Weak notions of jacobian determinant and relaxation

DOI:10.1051/cocv/2010047 www.esaim-cocv.org

WEAK NOTIONS OF JACOBIAN DETERMINANT AND RELAXATION

Guido De Philippis

Abstract. In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely thedistributional Jacobianand therelaxed total variation, which in general could be different.

We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals.

Mathematics Subject Classification.49J45, 28A75.

Received March 15, 2010. Revised July 15, 2010.

Published online December 2, 2010.

1. Introduction

The aim of this paper is to study weak notions of Jacobian determinant, detDu, for mapsu: Ω⊂Rⁿ→Rⁿ in the Sobolev classW^1,p for somep.

If uis a diffeomorphism, the change of variable formula and Lebesgue differentiation Theorem give a clear geometric meaning to detDu(x), it is the “infinitesimal” change of volume due to the deformationu. If uis not a bijective map the area formulas (in the unoriented version (1.1) or in the oriented one (1.2))

Ω|detDu(x)|dx=

RⁿN(u,Ω, y)dy (1.1)

Ω

detDu(x)dx=

Rⁿ

deg(u,Ω, y)dy (²) (1.2)

relate the integral of the Jacobian determinant to how many times the image ofucovers the target space.

If uis merely a Sobolev map it is still possible to consider the area formula and the “pointwise” Jacobian detDu (see [26]), however ifuis not sufficiently regular (more precisely ifu∈W^1,p andp < n) this gives only a partial information about the behaviour ofu.

Ifp≥n, H¨older inequality implies detDu∈L^np; moreover the map W^1,p→L^np

u→detDu

Keywords and phrases. Distributional determinant, topological degree, relaxation.

1 Scuola Normale Superiore, P.za dei Cavalieri 7, 56100 Pisa, Italy. guido.dephilippis@sns.it

2N(u,Ω, y) = #{x∈Ω :u(x) =y}is the Banach indicatrix function, while deg(u,Ω, y) is the Brouwer degree ofu(see Sect.2.2).

Article published by EDP Sciences c EDP Sciences, SMAI 2010

is continuous if we endow both spaces with the strong topology. What is more surprising is that if p > nthis map is still (sequentially) continuous also if we endow the spaces with the weak topology (see Thm. 2.2). If p= nwe don’t have continuity if we consider the L¹ weak topology for Jacobians, however we have it if we consider the weak-∗ topology (see for example [12], Chap. 8), however if detDu_k ≥0 we still have continuity with respect to theL¹weak topology (see [10,35]).

This implies the semicontinuity with respect to the weak convergence of the functional:

u→T V(u,Ω) :=

Ω|detDu| (1.3)

and, more in general, of polyconvex functionals,i.e. the ones that can be represented as the integral of a convex function of the minors of the gradient:

F(u,Ω) =

Ωg(Du,M1(Du), . . . ,detDu).

Ifp < n, in general detDuis not a summable function; moreover also if detDu∈L¹we lose continuity and semicontinuity properties. In particular it is possible to see that:

• the functionu(x) =_|x|^x is inW^1,p(B) for anyp < n, detDu= 0 almost everywhere but for any sequence of smooth functions strongly converging touin W^1,p(B)∩L^∞ withp > n−1 we have:

detDu_k ω^* _nδ₀ (³) in the sense of distributions.

• For any smooth function u there exists a sequence{u_k} ∈W^1,p, withp < n, weakly converging to u and such that detDu_k= 0 almost everywhere (see Ex. 3 at p. 284 in [26] where it is shown for the map u(x) =x, this implies the result for any smooth function).

The main reason for this behaviour is that the pointwise Jacobian detDu doesn’t count in any way the presence of fractures in the image ofu.

We now introduce the two weak formulations we are going to study. The first one is based on the particular structure of the Jacobian determinant and it leads to the notion of distributional determinant (introduced by Ball [3] in the context of non-linear elasticity and widely studied also in different contexts, see for example [1,9,11,14,15,28,34,36–38] and references therein). The second one is based on the Lebesgue-Serrin extension and it leads to therelaxed total variation (first introduced by Marcellini in [30] and then systematically studied in [6,20–23,31–33,41]).

Letube a C² map, thanks to the divergence free property of the cofactor matrix (see Sect.2 for a precise

definition)

i

∂_j(adjDu)^j_i = 0 (1.4)

we can express the Jacobian as a divergence detDu(x) = 1

n

i

∂_j

j

uⁱ(x)(adjDu(x))^j_i

= 1

ndiv(adjDu·u).

Therefore formal integration by parts suggests that we may define the distribution:

J u, ϕ:=−1 n

Ω(adjDu·u)·Dϕ ϕ∈ D(Ω) (⁴). (1.5)

3We denote withωnthe Lebesgue measure of the unit ballBand withδ0 the usual Dirac distribution.

4D(Ω) is the usual space of test functions.

Notice that for smooth maps we haveJ u= detDuand this can be extended by density tou∈W_loc^1,n. However the distribution (1.5) is well defined as far asuadjDu∈L¹_loc, for example ifu∈W^1,p∩L^∞andp∈(n−1, n).

