Open Access Pre-published online by De Gruyter February 11, 2020

# Rigidity and trace properties of divergence-measure vector fields

Gian Paolo Leonardi and Giorgio Saracco

# 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)=ct2) in dimension n4. 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 [ξνS] 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=(η1,,ηn)L(n;n) such that

1. (i)

η=0 on {xn:xn<0},

2. (ii)

divη=0 on n in distributional sense,

3. (iii)

ηnφ(|η|) almost everywhere on n,

one has that η0 almost everywhere on n.

We state here two results that establish this rigidity property.

## Theorem 2 (Linear rigidity).

Let n2. Then the φ-rigidity property holds in Rn when φ(t)=ct for some constant c>0.

## Theorem 3 (φ-rigidity in the plane).

Let n=2. Then the φ-rigidity holds in R2 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)=ct2 in any dimension n4, 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 n4 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)=t22 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 S2, 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 Ωn, 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 [ξνΩ]L(Ω;n-1) such that the following Gauss–Green formula holds for all ψCc1(n):

(1.1)Ωψ(x)ddivξ+Ωξ(x)ψ(x)𝑑x=Ωψ(y)[ξνΩ](y)𝑑n-1(y),

where we have denoted by n-1 the (n-1)-dimensional Hausdorff measure and by νΩ the measure-theoretic outer normal to Ω. Such a function [ξνΩ] 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, ξC1 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 ξ1 and that the weak normal trace [ξνS] attains the maximal value 1 at some point xS, 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

[ξνS](x)=νS(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)=t22.

## Theorem 4.

Let ξDM(R2) and SR2 an oriented H1-rectifiable with locally bounded H1-measure. Then, for H1-a.e. x0S such that [ξνS](x0)=ξ one has

ap-limxx0-ξ(x)=ξνS(x0).

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 x0 such that [ξνS](x0)|ξ(x)| for almost every xU. Of course, by a scaling argument one can always restrict to vector fields ξ with ξ1.

The statement of Theorem 4 holds for 1-a.e. x0S. More precisely, one needs x0S to be such that: the normal of S at x0 is defined; it is a Lebesgue point for the weak normal trace; and divξ does not concentrate around it. Asking x0 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 n4, 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 x0 is equivalent to the uniqueness of the blow-up of ξ at x0 (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 L1-type convergence, in analogy with what happens to a sequence of negative functions that weakly converge to zero. The fact that, in turn, L1-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

divu1+|u|2=Hon Ω

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

|EH(x)𝑑x|<P(E)for all EΩ

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. Ω2, 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

Tu=u1+|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 νΩ) 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 ct2-rigidity for n4. 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 n2, n denotes the Euclidean n-dimensional space, +n the upper half-space {xn:xn>0}, and 0n the boundary of +n. For any xn and r>0, Br(x) denotes the Euclidean open ball of center x and radius r. Given a unit vector vn we set

Brv(z)={xBr(z):(x-z)v>0}.

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

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

where En, xn, and r>0. Given Ωn an open set, we write EΩ 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 {vk}k𝒟(Ω), we say that vk converges to v𝒟(Ω) in the weak-* topology of L, and write vkv in L-w*, if for every fL1(Ω;n) one has

Ωfvk𝑑xΩfv𝑑xas 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, xn. If the limit

θ(E)(x):=limr0+|EBr(x)||Br(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):={xn:θ(E)(x)=α}.

### Definition 2 (One-sided approximate limit).

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

ap-limxz-f(x)=w

if for every α>0 the set {xΩB1-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 Ωn as

P(E;Ω):=sup{ΩχE(x)divψ(x)𝑑x:ψCc1(Ω;n),ψ1}.

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 |DχE| of the vector-valued Radon measure DχE (the distributional gradient of χE), which is defined for all Borel subsets of Ω thanks to Riesz Theorem. We also recall that P(E;Ω)=n-1(EΩ) when EΩ 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:={xE:limr0+DχE(Br(x))|DχE|(Br(x))=-νE(x)𝕊n-1}.

