Short-time heat flow and functions of bounded variation in <span class="mathjax-formula">$\mathbf{R}^N$</span>

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

MICHELEMIRANDA JR, DIEGOPALLARA, FABIOPARONETTO, MARCPREUNKERT

Short-time heat flow and functions of bounded variation inR^N

Tome XVI, n^o1 (2007), p. 125-145.

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Annales de la Facult´e des Sciences de Toulouse Vol. XVI, n^◦1, 2007 pp. 125–145

Short-time heat flow and functions of bounded variation in

Michele Miranda Jr⁽¹⁾, Diego Pallara⁽¹⁾, Fabio Paronetto⁽¹⁾, Marc Preunkert⁽²⁾

ABSTRACT.— We prove a characterisation of sets with finite perimeter andBV functions in terms of the short time behaviour of the heat semi- group inR^N. For sets with smooth boundary a more precise result is shown.

R^´ESUM´E.— On prouve une caract´erisation des ensembles avec p´erim`etre fini et des fonctions `a variation born´ee en termes du comportement du semi-groupe de la chaleur dansR^Nau voisinage det= 0. On prouve aussi un r´esultat plus pr´ecis pour les ensembles avec fronti`ere assez r´eguli`ere.

1. Introduction

Sets with finite perimeter have been introduced by E. De Giorgi in the fifties (see [5],[6]),as a part of the theory of functions of bounded variation, in order to deal with geometric variational problems and have proved to be very useful in several contexts. The first researches of De Giorgi were connected with the investigations of R. Caccioppoli,and in fact sets with finite perimeter are also called Caccioppoli sets. Let us refer to [2] for the properties of sets with finite perimeter andBV functions. De Giorgi’s orig- inal definition of the perimeter of a (measurable) setE⊂R^N was based on the heat semigroup (T(t))t0 inR^N,because of its regularising effects,and can be phrased as follows:

P(E) = lim

t→0∇xT(t)χ_EL¹(R^N),

(∗) Re¸cu le 21 f´evrier 2005, accept´e le 22 juin 2005

(1) Dipartimento di Matematica “Ennio De Giorgi”, Universit`a di Lecce, C.P.193, 73100, Lecce, Italy.

[email protected], [email protected], [email protected]

(2) Mathematisches Institut, Universit¨at T¨ubingen, Auf der Morgenstelle 10, D-72076 T¨ubingen, Germany.

[email protected]

where χ_E denotes the characteristic function of E. We denote by p_N :R^N ×R^N×R₊→Rthe Gauss-Weierstrass kernel,defined by

p_N(x, y, t) := 1

(4πt)^N/2e^−|x−y|

2 4t , so thatT(t)u(x) =

R^Np_N(x, y, t)u(y)dyfor every u∈L¹(R^N).

In [8] M. Ledoux investigated in a different perspective some connections between the heat semigroup (T(t))t0 on L²(R^N) and the isoperimetric inequality,observing that theL²-inequality

T(t)χEL²(R^N)T(t)χBL²(R^N) t0 (1.1) for all setsEwith smooth boundary with the same volume|E|as the ballB implies the isoperimetric inequality. By the self-adjointness of the operators T(t) and

T(t)χE²_L²_(R^N₎=T(t)χE, T(t)χE =T(2t)χE, χE , (1.2) the behaviour of T(t)χE, χE is related to the L²-norm ofT(t)χE,where we use the notationf, g =

R^Nf gdxwhenever the integral is finite. Notice that (1.1) can be easily deduced from the Riesz-Sobolev inequality (see e.g.