The distributional Jacobian enjoys continuity properties with respect to weak convergence and may differ from detDu ifp < n. For example ifu(x) =_|x|^x we have

J u=ω_nδ₀, while detDu= 0.

IfJ u is a Radon measure andu∈W^1,p withp≥n−1 we have the following theorem (see [36] for the case p > _n+1ⁿ² and [16] for the casen−1≤p≤ _n+1ⁿ² ).

Theorem 1.1. Let u∈W^1,p(Ω)∩L^∞_loc, if J uis a Radon measure then J u= detDudLⁿ+J u^S.

The second formulation we consider is the Lebesgue-Serrin extension of the functional (1.3) (see Sect.2.4for a more detailed discussion). We define for everyp∈(n−1, n)

T V^p(u,Ω) = inf

lim inf

k→∞

Ω|detDu_k|: u_k∈W_loc^1,n, u_k uinW^1,p

. (1.6)

If u∈ W^1,n and p > n−1 semicontinuity results below the natural growth exponent (see [13,30,31]) ensure that:

T V^p(u,Ω) :=

Ω|detDu|.

Observe we had underlined the dependence on p, however in all known example we have that T V^p(u,Ω) = T V^q(u,Ω), as long asp, q∈(n−1, n) and u∈W^1,p∩W^1,q, it seems natural to conjecture this to be always true.

Recently it has been shown by Schmidt [43] that the relaxed total variation enjoys the following quasi- convexity property⁵:

T V^p(u_A,Ω)≤T V^p(u_A+ϕ,Ω) for any affine mapu_A and deformationϕ∈W₀^1,p.

When p∈ (n−1, n), u ∈L^∞ and T V^p(u,Ω) <∞, then T V^p(u,·) can be extended to a Radon measure on Ω. Moreover in this case it is possible to show that alsoJ uis a Radon measure and that, for any open subset A⊆Ω

|J u|(A)≤T V^p(u, A) (1.7)

where|J u|is the total variation of the measureJ u(see [22,23] and Prop.3.2below).

In this paper we discuss equality cases in previous inequality (Thms.4.3and5.6) and we show how to obtain explicit representation formulas for the extension of general polyconvex functionals satisfying suitable growth conditions (Thms.6.1and6.2).

The paper is structured as follows. In Section2 we recall some preliminary results, in particular the notion of Brouwer degree for Sobolev maps, the measure representation theorem and a theorem of Brian White about minimal mass currents. In Section 3 we begin the comparison between|J u| and T V^p showing which are the main obstructions to equality in (1.7). In Section 4 we prove that equality holds for maps with values in the (n−1)-dimensional sphere. In Section 5, inspired by M¨uller and Spector INV condition [38], we introduce

5Recall that a mapf:M^n×n→Ris said to be quasi-convex if f(ξ)|B| ≤

Bf(ξ+Dϕ) ∀ϕ∈C₀¹(B) i.e. every affine map is a minimizer with respect to its boundary condition.

a particular class of Sobolev maps for which we are able to obtain equality in (1.7). In Section6 we show the general relaxation result.

2. Notations and preliminary results

Throughout this paper Ω is an open subset ofRⁿ with Lipschitz boundary. Points in Rⁿ will be denoted byx, theirith component byxⁱ,|x|is the usual Euclidean norm and we will also consider the ∞-norm:

|x|_∞= max{|xⁱ|: i= 1, . . . , n}.

We writeB(x₀, r) for{x:|x−x₀|< r}andQ(x₀, r) for{x: |x−x₀|∞< r}, if it is clear from the context we will drop the dependence onx₀and simply writeB_randQ_r. CandM will be constants, possibly depending on previous ones, whose value could change from line to line. Lⁿ andH^k will denote the n-dimensional Lebesgue measure and thek-dimensional Hausdorff measure, we will write dxto mean dLⁿ. Given a measureμwe denote with μ U the restriction of μ to U and with |μ| its total variation, we write μ^a and μ^S respectively for its absolutely continuous and singular part with respect to the Lebesgue measure. If f is a functionf U means its restriction toU.

For any matrixA∈M^n×n and vectorsv, w∈Rⁿ we will denote with:

• A^j_i the entry in thejth row andith column,A^T is the transpose ofA,A·v the usual action of a matrix on a vector andw·v the Euclidean scalar product betweenv andw;

• Mh(A) ∈ R^τ, with τ = _n

h 2

, the vector of minors of order h of A (i.e. the vector formed by the determinants of allh×hsubmatrices ofA);

• adjAthen×nmatrix that satisfies:

(adjA)A= (detA)Id i.e. the transpose of the cofactor matrix ofA.

2.1. Sobolev maps and precise representative

We will denote withW^1,p(Ω;Rⁿ) the space of Sobolev maps and we refer to [17] for its main properties and notations. In the sequel we will need to consider the restriction of a function uto lower dimensional subsets, mainly to the boundary∂Dof Lipschitz subsets. This can be done in two ways:

(1) considering the traceγ(u)∈L^p(∂D);

(2) considering the pointwise value of the precise representative of the class ofudefined by:

u(x) =

⎧⎨

⎩

r→0lim 1

|B(x, r)|

B(x,r)u(y)dy if the limit exists

0 otherwise.

