ] L ( Ω ; n - 1 ) such that the following Gauss–Green formula holds for all ψ C c 1 ( n ) :

(1.1) Ω ψ ( x ) d div ξ + Ω ξ ( x ) ψ ( x ) 𝑑 x = Ω ψ ( y ) [ ξ ν Ω ] ( y ) 𝑑 n - 1 ( y ) ,

where we have denoted by n - 1 the ( n - 1 ) -dimensional Hausdorff measure and by ν Ω the measure-theoretic outer normal to Ω . Such a function [ ξ ν Ω ] is called weak normal trace of ξ on Ω . For the sake of completeness we recall that a first, fundamental weak version of the Gauss–Green formula is the classical result by De Giorgi [16, 17] and Federer [19], which states that (1.1) holds true in the case ψ = 1 , ξ C 1 and Ω with finite perimeter. A further extension due to Vol’pert [36, 37] holds when ξ is weakly differentiable or BV. In the seminal works of Anzellotti [4, 3], the concepts of weak normal trace and of pairing between vector fields and (gradients of) functions are introduced for the first time. Since then, the class 𝒟 ( Ω ) of divergence-measure, bounded vector fields has been widely studied in view of applications to hyperbolic systems of conservation laws [7, 9, 10], to continuum and fluid mechanics [8], and to minimal surfaces [28, 34] among many others. In particular, the weak normal trace has been studied in different directions, see for instance [1, 11, 12, 13, 15, 14].

Despite some explicit characterizations of the weak normal trace are available (see the discussion in Section 2.3), a crucial issue coming with this distributional notion is that it is not possible to recover the pointwise value of such a trace by a standard, measure-theoretic limit, see Example 7. However, assuming that ξ 1 and that the weak normal trace [ ξ ν S ] attains the maximal value 1 at some point x S , it would seem quite natural to expect that ξ cannot oscillate too much around x, to ensure a maximal outflow at x. Thus, one is led to conjecture the existence of the classical trace, i.e. the validity of the formula

[ ξ ν S ] ( x ) = ν S ( x ) .

Indeed, this is exactly what we are able to prove, limitedly to the case n = 2 , by employing Theorem 3 with the specific choice φ ( t ) = t 2 2 .

Theorem 4.

Let ξ D M ( R 2 ) and S R 2 an oriented H 1 -rectifiable with locally bounded H 1 -measure. Then, for H 1 -a.e. x 0 S such that [ ξ ν S ] ( x 0 ) = ξ one has

ap-lim x x 0 - ξ ( x ) = ξ ν S ( x 0 ) .

In the above theorem the approximate limit is “one-sided” according to Definition 2. Notice that the same conclusion of the theorem applies when the weak normal trace attains a local maximum for the modulus of the vector field, that is, when there exists an open set U containing x 0 such that [ ξ ν S ] ( x 0 ) | ξ ( x ) | for almost every x U . Of course, by a scaling argument one can always restrict to vector fields ξ with ξ 1 .

The statement of Theorem 4 holds for 1 -a.e. x 0 S . More precisely, one needs x 0 S to be such that: the normal of S at x 0 is defined; it is a Lebesgue point for the weak normal trace; and div ξ does not concentrate around it. Asking x 0 to satisfy these hypotheses, makes us indeed discard an 1 -negligible set of S.

As the proof heavily relies on Theorem 3, we are able to show it only in dimension 2. Despite there exist counterexamples to the quadratic rigidity in n when n 4 , we cannot exclude that Theorem 4 might be true in any dimension. Were it false, one should be able to construct a vector field with maximal normal trace at some ( n - 1 ) -submanifold S, whose blow-ups at most points of S are not unique. Indeed, the existence of the classical, one-sided trace of ξ at x 0 is equivalent to the uniqueness of the blow-up of ξ at x 0 (see Proposition 2). This essentially corresponds to a pointwise almost-everywhere convergence property. However, a maximality condition on the value of the weak normal trace (viewed as a weak-limit of measures induced by classical traces on approximating smooth surfaces) might enforce no more than a L 1 -type convergence, in analogy with what happens to a sequence of negative functions that weakly converge to zero. The fact that, in turn, L 1 -convergence does not imply almost everywhere convergence (unless one extracts suitable subsequences) explains why proving Theorem 4 is so delicate.

Finally, we mention that Theorem 4 allows us to strengthen the main result of our previous paper [28]. In there we considered solutions to the prescribed mean curvature equation in a set Ω, in the so-called extremal case. The extremality condition for the prescribed mean curvature equation

div u 1 + | u | 2 = H on  Ω

is a critical situation for the existence of solutions, occurring when the following, necessary condition for existence

| E H ( x ) 𝑑 x | < P ( E ) for all  E Ω

becomes an equality precisely at E = Ω . When Ω is smooth, Giusti proved in his celebrated paper [22] that the above necessary condition is also sufficient for existence, and moreover that the extremal case can be characterized by other properties, among which the uniqueness of the solution up to vertical translations and the vertical contact of its graph with the boundary of the domain. In the physically meaningful case, i.e. Ω 2 , the extremal case for the prescribed mean curvature equation corresponds to the capillarity phenomenon for perfectly wetting fluids within a cylindrical container put in a zero gravity environment.