[9,Theorem 3.7])

R^N×R^N

f(x)g(x−y)h(y)dxdy

R^N×R^N

f^∗(x)g^∗(x−y)h^∗(y)dxdy, (1.3) where φ^∗ denotes the spherical symmetrisation of φ. Taking f = h= χE

andg=g^∗=pN(·,·, t) in (1.3),so that f^∗=h^∗=χB,the inequality (1.1) follows immediately:

T(t)χE²_L²_(R^N₎ = T(2t)χE, χE

=

R^N×R^Nχ_E(x)χ_E(y)p_N(x, y,2t)dxdy

R^N×R^N

χ_B(x)χ_B(y)p_N(x, y,2t)dxdy

= T(2t)χB, χB =T(t)χB²_L2(R^N)

In [8] an important point has been the formula

tlim→0

π

t T(t)χB, χB^c =P(B), (1.4)

Short-time heat flow and functions of bounded variation inR

whereB is a ball,and the inequality π

tT(t)χ_E, χ_E^c P(E) for everyt0, (1.5) which has been generalised in [12] for all E ⊂ R^N such that either E or its complementary set E^c has a finite volume (otherwise,both terms are infinite). IfE andB have the same volume,from the elementary relation

|E|=T(t)χE, χE +T(t)χE, χE^c for everyt0 and (1.2) it follows that theL²-inequality (1.1) is equivalent to

T(t)χE, χE^c T(t)χB, χB^c for everyt0,

and the semigroup inequality (1.1) implies the isoperimetric inequality for Caccioppoli sets inR^N. In connection with these results,it seems to be inter- esting to pursue the investigation of the relationships between the perimeter of a set and the short-time behaviour of the heat semigroup. The asymptotic expansion of the heat semigroups on regular manifolds or submanifolds has been deeply investigated (see e.g. [7],and also [10]),and in fact alocalised version of (1.4) can be proved for sets with smooth boundary (i.e.,such that the unique projection property in a tubular neighbourhood of the boundary holds),see Theorem 2.1. This result is in fact stronger than (1.4) itself,see Theorem 2.4. In Section 3 we prove that equality (1.4) holds true not only for smooth sets,but for all Caccioppoli sets B. Our proof is based upon the measure-theoretic properties of the reduced boundary. We also show that the finiteness of the limit on the left hand side characterises sets of finite perimeter. Let us point out that the same characterisation of finite perimeter sets is also proved,following a different approach based on the study of the behaviour of the difference quotients ofu,in the papers [3],[4], [11] (see also [1]),where convolution kernels more general than the Gauss- Weierstrass one are considered. Our approach is more geometric in spirit, and gives directly the optimal constants.

Section 4 is devoted to some remarks concerning the short-time be- haviour of the heat semigroup for general BV functions. Recalling that u ∈ BV(R^N) if u ∈ L¹(R^N) and its distributional gradient is a (R^N- valued) Radon measure with finite total variation given by

|Du|(R^N) = sup

R^N

udivgdx: g∈[C_c¹(R^N)]^N, gL^∞(R^N)1

,

we show that the equality

|Du|(R^N) = lim

t→0

√π 2√

t

R^N×R^N

|u(x)−u(y)|p_N(x, y, t)dxdy

holds. SinceP(E) =|Dχ_E|(R^N) ifu=χ_E,the above equality is equivalent to (1.4),and the finiteness of the right hand side characterises the BV functions. As a consequence,defining (as in [14]) the interpolation space (L¹(R^N), D(∆))_1/2,∞=

u∈L¹(R^N) : sup

0<t<1

√1

tT(t)u−uL¹(R^N)<+∞

, (1.6) we have thatBV(R^N)⊂(L¹(R^N), D(∆))_1/2,∞ and,from a known charac- terisation of the above interpolation space,an embedding theorem forBV into a Besov space follows.

Notation. — We denote byH^kthek-dimensional Hausdorff measure (which coincides with the classical measure for k-dimensional smooth submani- folds). For every measureµand measurable setE,we denote by µ E the restriction measure,i.e.,µ E(B) =µ(E∩B) for all measurableB. ForE measurable andt∈[0,1] we denote byE^tthe set of points of densityt,i.e., we set

x∈E^t ⇐⇒ lim

→0

|E∩B(x)|

|B| =t. (1.7)

Recall that the essential boundary of E is ∂^∗E = R^N \(E⁰∪E¹),and the reduced boundary FE is defined as follows. For E ⊂ R^N such that χE ∈BVloc(R^N),i.e.,Ehas locally finite perimeter,x∈supp|DχE|belongs toFEif the limit