It turns out that u =u Hⁿ⁻¹-almost everywhere and γ(u) equals u ∂D. Being this understood we will simply refer to the restriction ofuto∂D.

Given a domainDwith Lipschitz boundary we say that a function is inW^1,p(∂D) if its restriction satisfies:

∂D|u|^p+|D^Tu|^pdHⁿ⁻¹<∞

whereD^Tu(x) =Du(x) T_x(∂D)⁶. The definition could equivalently be given locally by flattening the boundary.

With an abuse of notation we say thatu∈W^1,p(D) ifu∈W^1,p(D)∩W^1,p(∂D).

6Tx(∂D) is the tangent space definedHⁿ⁻¹ ∂D-almost everywhere.

Proposition 2.1. If u∈W^1,p(D) there exists a sequence of Lipschitz maps u_k converging to uin W^1,p(D), i.e.

u−u_kW1,p(D):=u−u_kW^1,p_(D)+u−u_kW^1,p_(∂D)→0.

Proof. By standard arguments we can reduce to the case whereu∈W^1,p(B⁺) and sptuis a compact subset of B⁺={x= (x, xⁿ) :|x|<1, xⁿ≥0}. Moreover if we consider the sequence:

u_k(x) =

u(x,0) if 0≤xⁿ≤ ¹_k u(x, xⁿ−_k¹) ifxⁿ≥ ¹_k

we have thatu_k →uinW^1,p(B⁺) so we may supposeuto be a constant function ofxⁿfor smallxⁿ, say smaller thanδ. Consider now a sequence of smooth maps{v_k}converging to uinW^1,p(D). Then we have

_δ

0

Rⁿ⁻¹|u(x, t)−v_k(x, t)|^p+|Du(x, t)−Dv_k(x, t)|^pdxdt→0.

Hence there exists ans∈(0, δ) such that, up to subsequences:

Rⁿ⁻¹|u(x, s)−v_k(x, s)|^p+|Du(x, s)−Dv_k(x, s)|^pdx→0.

Recalling thatu(x,0) =u(x, s) it is easy to see that the sequence defined by:

u_k(x) =

v_k(x, s) if 0≤xⁿ ≤s v_k(x, xⁿ) ifxⁿ ≥s

satisfiesu_k−u_W1,p(B⁺)→0.

We end this section by stating the following theorem about continuity of minors (see [12], Thm. 8.20).

Theorem 2.2 (Reshetnyak). Let u_k ∈W^1,p(Ω) weakly converging tou, ifh < p we have:

Mh(Du_k)Mh(Du) in L^ph(Ω). (2.1)

2.2. Brouwer degree

In [38] (see also [7,8,19,25,39,40]) a definition of Brouwer degree has been given for the class of, possibly discontinuous, Sobolev mapsW^1,pwithp > n−1 (see [11] for the casep=n−1). Here we report some of its main properties.

Recall that for a smooth mapv: Ω→Rⁿ the Brouwer degree is defined for anyy∈Rⁿ\v(∂Ω) by:

deg(v,Ω, y) :=

x∈v⁻¹({y})

sign(detDv(x)).

Moreover the oriented area formula (1.2) says that for anyh∈L¹(Rⁿ;R):

Rⁿh(y) deg(v,Ω, y)dy=

Ωh(v(x)) detDv(x)dx. (2.2)

Recalling equation (1.4) we have for anyg∈C¹(Rⁿ;Rⁿ)∩W^1,∞:

Rⁿdeg(v,Ω, y) div_yg(y) dy=

Ωdiv_yg(v(x)) detDv(x) dx

=

Ω

i

∂gⁱ

∂yⁱ(v(x)) detDv(x) dx=

Ω

i,m

∂gⁱ

∂y^m(v(x))δ^m_i detDv(x) dx

=

Ω

i,m,k

∂gⁱ

∂y^m(v(x))∂v^m

∂x_k(x)(adjDv(x))^k_i dx

=

Ω

k,i

∂

∂x^k

gⁱ(v(x)) (adjDv(x))^k_i dx

=

Ω

k,i

∂

∂x^k

gⁱ(v(x))(adjDv(x))^k_i

dx

=

Ωdiv_x(adjDv(x)·g(v(x))) dx=

∂Ω(adjDv(x)·g(v(x)))·ν_∂ΩdHⁿ⁻¹. The previous equations reveal the following facts:

(1) deg(v,Ω, y) depends only onv ∂Ω;

(2) deg(v,Ω,·) is a BV function (see [17] for the definition and for the main properties ofBV functions), moreover:

|D_y(deg(v,Ω, y))|(Rⁿ)≤

∂Ω|adjDu| ≤c

∂Ω|Du|ⁿ⁻¹ (2.3)

and sptD_y(deg(v,Ω, y))⊂v(∂Ω).

According to this we give the following definition:

Definition 2.3. Letu∈W^1,p(Ω,Rⁿ) withp∈(n−1, n). For everyD⊂⊂Ω with Lipschitz boundary and such that u∈W^1,p(D), we define the degree ofuonD, Deg(u, D, y), as the onlyBV(Rⁿ,Z) function satisfying:

RⁿDeg(u, D, y) divg(y)dy=

∂D(adjDu·g(u(x)))·ν_∂DdHⁿ⁻¹ (2.4) for everyg∈C¹(Rⁿ;Rⁿ).