In our previous work [28] we extended Giusti’s result to the wider class of weakly-regular domains, obtaining an analogous characterization that involves the weak normal trace of the vector field

T u = u 1 + | u | 2

on Ω . In virtue of the result proved here, we obtain that the boundary behavior of the capillary solution in the cylinder Ω × actually improves to a vertical contact realized in the classical trace sense, rather than just in the sense of the weak normal trace. In other words, the normalized gradient Tu is shown to admit a classical trace (equal to the outer normal ν Ω ) at 1 -almost-every point of Ω . We remark that, at present, no other technique, like the one based on regularity for almost-minimizers of the perimeter, seems to be applicable in the case of a generic weakly-regular domain. The reason is that the boundary of the cylinder Ω × is not smooth enough to let the approaching boundary of the subgraph of the solution u be uniformly regularized via the standard excess-decay mechanism for almost-minimizers of the perimeter.

Briefly, the paper is organized as follows. In Section 2 we lay the notation, recall some basic facts from Geometric Measure Theory and weak normal traces. In Section 3 we give the proofs of Theorem 2 and of Theorem 3. Section 4 presents the construction of a counterexample to the c t 2 -rigidity for n 4 . Finally, Section 5 is devoted to the weak normal trace and to the proof of Theorem 4.

2 Preliminary notions and facts

2.1 Notation

We first introduce some basic notations. Given n 2 , n denotes the Euclidean n-dimensional space, + n the upper half-space { x n : x n > 0 } , and 0 n the boundary of + n . For any x n and r > 0 , B r ( x ) denotes the Euclidean open ball of center x and radius r. Given a unit vector v n we set

B r v ( z ) = { x B r ( z ) : ( x - z ) v > 0 } .

Given a Borel set E n , we denote by χ E its characteristic function, by | E | its n-dimensional Lebesgue measure, and by d ( E ) its Hausdorff d-dimensional measure. We set

E x , r = r - 1 ( E - x ) ,

where E n , x n , and r > 0 . Given Ω n an open set, we write E Ω whenever the topological closure E ¯ of E is a compact subset of Ω. Whenever a measurable function, or vector field, f is defined on Ω, we denote by f its L -norm on Ω. We denote by 𝒟 ( Ω ) the space of bounded vector fields defined in Ω whose divergence is a Radon measure. For brevity we set 𝒟 := 𝒟 ( n ) . It is convenient to consider the restriction to 𝒟 ( Ω ) of the weak- * topology of L : given a sequence { v k } k 𝒟 ( Ω ) , we say that v k converges to v 𝒟 ( Ω ) in the weak- * topology of L , and write v k v in L - w * , if for every f L 1 ( Ω ; n ) one has

Ω f v k 𝑑 x Ω f v 𝑑 x as  k .

2.2 Basic definitions in Geometric Measure Theory

We now recall some basic definitions and facts from Geometric Measure Theory and, in particular, from the theory of sets of locally finite perimeter.

Definition 1 (Points of density α).

Let E be a Borel set in n , x n . If the limit

θ ( E ) ( x ) := lim r 0 + | E B r ( x ) | | B r ( x ) |

exists, it is called the density of E at x. In general θ ( E ) ( x ) [ 0 , 1 ] , hence, we define the set of points of density α [ 0 , 1 ] for E as

E ( α ) := { x n : θ ( E ) ( x ) = α } .

Definition 2 (One-sided approximate limit).

Let Ω n be an open set, and let S Ω be an oriented n - 1 -rectifiable set. Take z S such that the exterior normal ν = ν S ( z ) is defined, and choose a measurable function, or vector field, f defined on Ω. We write

ap-lim x z - f ( x ) = w

if for every α > 0 the set { x Ω B 1 - ν ( z ) : | f ( x ) - w | α } has density 0 at z.

Definition 3 (Perimeter).

Let E be a Borel set in n . We define the perimeter of E in an open set Ω n as

P ( E ; Ω ) := sup { Ω χ E ( x ) div ψ ( x ) 𝑑 x : ψ C c 1 ( Ω ; n ) , ψ 1 } .

We set P ( E ) = P ( E ; n ) . If P ( E ; Ω ) < , we say that E is a set of finite perimeter in Ω. In this case (see [30]) one has that the perimeter of E coincides with the total variation | D χ E | of the vector-valued Radon measure D χ E (the distributional gradient of χ E ), which is defined for all Borel subsets of Ω thanks to Riesz Theorem. We also recall that P ( E ; Ω ) = n - 1 ( E Ω ) when E Ω is Lipschitz.

Theorem 4 (De Giorgi Structure Theorem).

Let E be a set of finite perimeter and let * E be the reduced boundary of E defined as

* E := { x E : lim r 0 + D χ E ( B r ( x ) ) | D χ E | ( B r ( x ) ) = - ν E ( x ) 𝕊 n - 1 } .