Then:

1. (i)

*E is countably n-1-rectifiable,

2. (ii)

for all x*E, χEx,rχHνE(x) in Lloc1(n) as r0+, where HνE(x) denotes the half-space through 0 whose exterior normal is νE(x),

3. (iii)

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

4. (iv)

Edivψ=*EψνE𝑑n-1 for any ψCc1(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(ΩΩ(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 L1(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 [ξνΩ] can be defined as the distribution

[ξνΩ],ψ:=Ωψddivξ+Ωξψdn,ψCc(n).

Taking EΩ of class C1, and defining [ξνE] in the same way, one can show that the distribution is represented by an L function defined on E, that we still denote by [ξνE] with a slight abuse of notation, so that

[ξνE],ψ=Eψ[ξνE]𝑑n-1.

Then, up to showing a locality property of the weak normal trace on C1 domains (see [1]), it is possible to define [ξνS] 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 Cc1(n). Another version of the formula with a slightly different pairing can be found in [28].

### Theorem 6 (Generalized Gauss–Green formula).

Let ΩRn be a weakly-regular set. For any ξDM(Ω) and ψCc1(Rn) one has

(2.1)Ωψddivξ+Ωξψdx=*Ωψ[ξνΩ]𝑑n-1,

where [ξνΩ]L(Ω;Hn-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 ξ𝒟(n), ψBVloc(n)Lloc(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 νΩ 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 νS. This actually corresponds to [4, Proposition 2.1], which says that, whenever S is of class C1 and divξL1, then

1ωnrnyBr()ξ(y)νS()𝑑n(y)[ξ,νS]in L(S)-w*as r0+.

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 Br(x) for xS. Taking x in a suitable subset of S of full n-1-measure one obtains

[ξ,νS](x)=ap-limr01ωn-1rnyBr-(x)ξ(y)(x-y)𝑑n-1(y)for n-1-a.e. xS,

where, for r>0 small enough, Br-(x) denotes the part of Br(x) which is on the side of S pointed by -νS(x). A second characterization given in [4, Proposition 2.2], assuming S of class C1, is the following. Given xS and r,ρ>0 small enough, let Sρ(x)=SBρ(x) and set

Qr,ρ(x)={z=y-tνS(x):ySρ(x),t(0,r)},
νx(z)=νS(y)if z=y-tνS(x).

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

[ξ,νS](x)=limρ0limr01ωn-1ρn-1rzQr,ρ(x)ξ(z)νx(z)𝑑n(z),

for n-1-almost-every xS. This result can be also obtained by testing (2.1) with a C1 function ψ that satisfies ψ=1 on Sρ, ψ=0 on Sρ-rνS(x), and which is linear on any segment (y,y-rνS(x)), ySρ.

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 i1 and j=1,,2i-1 we set xij=j2i, yi=12i, ri=12i+2. Then, for such i and j we take fiCc() with compact support in (0,ri), so that in particular fi(0)=fi(ri)=0, and define pij=(xij,yi). Notice that by our choice of parameters, the balls {Bij=Bri(pij)}i,j are pairwise disjoint. Whenever pBij, we set

ξ(p)=fi(|p-pij|)(p-pij),

while ξ(p)=0 otherwise (see Figure 1). One can suitably choose fi so that ξL(Bij)=1 for all i,j. Moreover, divξ=0 on +2 and thus for any ψCc1(2), by the Gauss–Green formula and owing to the definition of ξ, one has

Sψ[ξν]𝑑1=+2ξDψ=i,jBijξDψ=i,jBijψξνij=0,

so that [ξν]=0 on S. At the same time, ξ twists in any neighborhood of any point p0=(x0,0), x0(0,1), and the second component of the average of ξ on half balls centered at p0 has a lim inf strictly smaller than its lim sup as r0. In conclusion, the scalar product ξ(p)ν(p0) 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 ρε be the standard mollifier supported in a ball of radius ε>0. Then the regularized vector field ηε=ρεη is globally Lipschitz and satisfies (i)–(iii) of Definition 1 up to an ε-translation in the variable xn.

## Proof.

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

φ(|ηε(x)|)=φ(|Bεη(y)ρε(x-y)𝑑y|)
Bεφ(|η(y)|)ρε(x-y)𝑑y
Bεηn(y)ρε(x-y)𝑑y=ηnε(x).