Using Proposition2.1and equation (2.3) it is easy to see that Deg(u, D, y) is well defined.

It is clear from the definition that Deg(u, D, y) depends only on the value ofurestricted to∂D. By Sobolev embedding Theorem, we have that u∈ C⁰(∂D), and classical theorems imply that there exists a continuous functionϕ:D→Rⁿ such thatϕ ∂D=u ∂D. It is possible to show (see [38]) that:

deg(ϕ, D, y) = Deg(u, D, y).

The Brouwer degree enjoys the following continuity property:

Proposition 2.4. Letu_k, u∈W^1,p(Ω,Rⁿ), then for every domainD⊂⊂Ωwith Lipschitz boundary such that u_k uinW^1,p(∂D)

we have:

Deg(u_k, D,·)→Deg(u, D,·) in L¹(Rⁿ). (2.5)

Proof. Thanks to the weak convergence and the Sobolev embedding Theorem we have supu_kL^∞(∂D)+DuL^p(∂D)≤C.

Equation (2.3) ensures that the sequence{Deg(uk, D,·)}is relatively compact inL¹(Rⁿ). Calld(y)∈BV(Rⁿ,Z) its limit. Recalling that, thanks to our hypothesis,

(adjDu_k)^T ·ν_∂D(adjDu)^T ·ν_∂D

inL^n−1p (∂D) andg(u_k)→g(u) in anyL^q(∂D) (see Thm.2.2) we can pass to the limit in both sides of (2.4) to show

d(y) = Deg(u, D, y).

2.3. White’s Theorem

In this section we report a theorem about approximation of minimal mass rectifiable current due to White [44].

We will use it only in the simplified form expressed by Proposition2.6which doesn’t involve currents.

Recall that ak-current inRⁿ,T ∈ Dk(Rⁿ) is defined as linear continuous functional on the space of compactly supportedk-formsD^k(Rⁿ). A currentTis saidrectifiable,T ∈ Rk(Rⁿ), if its action on ak-form can be expressed as:

T(ω) =

M

ω(x), τ_M(x)θ(x)dH^k

where M is an H^k-rectifiable set, τ_M(x) is a k-vector orienting T_xM and θ(x) is an integer-valued positive function (see [18,26] for a precise definition and the statements of the theorems we use in the sequel). If T ∈ D_k(Rⁿ) we define:

• theboundary ∂T ∈ D_k−1(Rⁿ) as∂T(ω) =T(dω);

• themass M(T) = sup{T(ω) :|ω| ≤1};

• thepush-forward by a smooth proper map f as f_#T(ω) =T(f^#ω)⁷. We can now state:

Theorem 2.5 (White). Let f :∂B(0,1)⊂Rⁿ→R^k be a Lipschitz map and3≤n≤k then:

min

M(T), T ∈ Rn(R^k), ∂T=f_#[[∂B]]

= inf

BJ_ng(x)dx, g:B→R^k, g Lipschitz, g ∂B=f

whereJ_ng=

detDg^TDg.

Notice the restriction n ≥ 3, which is a consequence of Hurewicz Theorem about the relation between homotopy and homology groups (see [27]). Example3.5shows that this restriction is sharp.

As said before we will use the theorem only in the following form:

Proposition 2.6. Let u: B ⊂Rⁿ → Rⁿ be Lipschitz, n ≥3. Then for every σ > 0 there exists g:B →Rⁿ Lipschitz such that g=uon ∂B and:

B|detDg(x)|dx≤

Rⁿ|deg(u, B, y)|dy+σ.

Proof. We have only to show that:

Rⁿ|deg(u, B, y)|dy= min

M(T), T ∈ Rⁿ(Rⁿ)∂T =u_#[[∂B]]

.

7f^#ωis the usual pull-back of a differential form.

We will show more, namely that the current T =u_#[[B]] =Lⁿ deg(u, B, y)e₁∧. . .∧e_n is the only finite mass current in the class of competitors of the previous minimum.

First of all notice thatu_#[[∂B]] =∂u_#[[B]] and:

u_#[[B]](ψ(y)dy¹∧. . .∧dyⁿ) = [[B]](ψ(u(x))du¹∧. . .∧duⁿ)

=

Bψ(u(x)) detDudx=

Rⁿψ(y) deg(u, B, y)dy so that:

M(T) =

Rⁿ|deg(u, B, y)|dy.

Let nowS ∈ Rⁿ(Rⁿ) be such that∂S =∂T, then by Constancy Theorem (see [18] 4.1.7) we have that there existsc∈Rsuch that:

S=T +c[[Rⁿ]]

and so M(S) =∞unlessc= 0.

2.4. Lebesgue-Serrin extension

We briefly recall the definition of Lebesgue-Serrin extension of a functional below its natural growth exponent.

See [21,30,31] and references therein for a more detailed exposition and [6,20] for the proof of the cited results.

Suppose we have a continuous functionf :M^n×n→[0,∞) satisfying the growth assumption:

f(ξ)≤c(1 +|ξ|)^q and define:

F(u,Ω) =

Ω

f(Du(x)).