Then:

  1. * E is countably n - 1 - rectifiable,

  2. for all x * E , χ E x , r χ H ν E ( x ) in L loc 1 ( n ) as r 0 + , where H ν E ( x ) denotes the half-space through 0 whose exterior normal is ν E ( x ) ,

  3. for any Borel set A, P ( E ; A ) = n - 1 ( A * E ) , thus in particular P ( E ) = n - 1 ( * E ) ,

  4. E div ψ = * E ψ ν E 𝑑 n - 1 for any ψ C c 1 ( n ; n ) .

Finally, we recall the notion of weakly-regular set which is useful for the next section.

Definition 5 (Weakly-regular set).

An open, bounded set Ω of finite perimeter, such that P ( Ω ) = n - 1 ( Ω ) , is said to be weakly-regular.

For the sake of completeness, we recall that examples of weakly-regular sets are, for instance, Lipschitz sets, or minimal Cheeger sets with n - 1 ( Ω Ω ( 1 ) ) = 0 (see [28, 33] for an account of these facts, [26, 31, 32] for an introduction to Cheeger sets, and [27, 29] for recent results and examples).

2.3 The weak normal trace and the Gauss–Green formula

The weak normal trace was first defined in [3] for a vector field with divergence in L 1 ( Ω ) , when Ω is bounded and Lipschitz. When Ω is a generic open set, and denoting by n the Lebesgue measure on n , the weak normal trace [ ξ ν Ω ] can be defined as the distribution

[ ξ ν Ω ] , ψ := Ω ψ d div ξ + Ω ξ ψ d n , ψ C c ( n ) .

Taking E Ω of class C 1 , and defining [ ξ ν E ] in the same way, one can show that the distribution is represented by an L function defined on E , that we still denote by [ ξ ν E ] with a slight abuse of notation, so that

[ ξ ν E ] , ψ = E ψ [ ξ ν E ] 𝑑 n - 1 .

Then, up to showing a locality property of the weak normal trace on C 1 domains (see [1]), it is possible to define [ ξ ν S ] for any given, oriented ( n - 1 ) -rectifiable set S contained in Ω. We remark, however, that some particular care has to be taken when Ω is a bounded open set with finite perimeter (and ξ is a-priori defined only on Ω). In order to guarantee that the distributional weak normal trace is represented by an L function defined on S = Ω one has to additionally assume that Ω is weakly-regular.

Thus, when Ω is weakly-regular one obtains the Gauss–Green formula below (see [34, Section 3]) exploiting the pairing between vector fields in 𝒟 ( Ω ) and functions in C c 1 ( n ) . Another version of the formula with a slightly different pairing can be found in [28].

Theorem 6 (Generalized Gauss–Green formula).

Let Ω R n be a weakly-regular set. For any ξ D M ( Ω ) and ψ C c 1 ( R n ) one has

(2.1) Ω ψ d div ξ + Ω ξ ψ d x = * Ω ψ [ ξ ν Ω ] 𝑑 n - 1 ,

where [ ξ ν Ω ] L ( Ω ; H n - 1 ) is the so-called weak normal trace of ξ on * Ω .

We remark for completeness that more general formulas have been recently obtained for pairings between ξ 𝒟 ( n ) , ψ BV loc ( n ) L loc ( n ) and Ω bounded with finite perimeter (see in particular [14, 35]).

As already observed by Anzellotti in [4] (see also [28]), the weak normal trace is a proper extension of the scalar product between the exterior normal ν Ω and the trace of the vector field ξ, assuming that the latter exists in the classical sense. More generally, one expects that the weak normal trace is obtained as a weak-limit of scalar products between (suitable averages of) the vector field ξ and the normal field ν S . This actually corresponds to [4, Proposition 2.1], which says that, whenever S is of class C 1 and div ξ L 1 , then

1 ω n r n y B r ( ) ξ ( y ) ν S ( ) 𝑑 n ( y ) [ ξ , ν S ] in  L ( S ) - w * as  r 0 + .

It is worth recalling that there exist also some pointwise characterizations of the weak normal trace. A first one is obtained by testing (2.1) with a function ψ that approximates the characteristic function of a ball B r ( x ) for x S . Taking x in a suitable subset of S of full n - 1 -measure one obtains

[ ξ , ν S ] ( x ) = ap-lim r 0 1 ω n - 1 r n y B r - ( x ) ξ ( y ) ( x - y ) 𝑑 n - 1 ( y ) for  n - 1 - a.e.  x S ,

where, for r > 0 small enough, B r - ( x ) denotes the part of B r ( x ) which is on the side of S pointed by - ν S ( x ) . A second characterization given in [4, Proposition 2.2], assuming S of class C 1 , is the following. Given x S and r , ρ > 0 small enough, let S ρ ( x ) = S B ρ ( x ) and set

Q r , ρ ( x ) = { z = y - t ν S ( x ) : y S ρ ( x ) , t ( 0 , r ) } ,
ν x ( z ) = ν S ( y ) if  z = y - t ν S ( x ) .