Then we observe that ηε is globally Lipschitz, as a consequence of the boundedness of η, and verifies divηε=0 on n, that is, (ii). Moreover, for every x=(y,t) such that t-ε one has ηε(x)=0. Finally, up to a translation in the variable x, we can assume ηε(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+εen, defined for every t and pn by the Cauchy problem

{Φ(0,p)=p,tΦ(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(,):×nn is smooth, DpΦ|t=0=Id and the map Φ(t,):nn is a diffeomorphism for every t. Let us denote by Φn(t,p) the n-th component of Φ(t,p). Since Xn(p)ε for all pn, we have that tΦn(t,p)ε, hence the function tΦn(t,p) is strictly monotone and surjective. Therefore, for every h and pn there exists a unique T=T(p,h) such that Φn(T,p)=h. By the Implicit Function Theorem, T(p,h) is smooth and one has

hT(p,h)=Xn(Φ(T,p))-1,DpT(p,h)=-Xn(Φ(T,p))-1DpΦn(T,p).

Fix now an open, bounded and smooth set An-1 and h0>0. Let us set Ah0=A×{h0} and define the map

Ψ:A×(0,h0)n,Ψ(q,h)=Φ(T(p,h),p),

where we have set p=(q,h0). Before proceeding it is convenient to introduce some more notation. We write

Ψ=(Ψ1,,Ψn-1,Ψn)=(Ψ^,Ψn)
X=(X1,,Xn-1,Xn)=(X^,Xn)
y=(y1,,yn-1,yn)=(y^,yn)for yn.

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

hΨ(q,h)=Xn(Ψ)-1X(Ψ).

Note that Ψ(p)=Ψ(q,h0)=p and Ψn(q,h)=h by definition. Owing to the smoothness of Ψ, we first compute the partial derivative with respect to h of Ψji:=qjΨi, for i,j=1,,n-1:

hΨji=qjhΨi=qj[Xn(Ψ)-1Xi(Ψ)]
=Xn(Ψ)-1r=1nyrXi(Ψ)Ψjr-Xn(Ψ)-2Xi(Ψ)r=1nyrXn(Ψ)Ψjr
=r=1n[Xn(Ψ)-1yrXi(Ψ)-Xn(Ψ)-2Xi(Ψ)yrXn(Ψ)]Ψjr.

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

hDqΨ^=B(Ψ)DqΨ^,

where

B=Xn-1Dy^X^-Xn-2X^Dy^Xn.

Moreover, the determinant of DΨ coincides with that of DqΨ^. If we assume that qn-1 is fixed, and define δ(h)=detDqΨ^(q,h), we find by standard calculations that

δ(h)=tr[B(Ψ(q,h))]δ(h),

with initial condition δ(h0)=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 Fε(A):=Ψ(A×(0,h0)), depicted in Figure 2, is Lipschitz. Notice moreover that

|Ψ(q,h0)-Ψ(q,0)|0h0|hΨ(q,h)|𝑑h0h0Xn-1|X|𝑑hh0(c|η|+ε)-1(|η|+ε)max{h0,h0c},

thanks to the properties of η. Notice now that given a bounded An-1, there exists a constant R>0 depending only on A and h0, such that Fε(A) is contained in (-R,R)n for every ε>0. Setting Lε(A)=Ψ(A×{0}), by applying the Divergence Theorem on Fε(A) to the vector field X=η+εen we find the identity

0=Fε(A)divX=A(ηn(q,h0)+ε)𝑑q-εn-1(Lε(A)),

since the integral on the “lateral” boundary of Fε(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 XνFε(A)0. Using the fact that n-1(Lε(A))(2R)n-1 we can pass to the limit as ε0 in the identity above, obtaining that ηn vanishes on A×{h0}. By the arbitrary choice of both A and h0, and by property (iii) of Definition 1, we conclude that η=0 on n. ∎

### Figure 2

The evolution of Ah0 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,h0), i.e. the fact that their “lateral” perimeter is constantly equal to 2h0 (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 η20, by the Divergence Theorem we find

(3.1)-rr|η2(x1,t)|𝑑x1=-rrη2(x1,t)𝑑x1=0tη1(-r,x2)-η1(r,x2)dx22tη.

Therefore, η2(,t)L1(;) for all t>0 and

(3.2)η2(,t)12tη.