Then F is finite and strongly continuous only onW^1,q (in our case we are considering f(ξ) = |detξ| and so q=n). There is a standard way to extendF toW^1,p forp < qby defining:

F^p(u,Ω) = inf

lim inf

k→∞ F(u_k,Ω) u_k∈W_loc^1,q, u_k uin W^1,p

.

We remark that the choice of sequences in W_loc^1,q prevents some problems at the boundary of Ω if this is not smooth, see [20]. Observe that if F is p-coercive (i.e. F(u) ≥ cu_W^1,p) then F^p is the largest lower- semicontinuous functional belowF.

In generalF cannot be (extended to) a measure in the second variable, however we have the following result:

Theorem 2.7. Let p < q < p_n−1ⁿ and let u∈W^1,p be such that F^p(u,Ω) <∞. Then there exists a Radon measureμ^p_u on Ωsuch that:

μ^p_u(A) =F^p(u, A)

for any open set A⊆Ω. Moreover if f is a quasi-convex function we have that the Radon-Nykodim derivative of μ^p_u equalsf, that is:

dμ^p_u

dLⁿ(x) =f(Du(x)) almost everywhere in Ω.

In the sequel we show, under appropriate hypotheses, how to characterize the singular part ofμ^p_u.

We end this section by recalling two classical theorems about continuity and semi-continuity for functionals defined on the space of Radon measures (see [2] for a proof).

Ifg:R→[0,∞) is a convex function, we define therecession function as:

g_∞(s) := lim

t→+∞

g(st)−g(0)

t ,

which turns out to be a positively homogeneous, lower semicontinuous function. Moreover if g(t)≤c(1 +|t|) we have thatg_∞ is locally bounded and hence continuous.

Theorem 2.8. Let μbe a positive Radon measure andg:R→[0,∞)be a convex function. Then the functional defined on M(Ω;R)⁸by:

G(λ) =

Ωg

dλ dμ

dμ+

Ωg_∞

dλ d|λ|

d|λ|

is lower semicontinuous with respect to weak-* convergence.

Theorem 2.9 (Reshetnyak). Leth:Rⁿ→[0,∞)be a continuous, convex and positively homogeneous function and for any λ∈ M(Ω;Rⁿ)consider:

H(λ) =

Ωh

dλ d|λ|

d|λ|.

Then if μ_k μ^* and|μ_k|(Ω)→ |μ|(Ω) we have:

H(μ_k)→H(μ).

3. T V ^VS. Ju

First of all we show the continuity of the distributional Jacobian:

Proposition 3.1. Let {u_k} ⊂W^1,p(Ω)such that u_k u, supposep > n−1 and supk uk_L^∞_(K)≤C(K)

for any compactK⊂Ω, then

J u_k J u^*

as distributions.

Proof. We have to prove that for everyϕ∈ D(Ω):

j

sptϕuⁱ_k(adjDu_k)^j_i∂_jϕ→

j

sptϕuⁱ(adjDu)^j_i∂_j, ϕ.

From Theorem2.2we have adjDu_kadjDuin L^n−1p , moreover the sequence isL^∞bounded on sptϕ, hence we haveuⁱ_k∂_jϕ→uⁱ∂_jϕin L

p−(n−1)p =

L

n−1p _∗

, so the thesis follows.

8M(Ω;R) is the set of finite Radon measures on Ω.

It is clear from the proof that if u_k uinW^1,pwithp > _n+1ⁿ² thenJ u_k J u^* as distributions without any boundedness assumptions, in fact in this case Sobolev embedding theorem implies thatu_k →uin L

p−(n−1)p . Consequence of this is the following comparison result (see [22,23]):

Proposition 3.2. Suppose p∈(n−1, n) andu∈W^1,p(Ω;Rⁿ)∩L^∞_loc satisfiesT V^p(u,Ω)<∞, then J u is a Radon measure and for any open subset A⊂Ω:

|J u|(A)≤T V^p(u, A). (3.1)

Proof. Supposeu∈L^∞(Ω) and letu_k∈W_loc^1,n(Ω) be a sequence weakly converging touand satisfying lim inf

k→∞

Ω|detDu_k|<∞.

Applying LemmaA.1we can find a sequence{v_k}, also weakly converging tou, such thatv_k∞≤2u∞and, for any open setA⊂Ω,

lim inf

k→∞

A|detDv_k| ≤lim inf

k→∞

A|detDu_k|. (3.2)

Then, thanks to Proposition3.1, detDv_kdLⁿis a sequence of Radon measures with equi-bounded total variation which converges to J u in the sense of distributions, hence the first part of the theorem follows. The second part is a consequence of equation (3.2) and of the semicontinuity of the total variation with respect to weak-∗

convergence of measures.

Let nowube inL^∞_loc(Ω), by the previous part the distributionJ uis a Radon measure when restricted to any open setA ⊂⊂Ω, moreover for anyϕ∈ D(Ω) we can choose an open set U such that sptϕ⊂U ⊂⊂ Ω, then we have

J u, ϕ ≤ ϕ_∞T V^p(u, U)≤ ϕ_∞T V^p(u,Ω),

and hence, thanks to the Riesz theorem,J uis a Radon measure also in Ω. Equation (3.1) can now be extended

to any open setA⊂Ω by approximation.