Note that ν x ( z ) is well-defined on Q r , ρ ( x ) as soon as r , ρ are small enough. In other words, Q r , ρ ( x ) is a curvilinear rectangle foliated by translates of S B ρ ( x ) in the - ν S ( x ) direction. It is proved in [4] that

[ ξ , ν S ] ( x ) = lim ρ 0 lim r 0 1 ω n - 1 ρ n - 1 r z Q r , ρ ( x ) ξ ( z ) ν x ( z ) 𝑑 n ( z ) ,

for n - 1 -almost-every x S . This result can be also obtained by testing (2.1) with a C 1 function ψ that satisfies ψ = 1 on S ρ , ψ = 0 on S ρ - r ν S ( x ) , and which is linear on any segment ( y , y - r ν S ( x ) ) , y S ρ .

Despite the characterizations described above, one can easily construct examples of vector fields like the one presented below (a variant of a piece-wise constant one defined in [4]) that illustrate possible wild behaviors of divergence-free vector fields for which the weak normal trace on S is well-defined. More precisely, the example below shows that, in general, the weak normal trace does not coincide with any classical, or measure-theoretic, limit of the scalar product of the vector field with the normal to S, even when S is of class C and the vector field is divergence-free and smooth in a neighborhood of S (minus S itself).

Example 7.

Let us set + 2 = { ( x , y ) : y > 0 } , S = { ( x , y ) : y = 0 } , and ν = ( 0 , - 1 ) . For i 1 and j = 1 , , 2 i - 1 we set x i j = j 2 i , y i = 1 2 i , r i = 1 2 i + 2 . Then, for such i and j we take f i C c ( ) with compact support in ( 0 , r i ) , so that in particular f i ( 0 ) = f i ( r i ) = 0 , and define p i j = ( x i j , y i ) . Notice that by our choice of parameters, the balls { B i j = B r i ( p i j ) } i , j are pairwise disjoint. Whenever p B i j , we set

ξ ( p ) = f i ( | p - p i j | ) ( p - p i j ) ,

while ξ ( p ) = 0 otherwise (see Figure 1). One can suitably choose f i so that ξ L ( B i j ) = 1 for all i , j . Moreover, div ξ = 0 on + 2 and thus for any ψ C c 1 ( 2 ) , by the Gauss–Green formula and owing to the definition of ξ, one has

S ψ [ ξ ν ] 𝑑 1 = + 2 ξ D ψ = i , j B i j ξ D ψ = i , j B i j ψ ξ ν i j = 0 ,

so that [ ξ ν ] = 0 on S. At the same time, ξ twists in any neighborhood of any point p 0 = ( x 0 , 0 ) , x 0 ( 0 , 1 ) , and the second component of the average of ξ on half balls centered at p 0 has a lim inf strictly smaller than its lim sup as r 0 . In conclusion, the scalar product ξ ( p ) ν ( p 0 ) does not converge to 0 in any pointwise or measure-theoretic sense.

Figure 1

A twisting vector field defined on + 2 .

3 Proofs of the rigidity results

We start by proving that the convexity assumption on the function φ makes Definition 1 essentially equivalent to a weaker one, in which the C smoothness and the global Lipschitz-continuity of the vector field η are required.

Lemma 1.

Let φ and η be as in Definition 1. Let ρ ε be the standard mollifier supported in a ball of radius ε > 0 . Then the regularized vector field η ε = ρ ε η is globally Lipschitz and satisfies (i)–(iii) of Definition 1 up to an ε-translation in the variable x n .

Proof.

We note that η n φ ( | η | ) almost everywhere on n by (iii), therefore Jensen’s inequality implies that for all x n and ε > 0

φ ( | η ε ( x ) | ) = φ ( | B ε η ( y ) ρ ε ( x - y ) 𝑑 y | )
B ε φ ( | η ( y ) | ) ρ ε ( x - y ) 𝑑 y
B ε η n ( y ) ρ ε ( x - y ) 𝑑 y = η n ε ( x ) .

Then we observe that η ε is globally Lipschitz, as a consequence of the boundedness of η, and verifies div η ε = 0 on n , that is, (ii). Moreover, for every x = ( y , t ) such that t - ε one has η ε ( x ) = 0 . Finally, up to a translation in the variable x, we can assume η ε ( x ) = 0 for all x - n , hence property (i) is also satisfied. ∎

Proof of Theorem 2.

Without loss of generality, by Lemma 1 we can assume that η is smooth and globally Lipschitz. Let us fix ε > 0 and consider the one-parameter flow associated with the vector field X = η + ε e n , defined for every t and p n by the Cauchy problem

{ Φ ( 0 , p ) = p , t Φ ( t , p ) = X ( Φ ( t , p ) ) .

Note that in the notation above we dropped the dependence upon the parameter ε for the sake of simplicity. Due to the smooth dependence from the initial datum, the map Φ ( , ) : × n n is smooth, D p Φ | t = 0 = Id and the map Φ ( t , ) : n n is a diffeomorphism for every t . Let us denote by Φ n ( t , p ) the n-th component of Φ ( t , p ) . Since X n ( p ) ε for all p n , we have that t Φ n ( t , p ) ε , hence the function t Φ n ( t , p ) is strictly monotone and surjective. Therefore, for every h and p n there exists a unique T = T ( p , h ) such that Φ n ( T , p ) = h . By the Implicit Function Theorem, T ( p , h ) is smooth and one has

h T ( p , h ) = X n ( Φ ( T , p ) ) - 1 , D p T ( p , h ) = - X n ( Φ ( T , p ) ) - 1 D p Φ n ( T , p ) .