By combining (iii) of Definition 1 with (3.2) and the fact that η2 is Lipschitz, we infer that

(3.3)φ(|η1(x1,t)|)η2(x1,t)0as x1±,

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

limx1±η1(x1,t)=0.

This implies that

2ηη1(-r,t)-η1(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

limr+-rr|η2(x1,t)|𝑑x1=limr+0tη1(-r,x2)-η1(r,x2)dx2
=0tlimr+(η1(-r,x2)-η1(r,x2))dx2=0.

Hence, the L1-norm of η2(,t) is zero, thus η2(,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 xn, we set r=(x1,,xn-1) and z=xn. We consider vector fields ηC0(n) such that for r0 one has

(4.1)η(r,z)=(rf(r,z),h(r,z))

with f,hC1(+n{(0,z):z}). We have the following:

## Proposition 1.

Let ηC0(Rn) be as in (4.1). Define

(4.2)V(r,z):=-|r|n-10zf(r,s)𝑑s.

Then η satisfies

(4.3)η(r,z)=0for all z0,
(4.4)|η(x)|1for all xn,
(4.5)divη=0on n,
(4.6)ηn(x)c1|η(x)|2for all xn,

if and only if V(r,z) satisfies

(4.7)V(r,z)=0for all z0,
(4.8)|V(r,z)||r|n-2for all r>0 and z,
(4.9)rrV(r,z)c2|r|3-n(zV(r,z))2for all r>0 and z.

Moreover, one has

(4.10)η(r,z)=|r|1-n(-(zV)r,rrV).

## Proof.

We show in full detail the only if part. Since

divη(r,z)=divr(rf(r,z))+zh(r,z)=(n-1)f+rrf(r,z)+zh(r,z),

equation (4.5) is equivalent to

(4.11)zh=-(rrf+(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)=-0z(rrf(r,s)+(n-1)f(r,s))𝑑s.

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

(4.13)hc|r|2f2.

Let us set V as in (4.2) and observe that (4.3) implies V(r,z)=0 when z0. Then using (4.12) we obtain

rV=-|r|n-10z[rf(r,s)+(n-1)f(r,s)|r|-2r]𝑑s

and

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

so that in particular

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

and inequality (4.13) implies

rrVc|r|n+1f2c|r|3-n(zV)2.

Then by observing that (4.4) is equivalent to |r|2f2+h21, 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 n4.

## Proof.

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

V(r,z)={γ[(1+|r|n-1)1n-1-1]arctan(z2)if z0,0otherwise.

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

ρV=γarctan(z2)(1+ρn-1)2-nn-1ρn-2

and

zV=2γ[(1+ρn-1)1n-1-1]z1+z4.

Notice that V is of class C1 and V(ρ,z)=0 for all ρ>0 and z0. We obtain

|V|γarctan(z2)(1+ρn-1)2-nn-1ρn-2+2γ[(1+ρn-1)1n-1-1]z1+z4
(4.16)γπρn-22+γ3342[(1+ρn-1)1n-1-1],

where the second inequality follows from the maximization of the function z(1+z4) for z[0,+). Let us consider the function

φ(ρ)=(1+ρn-1)1n-1-1ρn-2.

As ρ0+, we have φ(ρ)ρn-1, while as ρ+ we have φ(ρ)ρ3-n. Moreover, when 0<t<1, one has (1+t)1n-11+tn-1, hence we deduce that

φ(ρ)ρn-11

when ρ<1, and

φ(ρ)ρ3-n1

when ρ1. Therefore by (4.16) and the last inequalities we find that

|V|Cγρn-2,

where

C=π+3342.

Assuming γC-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 ρρVρ3-n(zV)2, after separation of variables, becomes

arctan(z2)(1+z4)2z24γφ(ρ)2(1+ρn-1)n-2n-1.

We argue as for the upper bound of φ(ρ) (more precisely, we discuss the two cases ρ<1 and ρ1; in the first case we use the bound φ(ρ)1, while in the second case we use the fact that n4, and the inequalities φ(ρ)ρ3-n and 1+ρn-12ρn-1), and find

(4.17)4γφ(ρ)2(1+ρn-1)n-2n-123n-4n-1γ.