Equality in (3.1) is not achieved in general. The reasons for this gap being essentially of two types:cancellation andtopological obstruction. In order to show this we prove two propositions which link the Brouwer degree for Sobolev maps to the Distributional Jacobian and the Total Variation. The first one is well known (see [28]) while the second one is an immediate generalization of Theorem 1.4 of [33].

We call a map u∈W^1,p(B(0,1);Rⁿ) 0-homogeneous if there exists a Lipschitz mapϕ: Sⁿ⁻¹=∂B →Rⁿ such thatu(x) =ϕ(_|x|^x). Observe thatu∈Lip_loc(B\ {0})∩W^1,pfor everyp < n.

Proposition 3.3. Let ube a0-homogeneous map, then:

J u=

RⁿDeg(u, B, y)dy

δ₀.

Proof. Letv:B →Rⁿ be a Lipschitz map coinciding withϕon∂B, then:

Deg(u, B, y) = deg(v, B, y).

Moreover the sequence:

u_ε(x) = ϕ_x

|x| ifε <|x|<1 v_x

ε if|x|< ε

converges touinW^1,p. Thanks to Proposition3.1we have:

detDu_ε J u^* in the sense of distributions. Letψbe a test function, then:

ψ(x) detDu_ε(x) = 1 εⁿ

B(0,ε)ψ(x) detDv x

ε

dx

=

B(0,1)ψ(εx) detDv(x)dx→ψ(0)

B(0,1)detDv and thanks to (1.1) we have:

BdetDv(x)dx=

Rⁿdeg(v, B, y)dy.

Proposition 3.4. Let u:B →Rⁿ be a0−homogeneous map andn≥3. Then for anyp∈(n−1, n):

T V^p(u, B) =

Rⁿ|Deg(u, B, y)|dy.

Proof. Letwbe a Lipschitz map that agrees withϕon∂B. Thanks to Proposition2.6, for any positiveσthere exists a Lipschitz mapg which agrees withwon∂B and such that:

B|detDg| ≤

Rⁿ|deg(w, B, y)|+σ=

Rⁿ|Deg(u, B, y)|+σ.

Considering

u_ε(x) = ϕ_x

|x| ifε <|x|<1 g_x

ε if|x|< ε we have

T V^p(u, B)≤

Rⁿ|Deg(u, B, y)|dy.

To prove the reverse inequality we recall that for 0-homogeneous maps the approximation sequences could be chosen to satisfyu_k =uon∂B(see [22], Lem. 22). So for any such sequence we have, thanks to the unoriented version of the area formula:

B|detDu_k|=

Rⁿ#{x∈Ω :u_k(x) =y}dy

≥

Rⁿ|deg(u_k, B, y)|dy=

Rⁿ|Deg(u, B, y)|dy.

It is clear now that for 0-homogeneous maps and n ≥ 3 a necessary and sufficient condition for equality in (3.1) is that Deg(u, B, y) does not change sign (i.e. there must not be cancellation).

The following example, considered for the first time by Mal´y in [29], shows that the situation for n= 2 is more involved (see also [22–24]). Some kind of homotopic topological obstructions prevents the equality in (3.1).

Example 3.5. Consider the following functionϕ:S¹→R²:

ϕ(ϑ) :=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

(−1 + cos 4ϑ,sin 4ϑ) ϑ∈[0,^π₂] (1−cos 4ϑ,sin 4ϑ) ϑ∈[^π₂, π]

(−1 + cos 4ϑ,−sin 4ϑ) ϑ∈[π,^3π₂ ] (1−cos 4ϑ,−sin 4ϑ) ϑ∈[^3π₂,2π]

which covers the “eight” curve:

X ={(x¹, x²)∈R²: (x¹−1)²+ (x²)²= 1}

∪ {(x¹, x²)∈R²: (x¹+ 1)²+ (x²)²= 1}

two times with opposite orientation. Define its 0-homogeneous extension to the unit ballu(, ϑ) :=ϕ(ϑ). We have that Deg(u, B, y) = 0 and so:

J u= 0 however:

T V^p(u,·) = 2πδ₀.

In fact, because of the non trivial homotopy type of ϕ(S¹) inR²\ {(1,0),(−1,0)}, any smooth map approxi- matingumust cover one of the disks bounded byX at least two times (see [41] for a precise discussion).

4. Sphere-valued maps

In this section we show how to get equality between J u and T V^p in the case of functions whose image is contained in the (n−1)-dimensional sphere Sⁿ⁻¹. This result, conjectured in [23] and proved in [41] for 0-homogeneous maps, is a simple consequence of Bethuel’s approximation Theorem4.2and of the structure of Distributional Jacobian for sphere-valued maps (Thm. 4.1).

The following can be found in [28] (see also [1] for a different proof with more general hypotheses).

Theorem 4.1. Let u ∈ W^1,p(Ω;Sⁿ⁻¹), p ≥ n−1 and suppose that J u is a Radon measure, then exist {xi}^m_i=1⊂Ωandd_i∈Zsuch that:

J u=ω_n m i=1

d_iδ_xi whered_i= deg(u, ∂B(x_i, r_i), Sⁿ⁻¹)with r_i small enough.