Fix now an open, bounded and smooth set A n - 1 and h 0 > 0 . Let us set A h 0 = A × { h 0 } and define the map

Ψ : A × ( 0 , h 0 ) n , Ψ ( q , h ) = Φ ( T ( p , h ) , p ) ,

where we have set p = ( q , h 0 ) . Before proceeding it is convenient to introduce some more notation. We write

Ψ = ( Ψ 1 , , Ψ n - 1 , Ψ n ) = ( Ψ ^ , Ψ n )
X = ( X 1 , , X n - 1 , X n ) = ( X ^ , X n )
y = ( y 1 , , y n - 1 , y n ) = ( y ^ , y n ) for  y n .

We start by computing the partial derivative of Ψ with respect to h:

h Ψ ( q , h ) = X n ( Ψ ) - 1 X ( Ψ ) .

Note that Ψ ( p ) = Ψ ( q , h 0 ) = p and Ψ n ( q , h ) = h by definition. Owing to the smoothness of Ψ, we first compute the partial derivative with respect to h of Ψ j i := q j Ψ i , for i , j = 1 , , n - 1 :

h Ψ j i = q j h Ψ i = q j [ X n ( Ψ ) - 1 X i ( Ψ ) ]
= X n ( Ψ ) - 1 r = 1 n y r X i ( Ψ ) Ψ j r - X n ( Ψ ) - 2 X i ( Ψ ) r = 1 n y r X n ( Ψ ) Ψ j r
= r = 1 n [ X n ( Ψ ) - 1 y r X i ( Ψ ) - X n ( Ψ ) - 2 X i ( Ψ ) y r X n ( Ψ ) ] Ψ j r .

Moreover, it is immediate to check that q j Ψ n ( q , h ) = 0 for all j = 1 , , n - 1 and that h Ψ n ( q , h ) = 1 , so that in particular the matrix form of the previous computation is

h D q Ψ ^ = B ( Ψ ) D q Ψ ^ ,

where

B = X n - 1 D y ^ X ^ - X n - 2 X ^ D y ^ X n .

Moreover, the determinant of D Ψ coincides with that of D q Ψ ^ . If we assume that q n - 1 is fixed, and define δ ( h ) = det D q Ψ ^ ( q , h ) , we find by standard calculations that

δ ( h ) = tr [ B ( Ψ ( q , h ) ) ] δ ( h ) ,

with initial condition δ ( h 0 ) = 1 . This shows that the Jacobian matrix of Ψ is uniformly invertible on compact subsets of n . Consequently, Ψ is a smooth diffeomorphism and the set F ε ( A ) := Ψ ( A × ( 0 , h 0 ) ) , depicted in Figure 2, is Lipschitz. Notice moreover that

| Ψ ( q , h 0 ) - Ψ ( q , 0 ) | 0 h 0 | h Ψ ( q , h ) | 𝑑 h 0 h 0 X n - 1 | X | 𝑑 h h 0 ( c | η | + ε ) - 1 ( | η | + ε ) max { h 0 , h 0 c } ,

thanks to the properties of η. Notice now that given a bounded A n - 1 , there exists a constant R > 0 depending only on A and h 0 , such that F ε ( A ) is contained in ( - R , R ) n for every ε > 0 . Setting L ε ( A ) = Ψ ( A × { 0 } ) , by applying the Divergence Theorem on F ε ( A ) to the vector field X = η + ε e n we find the identity

0 = F ε ( A ) div X = A ( η n ( q , h 0 ) + ε ) 𝑑 q - ε n - 1 ( L ε ( A ) ) ,

since the integral on the “lateral” boundary of F ε ( A ) vanishes. This happens because the lateral boundary of the flow-tube consists of integral curves of the flow X, and thus on this lateral boundary one has X ν F ε ( A ) 0 . Using the fact that n - 1 ( L ε ( A ) ) ( 2 R ) n - 1 we can pass to the limit as ε 0 in the identity above, obtaining that η n vanishes on A × { h 0 } . By the arbitrary choice of both A and h 0 , and by property (iii) of Definition 1, we conclude that η = 0 on n . ∎

Figure 2 
                The evolution of 
                     
                        
                           
                              A
                              
                                 h
                                 0
                              
                           
                        
                        
                        {A_{h_{0}}}
                     
                   through the diffeomorphism Ψ defines a “flow-tube”.
Figure 2

The evolution of A h 0 through the diffeomorphism Ψ defines a “flow-tube”.

We now proceed with the proof of Theorem 3. Here, instead of using the “flow-tube” method employed in the proof of Theorem 2, we take advantage of a special feature of 2-dimensional cylinders of the form ( - a , a ) × ( 0 , h 0 ) , i.e. the fact that their “lateral” perimeter is constantly equal to 2 h 0 (and in particular it does not blow up when a + ).

Proof of Theorem 3.