At the same time, by easy calculations we infer that the function arctan(t)(1+t2)2t is bounded from below by 1. Hence (4.9) is implied by the condition

γ24-3nn-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 n4 is given by

η(r,z)={γ|r|1-n(-2[(1+|r|n-1)1n-1-1]z1+z4r,arctan(z2)(1+|r|n-1)2-nn-1|r|n-1)if z>0,0if z0.

We remark that the vector field η is of class C0, 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 C1 function satisfying properties (4.7)–(4.9). Then V(r,z)=ψ1(|r|)ψ2(z) for suitable functions ψ1,ψ2 implies V0.

## Proof.

We set ρ=|r| and write (4.9) as

ρψ1(ρ)ψ2(z)cψ12(ρ)[ψ2(z)]2.

Let us assume by contradiction that ψ1 and ψ2 are not trivial, hence there exist z0>0 and ρ0>0 such that ψ2(z0)0, ψ2(z0)0, and ψ1(ρ0)0. Setting a=ψ2(z0) and b=ψ2(z0), we have four cases according to the sign of a and ψ1(ρ0). We discuss the first case a>0 and ψ1(ρ0)>0. We consider the differential inequality

ρψ1(ρ)γψ12(ρ)

with γ=cb2a>0. Therefore, the function ψ1 is increasing and by separation of variables and integration between ρ0 and ρ>ρ0 we get

(4.18)ψ1-1(ρ0)-ψ1-1(ρ)+ψ1-1(ρ0)γlog(ρρ0).

We denote by I=[ρ0,β) the maximal right interval of existence of the solution ψ1, for which ψ1>0. We can exclude the case β<+, as we would obtain by maximality that ψ1(ρ)+ as ρβ-, 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)=(rf(|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 z𝒟, without loss of generality (see [2, Theorem 2.56]) we shall assume x0S to be such that

1. (a)

the normal vector νS(x0) is defined at x0,

2. (b)

x0 is a Lebesgue point for the weak normal trace [zνS] of z on S, with respect to the measure n-1S,

3. (c)

|divz|(Br(x0)S)=o(rn-1) as r0+.

## Lemma 1.

Let η,{zk}k be vector fields in DM with supkzk<+. Let Σ,{Sk}k be oriented, closed Hn-1-rectifiable sets with locally finite Hn-1-measure, satisfying the following properties:

1. (i)

zkη in L-w*,

2. (ii)

n-1Skn-1Σ,

3. (iii)

|divzk|(nSk)0.

Then divη=0 in RnΣ.

## Proof.

Fix φCc1(nΣ) and set μk=divzk and μ=divη. By the Divergence Theorem coupled with the formula (see [1, Proposition 3.2])

μkSk=([zkνSk]+-[zkνSk]-)n-1Sk,

we have

nSkφ𝑑μk=-Skφ𝑑μk-nSkφzk
=-Skφ([zkνSk]+-[zkνSk]-)𝑑n-1-nφzk,

hence by (ii) and (iii)

|nφzk|nSk|φ|𝑑μk+2zk|φ|𝑑n-1Sk0as k.

This shows that

nφ𝑑μ=limknφ𝑑μk=limknφzk=0,

which proves the thesis. ∎

## Proposition 2.

Let zDM and S a closed, oriented Hn-1-rectifiable set with locally finite Hn-1-measure. Then, for Hn-1-almost-every x0S and for any decreasing and infinitesimal sequence {rk}k, the sequence zk of vector fields defined by zk(y)=z(x0+rky) converges up to subsequences to a vector field ηDM in L-w*, such that setting Σ=[νS(x0)], we have divη=0 on RnΣ and [ηνΣ]=[zνS](x0) on Σ.

## Proof.

We show that hypotheses (i)–(iii) of Lemma 1 are satisfied. Since zk=z for all k, thanks to Banach–Alaoglu Theorem (see also [5, Theorem 3.28]) we can extract a not relabeled subsequence converging to η𝒟, which gives (i). We set Sk=rk-1(S-x0), then thanks to (c) we have

(5.1)|divzk|(BRSk)=rk1-n|divz|(BRrk(x0)S)0as 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 C1 such that x0Ω and νS(x0)=νΩ. Defining Ωk=rk-1(Ω-x0), the proof of (ii) is reduced to showing that n-1Ωk weakly converge as measures to n-1H, where H is the tangent half-space to Ω at x0 (so that Σ=H). This fact is a consequence of Theorem 4. Now we can apply Lemma 1 and obtain divη=0 on nΣ. In order to prove the last part of the statement we have to show that for any ψCc1(n) one has

Hψddivη+Hψη=τ0Hψ𝑑n-1,

where τ0=[zνS(x0)]. Since we have proved that divη=0 on nΣ, we only have to show that

(5.2)Hψη=τ0Hψ𝑑n-1.