Observe that ifp > n−1, thanks to Sobolev immersion,u ∂B_r(x) is continuous for almost all radii, so the degree considered in the theorem is the classical Brouwer degree of a continuous map (in the casep=n−1 one should refer to the theory of Brouwer degree in BMO, see [7]).

Theorem 4.2. Let u∈W^1,p(Ω;Sⁿ⁻¹), p≥n−1 then there exists a sequence {u_k} ⊂ C^∞(Ω;Sⁿ⁻¹)strongly converging to uif and only if

J u= 0.

This theorem has been proved for the casep=n−1 in [4] but the technique used there cannot be generalized.

The proof for the case p∈ (n−1, n), instead, is an easy consequence of the techniques used in the proof of Theorem 1 of [5]. Another proof has recently been given in [42].

With these tools we are now able to prove the following:

Theorem 4.3. Let u∈W^1,p(Ω;Sⁿ⁻¹),p > n−1, such that T V^p(u,Ω)<∞, then:

|J u|(Ω) =T V^p(u,Ω).

Proof. Recall thatT V^p(u,Ω)<∞implies thatJ uis a Radon measure hence, thanks to Theorem4.1, we have

J u=ω_n m i=1

d_iδ_xi.

For simplicity we assume

J u=ω_nd δ_x0

for somex₀∈Ω withd= deg(u, ∂B(x₀, r), Sⁿ⁻¹), the proof in the general case being equal. By translation we can supposex₀= 0.

Choose anr < ¹₂dist(0,Ω), we clearly have

J u(Ω\B(0, r)) = 0

and so we can find a sequence u_k of smooth maps with values in Sⁿ⁻¹ strongly converging to u in W^1,p

Ω\B(0, r)

. Thanks to a Fubini type argument we can finds∈(r,2r) such that:

• u_k−u_W^1,p_(∂B(0,s))→0;

•

∂B(0,s)|Du|^p≤ 1 r

B(0,2r)|Du|^p.

From Sobolev Embedding Theorem we have thatuk−u_L^∞_(∂B(0,s))→0 and so, forklarge, we have:

deg(u_k, ∂B(0, s), Sⁿ⁻¹) = deg(u, ∂B(0, s), Sⁿ⁻¹) =d.

Denote withw:∂B(0, s)→Sⁿ⁻¹ a smooth map such that:

deg(w, ∂B(0, s), Sⁿ⁻¹) =d

#{x∈∂B(0, s) :w(x) =y}=|d|

where the last equality holds for all points in Sⁿ⁻¹ except two. Hopf Theorem implies that for everyk there exists a continuous homotopyH_k:∂B(0, s)×[0,1]→Sⁿ⁻¹such that

H_k(·,0) =w(·) andH_k(·,1) =u_k ∂B(0, s)(·).

MollifyingH_k and projecting again onSⁿ⁻¹we can suppose the homotopy to be smooth, see [41], Lemma 3.

Letsand consider the map:

v_k,s,=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

u_k(x) ifx∈Ω\B(0, s) u_k

s_|x|^x ifx∈B(0, s)\B(0,2) H_k

s_|x|^x,^|x|−

ifx∈B(0,2)\B(0, )

|x| w

s_|x|^x

ifx∈B(0, ).

Then (recall that the area formula (1.1) implies that ifv:A→Rⁿis Lipschitz andv(A)⊂Sⁿ⁻¹then detDv= 0 a.e. onA):

Ω|detDv_k,s,|=

B(0,)|detDv_k,s,|

=

B(0,)

detD

|x|

w

s x

|x|

dx=

B(0,1)

detD

|x|w

s x

|x|

dx

=

Rⁿ#

x∈B(0,1) :|x|w

s x

|x|

=y

dy

=

B(0,1)#

x∈∂B(0, s) :w(x) = y

|y|

dy=ω_n|d|=|J u|(Ω),

since Proposition3.2gives the opposite inequality to conclude the proof we just have to show thatv_k,s, u.

Actually we can show more, in fact we havestrong convergence.

Observing thatv_k,s,∞≤1 we obtain:

Ω|v_k,s,−u|^p≤

Ω\B(0,r)|u_k−u|^p+ 2^p|B(0, r)| →0 fork→ ∞ andr→0 (recall thats∈(r,2r)).

Moreover:

Ω|Dvk,s,−Du|^p≤

Ω\B(0,r)|Duk−Du|^p +C

B(0,s)|Du|^p+

B(0,s)

D

u_k

s x

|x|

p

+

B(0,2)|Dv_k,s,|^p

. (4.1) We now estimate the second and third terms inside the brackets.

The second one can be majorized as follows (see Lem. A.2):

B(0,s)

D(u_k

s x

|x|

p

≤Cs

∂B(0,s)|Du_k|^p

≤2Cs

∂B(0,s)|Du|^p≤2Cs r

B(0,2r)|Du|^p≤4C

B(0,2r)|Du|^p. Finally the last term can be estimated by:

B(0,2)|Dv_k,s,|^p≤

max{LipH_k,Lipw}

_p

B(0,2)

1 ^p +s 1

|x|^p

≤C(n, p)(max{LipH_k,Lipw})^pn−p. Hence, looking at (4.1), we have

Ω|Dv_k,s,−Du|^p≤α(k, r) +β(r) +δ(k, )

where β(r)→0 forr→0,α(k, r)→0 fork→ ∞(and fixed r) andδ(, k)→0 for→0 (and fixed k). It’s now clear that

u−v_k,s,W^1,p_(Ω)→0.