By Lemma 1 we can additionally assume that η is smooth and globally Lipschitz. Since η 2 0 , by the Divergence Theorem we find

(3.1) - r r | η 2 ( x 1 , t ) | 𝑑 x 1 = - r r η 2 ( x 1 , t ) 𝑑 x 1 = 0 t η 1 ( - r , x 2 ) - η 1 ( r , x 2 ) d x 2 2 t η .

Therefore, η 2 ( , t ) L 1 ( ; ) for all t > 0 and

(3.2) η 2 ( , t ) 1 2 t η .

By combining (iii) of Definition 1 with (3.2) and the fact that η 2 is Lipschitz, we infer that

(3.3) φ ( | η 1 ( x 1 , t ) | ) η 2 ( x 1 , t ) 0 as  x 1 ± ,

hence, owing to the properties of φ, we get for all t > 0 ,

lim x 1 ± η 1 ( x 1 , t ) = 0 .

This implies that

2 η η 1 ( - r , t ) - η 1 ( r , t ) 0

as r + , for all t > 0 . Therefore, we can take the limit in (3.1) as r and, thanks to the Dominated Convergence Theorem, we obtain that

lim r + - r r | η 2 ( x 1 , t ) | 𝑑 x 1 = lim r + 0 t η 1 ( - r , x 2 ) - η 1 ( r , x 2 ) d x 2
= 0 t lim r + ( η 1 ( - r , x 2 ) - η 1 ( r , x 2 ) ) d x 2 = 0 .

Hence, the L 1 -norm of η 2 ( , t ) is zero, thus η 2 ( , t ) = 0 , for all t > 0 . By (3.3) and the properties of φ we get η = 0 on 2 . ∎

4 Counterexamples to quadratic rigidity in dimension n 4

Given x n , we set r = ( x 1 , , x n - 1 ) and z = x n . We consider vector fields η C 0 ( n ) such that for r 0 one has

(4.1) η ( r , z ) = ( r f ( r , z ) , h ( r , z ) )

with f , h C 1 ( + n { ( 0 , z ) : z } ) . We have the following:

Proposition 1.

Let η C 0 ( R n ) be as in (4.1). Define

(4.2) V ( r , z ) := - | r | n - 1 0 z f ( r , s ) 𝑑 s .

Then η satisfies

(4.3) η ( r , z ) = 0 for all  z 0 ,
(4.4) | η ( x ) | 1 for all  x n ,
(4.5) div η = 0 on  n ,
(4.6) η n ( x ) c 1 | η ( x ) | 2 for all  x n ,

if and only if V ( r , z ) satisfies

(4.7) V ( r , z ) = 0 for all  z 0 ,
(4.8) | V ( r , z ) | | r | n - 2 for all  r > 0 and  z ,
(4.9) r r V ( r , z ) c 2 | r | 3 - n ( z V ( r , z ) ) 2 for all  r > 0 and  z .

Moreover, one has

(4.10) η ( r , z ) = | r | 1 - n ( - ( z V ) r , r r V ) .

Proof.

We show in full detail the only if part. Since

div η ( r , z ) = div r ( r f ( r , z ) ) + z h ( r , z ) = ( n - 1 ) f + r r f ( r , z ) + z h ( r , z ) ,

equation (4.5) is equivalent to

(4.11) z h = - ( r r f + ( n - 1 ) f ) .

By recalling that h ( r , 0 ) = 0 for all r by (4.3), from (4.11) we obtain that

(4.12) h ( r , z ) = - 0 z ( r r f ( r , s ) + ( n - 1 ) f ( r , s ) ) 𝑑 s .

Inequality (4.6), up to a change of the constant c, is equivalent to

(4.13) h c | r | 2 f 2 .

Let us set V as in (4.2) and observe that (4.3) implies V ( r , z ) = 0 when z 0 . Then using (4.12) we obtain

r V = - | r | n - 1 0 z [ r f ( r , s ) + ( n - 1 ) f ( r , s ) | r | - 2 r ] 𝑑 s

and

(4.14) z V ( r , z ) = - | r | n - 1 f ( r , z ) ,

so that in particular

(4.15) r r V = | r | n - 1 h ,

and inequality (4.13) implies

r r V c | r | n + 1 f 2 c | r | 3 - n ( z V ) 2 .

Then by observing that (4.4) is equivalent to | r | 2 f 2 + h 2 1 , we obtain by (4.14) and (4.15)

| V | | r | n - 2 .

We have thus proved that defining V as in (4.2) we get (4.7)–(4.9).

Conversely, one can easily check that given V satisfying (4.7)–(4.9), the vector field η defined as in (4.10) satisfies (4.3)–(4.6). ∎

Theorem 2.

The quadratic rigidity property does not hold in dimension n 4 .

Proof.

Let us consider a positive parameter γ (to be chosen later) and define the function

V ( r , z ) = { γ [ ( 1 + | r | n - 1 ) 1 n - 1 - 1 ] arctan ( z 2 ) if  z 0 , 0 otherwise .