Note that by the convergence of zk to η, the Lloc1-convergence of Ωk to H, and the convergence of the measures n-1Ωk to n-1H, we have

(5.3)Hψη-τ0Hψ𝑑n-1=limkΩkψzk-τ0Ωkψ𝑑n-1.

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

|Ωkψzk-τ0Ωkψ𝑑n-1||Ωkψ([zkνΩk]-τ0)𝑑n-1-Ωkψddivzk|
ψΩksptψ|[zkνΩk]-τ0|+nΩk|ψ||divzk|0as k.

Combining this last fact with (5.3) implies (5.2) at once, and concludes the proof. ∎

By relying on Proposition 2 and on Theorem 3, we are now able to prove the main result of the section, i.e. Theorem 4 which states the existence of the classical trace for a divergence-measure vector field having a maximal weak normal trace on a oriented 1-rectifiable set S.

## Proof of Theorem 4.

Without loss of generality, up to a translation we can suppose x0=0 and up to a rotation that νS(x0)=-e2. Moreover, up to rescaling ξ, we can suppose ξ=1. Let

B1+=B1+2.

We then want to show that the set

Nα:={xB1+:|ξ(x)+e2|α}

has density zero at 0 for all α>0. Argue by contradiction and suppose there exist α,β>0 and a sequence of radii {rk}k decreasing to zero such that

(5.4)|NαBrk|πrk2βfor all k.

Define z0(x):=ξ(x)+e2 and the sequence zk(y)=z0(rky) for k. Since the second component of zk is zk,2(x)=ξ2(rkx)+1, one easily sees that

(5.5)zk,2(x)|zk(x)|22,

for almost every x2. By the definition of Nα and by (5.5), the contradiction hypothesis (5.4) reads equivalently as

(5.6)|{xrk-1B1+:zk,2(x)>α22}B1|πβ.

On top of that, z0𝒟 with z02. By Proposition 2 the sequence zk defined above converges in L-w* (up to subsequences, we do not relabel) to a vector field η such that div(η)=0 on +2 and [η(-e2)]=[z0(-e2)](0) on 02. We aim to show that η satisfies hypotheses (i)–(iii) of Theorem 3. Were this the case, one would conclude η0 in +2 and this would yield a contradiction with (5.6). Indeed, taking χB1+ as a test function, we get

πα2β2B1+zk,2𝑘B1+η2.

On the one hand, we know that hypothesis (i) of Theorem 3 is satisfied as div(η)=0 on +2. On the other hand, as [ην]=-[z0e2](0) on 02, we get that

[ην]=-[ξe2](0)-e2e2=ξ-1=0,

so that hypothesis (ii) of Theorem 3 holds as well. We are left to show that hypothesis (iii) of Theorem 3 is satisfied. Since zk is equibounded, by (5.5) we infer as well that |zk|2 converges (up to subsequences, we do not relabel) to some function ζ in L-w*. Clearly, one has from (5.5) and the weak-* convergence of zk and of |zk|2 that ζ2η2 almost everywhere. We want to prove that the same holds with |η|2 in place of ζ so to retrieve hypothesis (iii) of Theorem 3 with the choice φ(t)=t22. Take a probability measure fdx with fL1(2). Then, by Jensen’s inequality,

|zk|2f𝑑x|zkf𝑑x|2=(zk,1f𝑑x)2+(zk,2f𝑑x)2.

As k, by the weak-* convergence we get

ζf𝑑x(η1f𝑑x)2+(η2f𝑑x)2.

Thus, by letting fdx toward the Dirac measure centered at x, (iii) follows at once for almost-every point x (more precisely, x must be a Lebesgue point for the functions ζ,η1,η2). A direct application of Theorem 3 yields the desired contradiction. ∎

## Corollary 3.

Let ΩR2 be weakly-regular and let ξDM(Ω). Then, for H1-a.e. x0Ω such that [ξνΩ](x0)=ξ one has

ap-limxx0-ξ(x)=ξνΩ(x0).