We remark that, thanks to the strong convergence and a standard diagonal procedure, the previous theorem gives a sequence of smooth maps such that for any open cube Q_R ⊂Ω such thatx_i∈/ ∂Q_R:

•

QR

|detDv_k| →T V^p(u, Q_R);

• lim sup

k→∞

QR

|Dv_k|^p ≤M

QR

|Du|^p.

Notice that this is a fine cover of Ω, this fact will be used in the proof of Theorems6.1and 6.2.

5. Maps with positive degree

According to the discussion in Section3we have to impose some conditions about the sign of Brouwer degree ofuif we want to get equality between|J u|andT V^p. The following one turns out to to be sufficient.

In the sequel we will always assumep∈(n−1, n).

Definition 5.1. We say thatu∈W^1,p(Ω)weakly preserves orientation (u∈W OP(Ω)) if for everyx∈Ω and almost every radiir:

Deg(u, B(x, r), y)≥0. (5.1)

W OP condition is inspired by theIN V condition of [38] and in some sense generalizes it. In fact in [38] it is showed that ifusatisfies the INV condition and detDu >0 almost everywhere, then Deg(u, B(x, r), y)∈ {0,1}

for almost allr.

If u is a smooth map then u ∈ W OP(Ω) if and only if detDu ≥ 0 in Ω. For a generical Sobolev map W OP condition is stronger that detDu≥0 almost everywhere. Roughly speaking we are asking to preserve orientation also in the discontinuity points.

Example 5.2. Consideru:B(0,1)→R²:

u(x) =1− |x|

|x| (−x¹, x²)

then detDu >0 a.e., but Deg(u, B(0, r), y)≤0, in fact a continuous map coinciding withuon∂B(0, r) is the affine function:

w(x) = (1−r)

r (−x¹, x²) for which deg(w, B(0, r), y)≤0.

Observe that the class W OP ∩ {u∞ ≤ C} is weakly closed. This is an immediate consequence of the continuity of degree with respect to weak convergence and of the following well-known result see [38], Lemma 2.9.

Lemma 5.3. Let {uk} ⊂W^1,p(Ω), u_k u then for every x∈Ω and almost every r∈ (0,dist(x, ∂Ω)) there exists a subsequence, depending onxandr, such that u_kj uin W^1,p(∂B(x, r)).

The following proposition gives some useful properties of the distributional Jacobian of W OP maps, the proof follows the one for INV maps in [38]:

Proposition 5.4. Let u∈W OP(Ω)∩W^1,p∩L^∞ then:

(1) J u is a positive Radon measure;

(2) for every xand almost every r:

J u(B(x, r)) =

RⁿDeg(u, B(x, r), y)dy. (5.2)

Proof. The first statement will follow if we are able to prove that (J u _ε)(x) ≥ 0 where _ε are standard mollifiers. Choose a function f ∈ C^∞([0,1)) with f ≤0 such that _ε(x) := f_ε(|x|) is a family of mollifiers, wheref_ε(r) := _ε¹nf(^r_ε)

(J u _ε)(x) = J u, ε(· −x)

=−1 n

B(x,ε)(adjDu(y)·u(y))·D_ε(x−y)dy

=−1 n

_ε

0

f_ε(s)

∂B(x,s)

(adjDu·u)·νdHⁿ⁻¹ds

=−1 n

_ε

0 f_ε(s) Deg(u, B(x, s), y)ds≥0 where in the last equality we have applied (2.4) withg(y) = y

n. To prove the second statement choose:

f_δ() =

⎧⎪

⎨

⎪⎩

1 < r−δ

r−δ r−δ < < r 0 > r

then for a.e. r:

J u(B(x, r)) = lim

δ→0 J u, f_δ(| · −x|)

= lim

δ→0−1 n

B(x,r)(adjDu(y)·u(y))·D_yf_δ(|y−x|)dy

= lim

δ→0

1 n 1 δ

_r

r−δ

∂B(x,s)(adjDu·u)·νdHⁿ⁻¹ds

= 1 n

∂B(x,r)(adjDu·u)·νdHⁿ⁻¹=

RⁿDeg(u, B(x, r), y)dy.

In the sequel we will need to have a criterion to establish if equality (5.2) holds. Suppose thatu∈W^1,p(Ω)∩ L^∞ andT V^p(u,Ω)<∞. Choose a sequencew_n ∈C¹(Ω)∩W^1,n weakly converging touand such that

lim sup

n→∞

Ω|detDw_n| ≤C.

Notice that you can also suppose supw_n∞ ≤C without loss of generality (see the proof of Thm. 5 in [23]).

Then (up to subsequences) there exists a positive Radon measureσsuch that

detDw_ndLⁿ J u^* and|detDw_n|dLⁿ σ.^* Now for any LipschitzD⊂Ω such that

σ(∂D) = 0 andw_n uinW^1,p(∂D)