Our aim is to verify properties (4.7)–(4.9) up to a suitable choice of γ, and then to use the equivalence stated in Proposition 1. Of course (4.7) is true by definition of V. Let us set ρ = | r | and write V = V ( ρ , z ) for simplicity. One has

ρ V = γ arctan ( z 2 ) ( 1 + ρ n - 1 ) 2 - n n - 1 ρ n - 2

and

z V = 2 γ [ ( 1 + ρ n - 1 ) 1 n - 1 - 1 ] z 1 + z 4 .

Notice that V is of class C 1 and V ( ρ , z ) = 0 for all ρ > 0 and z 0 . We obtain

| V | γ arctan ( z 2 ) ( 1 + ρ n - 1 ) 2 - n n - 1 ρ n - 2 + 2 γ [ ( 1 + ρ n - 1 ) 1 n - 1 - 1 ] z 1 + z 4
(4.16) γ π ρ n - 2 2 + γ 3 3 4 2 [ ( 1 + ρ n - 1 ) 1 n - 1 - 1 ] ,

where the second inequality follows from the maximization of the function z ( 1 + z 4 ) for z [ 0 , + ) . Let us consider the function

φ ( ρ ) = ( 1 + ρ n - 1 ) 1 n - 1 - 1 ρ n - 2 .

As ρ 0 + , we have φ ( ρ ) ρ n - 1 , while as ρ + we have φ ( ρ ) ρ 3 - n . Moreover, when 0 < t < 1 , one has ( 1 + t ) 1 n - 1 1 + t n - 1 , hence we deduce that

φ ( ρ ) ρ n - 1 1

when ρ < 1 , and

φ ( ρ ) ρ 3 - n 1

when ρ 1 . Therefore by (4.16) and the last inequalities we find that

| V | C γ ρ n - 2 ,

where

C = π + 3 3 4 2 .

Assuming γ C - 1 , we obtain (4.8).

We now show that (4.9) (with the constant c = 1 ) holds up to taking a smaller γ. Indeed, the relation ρ ρ V ρ 3 - n ( z V ) 2 , after separation of variables, becomes

arctan ( z 2 ) ( 1 + z 4 ) 2 z 2 4 γ φ ( ρ ) 2 ( 1 + ρ n - 1 ) n - 2 n - 1 .

We argue as for the upper bound of φ ( ρ ) (more precisely, we discuss the two cases ρ < 1 and ρ 1 ; in the first case we use the bound φ ( ρ ) 1 , while in the second case we use the fact that n 4 , and the inequalities φ ( ρ ) ρ 3 - n and 1 + ρ n - 1 2 ρ n - 1 ), and find

(4.17) 4 γ φ ( ρ ) 2 ( 1 + ρ n - 1 ) n - 2 n - 1 2 3 n - 4 n - 1 γ .

At the same time, by easy calculations we infer that the function arctan ( t ) ( 1 + t 2 ) 2 t is bounded from below by 1. Hence (4.9) is implied by the condition

γ 2 4 - 3 n n - 1 .

In conclusion, by taking γ small enough, and thanks to Proposition 1, a divergence-free vector field providing a counterexample to the rigidity property in dimension n 4 is given by

η ( r , z ) = { γ | r | 1 - n ( - 2 [ ( 1 + | r | n - 1 ) 1 n - 1 - 1 ] z 1 + z 4 r , arctan ( z 2 ) ( 1 + | r | n - 1 ) 2 - n n - 1 | r | n - 1 ) if  z > 0 , 0 if  z 0 .

We remark that the vector field η is of class C 0 , however one can obtain a C counterexample by mollification of η. ∎

Concerning the 3-dimensional case, the construction of a counterexample with cylindrical symmetry, as done in Theorem 2, does not work. Indeed we are unable to get an estimate like (4.17), as the function φ ( ρ ) tends to 1 as ρ + . More precisely we can prove the following result.

Proposition 3.

Let n = 3 and assume that V ( r , z ) is a C 1 function satisfying properties (4.7)–(4.9). Then V ( r , z ) = ψ 1 ( | r | ) ψ 2 ( z ) for suitable functions ψ 1 , ψ 2 implies V 0 .

Proof.

We set ρ = | r | and write (4.9) as

ρ ψ 1 ( ρ ) ψ 2 ( z ) c ψ 1 2 ( ρ ) [ ψ 2 ( z ) ] 2 .

Let us assume by contradiction that ψ 1 and ψ 2 are not trivial, hence there exist z 0 > 0 and ρ 0 > 0 such that ψ 2 ( z 0 ) 0 , ψ 2 ( z 0 ) 0 , and ψ 1 ( ρ 0 ) 0 . Setting a = ψ 2 ( z 0 ) and b = ψ 2 ( z 0 ) , we have four cases according to the sign of a and ψ 1 ( ρ 0 ) . We discuss the first case a > 0 and ψ 1 ( ρ 0 ) > 0 . We consider the differential inequality

ρ ψ 1 ( ρ ) γ ψ 1 2 ( ρ )

with γ = c b 2 a > 0 . Therefore, the function ψ 1 is increasing and by separation of variables and integration between ρ 0 and ρ > ρ 0 we get

(4.18) ψ 1 - 1 ( ρ 0 ) - ψ 1 - 1 ( ρ ) + ψ 1 - 1 ( ρ 0 ) γ log ( ρ ρ 0 ) .