## Proof.

Thanks to the Gauss–Green formula (2.1) in the special case ψ=1, one deduces that the vector field ξ~ defined as ξ~=ξ on Ω and ξ~=0 on 2Ω belongs to 𝒟(2). The conclusion is achieved by applying Theorem 4 with ξ~ and Ω in place of, respectively, ξ and S. ∎

### 5.1 An application to capillarity in weakly-regular domains

The trace property that we have studied in the last section is motivated by the study of the boundary behavior of solutions to the prescribed mean curvature equation in domains with non-smooth boundary (see [28]). Let us consider the vector field

Tu=u1+|u|2

associated with any given uWloc1,1(Ω). We say that u is a solution to the prescribed mean curvature equation if

(PMC)divTu=Hon Ω

in the distributional sense, where H is a prescribed function on Ω. One of the main results of [28] is the following theorem.

### Theorem ([28, Theorem 4.1]).

Let ΩRn be a weakly-regular domain and let H be a given Lipschitz function on Ω. Assume that the necessary condition for existence of solutions to (PMC) holds, that is,

|AH(x)𝑑x|<P(A)for all AΩ such that 0<|A|<|Ω|.

Then the following properties are equivalent:

1. (E)

(Extremality)|ΩHdx|=P(Ω).

2. (U)

(Uniqueness) ((PMC)) admits a solution u which is unique up to vertical translations.

3. (M)

(Maximality) Ω is maximal for ((PMC)), i.e. no solution can exist in any domain strictly containing Ω.

4. (V)

(Weak verticality) There exists a solution u which is weakly-vertical at Ω, i.e.

[Tuν]=1n-1-a.e. on Ω,

where [Tuν] is the weak normal trace of Tu on Ω.

We remark that, in the relevant case of H a positive constant, the extremality property (E) is equivalent to Ω being a minimal Cheeger set (i.e. Ω is the unique minimizer of the ratio P(A)|A| among all measurable AΩ with positive volume, see for instance [26, 27, 29, 31, 32]) and H equals the Cheeger constant of Ω. In dimension n=2, this extremal case corresponds exactly to capillarity in zero gravity for a perfectly wetting fluid that partially fills a cylindrical container with cross-section Ω. We also stress that the uniqueness property (U) holds in this case without any prescribed boundary condition; this means that the capillary interface in Ω× only depends upon the geometry of Ω. Another important remark should be made on the verticality condition (V), which corresponds to the tangential contact property that characterizes perfectly wetting fluids. In [28] this condition is obtained under the weak-regularity assumption on Ω, which somehow justifies the presence of the weak normal trace in the statement (see for instance in Figure 3 an example of non-Lipschitz, weakly-regular domain built in [29] and covered by [28, Theorem 4.1]).

### Figure 3

A non-Lipschitz, weakly-regular domain. Originally appeared in [28, 29].

Nevertheless, in the physical 3-dimensional case (i.e. when Ω2), the weak-verticality (V) improves to strong-verticality, that is, the trace of Tu exists and is equal to νΩ almost-everywhere on Ω thanks to Theorem 4 and to Corollary 3. However, we remark that the strategy of proof strongly relies on the rigidity property, which we have been able to prove only in dimension n=2. It is an open question whether the weak-verticality condition always improves to the strong-verticality given by the existence of the classical trace of Tu at n-1-almost-every point of Ω, and more generally if Theorem 4 holds in any dimension.

Communicated by Frank Duzaar

Funding source: Istituto Nazionale di Alta Matematica

Award Identifier / Grant number: U-UFMBAZ-2019-000473 11-03-2019

Funding statement: Gian Paolo Leonardi and Giorgio Saracco have been partially supported by the INdAM-GNAMPA 2019 project “Problemi isoperimetrici in spazi Euclidei e non” (n. prot. U-UFMBAZ-2019-000473 11-03-2019).

# Acknowledgements

The authors would like to thank Carlo Mantegazza for fruitful discussions on the topic, and Guido De Philippis for suggesting the “flow-tube” strategy used to prove Theorem 2.

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