We denote by I = [ ρ 0 , β ) the maximal right interval of existence of the solution ψ 1 , for which ψ 1 > 0 . We can exclude the case β < + , as we would obtain by maximality that ψ 1 ( ρ ) + as ρ β - , however this would contradict the fact that | ψ ( ρ ) | ρ for all ρ > 0 . Contrarily, if ρ + , one gets a contradiction with (4.18). The remaining three cases can be discussed in a similar way. ∎

Remark 4.

As a consequence of Proposition 3 we infer that in dimension n = 3 no counterexample to the rigidity property can be found in the class of vector fields of the form η ( r , z ) = ( r f ( | r | , z ) , h ( | r | , z ) ) .

5 The trace of a vector field with locally maximal normal trace

The results we shall discuss in this section are stated for n - 1 -almost-every point of S, being S an oriented, n - 1 -rectifiable set with locally finite n - 1 -measure. Therefore, given z 𝒟 , without loss of generality (see [2, Theorem 2.56]) we shall assume x 0 S to be such that

  1. the normal vector ν S ( x 0 ) is defined at x 0 ,

  2. x 0 is a Lebesgue point for the weak normal trace [ z ν S ] of z on S, with respect to the measure n - 1 S ,

  3. | div z | ( B r ( x 0 ) S ) = o ( r n - 1 ) as r 0 + .

Lemma 1.

Let η , { z k } k be vector fields in D M with sup k z k < + . Let Σ , { S k } k be oriented, closed H n - 1 -rectifiable sets with locally finite H n - 1 -measure, satisfying the following properties:

  1. z k η in L - w * ,

  2. n - 1 S k n - 1 Σ ,

  3. | div z k | ( n S k ) 0 .

Then div η = 0 in R n Σ .

Proof.

Fix φ C c 1 ( n Σ ) and set μ k = div z k and μ = div η . By the Divergence Theorem coupled with the formula (see [1, Proposition 3.2])

μ k S k = ( [ z k ν S k ] + - [ z k ν S k ] - ) n - 1 S k ,

we have

n S k φ 𝑑 μ k = - S k φ 𝑑 μ k - n S k φ z k
= - S k φ ( [ z k ν S k ] + - [ z k ν S k ] - ) 𝑑 n - 1 - n φ z k ,

hence by (ii) and (iii)

| n φ z k | n S k | φ | 𝑑 μ k + 2 z k | φ | 𝑑 n - 1 S k 0 as  k .

This shows that

n φ 𝑑 μ = lim k n φ 𝑑 μ k = lim k n φ z k = 0 ,

which proves the thesis. ∎

Proposition 2.

Let z D M and S a closed, oriented H n - 1 -rectifiable set with locally finite H n - 1 -measure. Then, for H n - 1 -almost-every x 0 S and for any decreasing and infinitesimal sequence { r k } k , the sequence z k of vector fields defined by z k ( y ) = z ( x 0 + r k y ) converges up to subsequences to a vector field η D M in L - w * , such that setting Σ = [ ν S ( x 0 ) ] , we have div η = 0 on R n Σ and [ η ν Σ ] = [ z ν S ] ( x 0 ) on Σ.

Proof.

We show that hypotheses (i)–(iii) of Lemma 1 are satisfied. Since z k = z for all k, thanks to Banach–Alaoglu Theorem (see also [5, Theorem 3.28]) we can extract a not relabeled subsequence converging to η 𝒟 , which gives (i). We set S k = r k - 1 ( S - x 0 ) , then thanks to (c) we have

(5.1) | div z k | ( B R S k ) = r k 1 - n | div z | ( B R r k ( x 0 ) S ) 0 as  k

for all R > 0 , which gives (iii). Owing to the localization property proved in [1, Proposition 3.2], we can replace S with the boundary of an open set Ω of class C 1 such that x 0 Ω and ν S ( x 0 ) = ν Ω . Defining Ω k = r k - 1 ( Ω - x 0 ) , the proof of (ii) is reduced to showing that n - 1 Ω k weakly converge as measures to n - 1 H , where H is the tangent half-space to Ω at x 0 (so that Σ = H ). This fact is a consequence of Theorem 4. Now we can apply Lemma 1 and obtain div η = 0 on n Σ . In order to prove the last part of the statement we have to show that for any ψ C c 1 ( n ) one has

H ψ d div η + H ψ η = τ 0 H ψ 𝑑 n - 1 ,

where τ 0 = [ z ν S ( x 0 ) ] . Since we have proved that div η = 0 on n Σ , we only have to show that

(5.2) H ψ η = τ 0 H ψ 𝑑 n - 1 .

Note that by the convergence of z k to η, the L loc 1 -convergence of Ω k to H, and the convergence of the measures n - 1 Ω k to n - 1 H , we have

(5.3) H ψ η - τ 0 H ψ 𝑑 n - 1 = lim k Ω k ψ z k - τ 0 Ω k ψ 𝑑 n - 1 .

Therefore by (5.1) and (b) we infer that

| Ω k ψ z k - τ 0 Ω k ψ 𝑑 n - 1 | | Ω k ψ ( [ z