ν_E(x) := lim

→0

Dχ_E(B(x))

|DχE|(B(x))

exists in R^N and satisfies |νE(x)| = 1. The function νE : FE → S^N−1 is called thegeneralised inner normaltoE,and,see e.g. [2,Theorem 3.59],for every x∈ FE the hyperplane π_x ={y :y·ν_E(x) = 0} is the approximate tangent planetoFE at x,i.e.,

lim→0

1 ^N−1

E

φ x−y

dy=

πx

φ(y)dH^N−1(y) ∀ φ∈Cc(R^N). (1.8) Moreover,see e.g. [2,Theorem 3.78],

H^N−1(∂^∗E\ FE) = 0. (1.9) and,as a consequence,the distributional derivative of χE is given by the R^N-valued measure

Dχ_E =ν_EH^N−1 FE, H^N−1(FE) =P(E). (1.10)

Short-time heat flow and functions of bounded variation inR

2. Diffusion for regular sets

In this section we study the short-time behaviour ofT(t)χA for setsA with smooth boundary. Bysmoothwe mean the minimal regularity ensuring the unique projection property in a tubular neighbourhood of the boundary.

To this end,the Lipschitz continuity of the unit normal vector field is the natural requirement. We say thatA⊂R^N isuniformlyC^1,1-regularif there are, L >0 such that for everyp∈∂Athe set ∂A∪B(p) is the graph of a C^1,1 functionψwith∇ψ_∞L. Setting

A^δ :={x∈A^c: dist(x, ∂A)δ}, Aε:={x∈A: dist(x, ∂A)ε}

(2.1) forδ, ε >0,we prove the following theorem.

Theorem 2.1. — LetA ⊂R^N be uniformlyC^1,1-regular. Let A_ε and A^δ be an inner and outer tubular neighborhood of∂Adefined in (2.1). Then for every continuous ϕ:R^N →Rwith compact support the equality

tlim→0

π

t T(t)χAε, ϕχ_A^δ =

∂A

ϕ dH^N−1

holds.

Equality (1.4) for uniformlyC^1,1-regular sets follows easily from The- orem 2.1 (see Theorem 2.4). Our proof of Theorem 2.1 is based on a pre- liminary one-dimensional computation and then on a partition of unity ar- gument,reflecting the physical insight that for short time the heat flow is approximately “transversal” to the boundary.

The first step in the proof of Theorem 2.1 is the one-dimensional com- putation below.

Lemma 2.2. — Fix δ, ε >0, and for t >0 define the functionsftby f_t(z) := 1

√t δ

0

χ_{−εr+^√_{2t z0}}(r)dr, z0. (2.2) Thenf_tf_s for every0< t < sand

t→0limf_t(z) =−√

2z, z0.

Proof. — In order to compute the integral in (2.2),notice that the inte- grand is nonzero if and only if

r∈It= [0, δ]∩[−ε−√

2t z,−√ 2t z],

and that the intervalI_tis

It=









[−ε−√

2tz, δ] ifzmin{−ε/√

2t,−δ/√ 2t},

[0, δ] ifz∈[−ε/√

2t,−δ/√ 2t], [−ε−√

2tz,−√

2tz] ifz∈[−δ/√

2t,−ε/√ 2t], [0,−√

2tz] ifzmax{−ε/√

2t,−δ/√ 2t}. Notice that only one between the second and third possibility for I_t may occur,according to the relative size ofεandδ. Consequently,since

ft(z) = 1

√t|It|, we have

f_t(z) =









(δ+ε)/√ t+√

2z −∞< zmin{−ε/√

2t,−δ/√ 2t}, min{ε, δ}/√

t min{−ε/√

2t,−δ/√ 2t}z max{−ε/√

2t,−δ/√ 2t},

−√

2z max{−ε/√

2t,−δ/√

2t}z0, and the assertion follows immediately.

In order to fix the notation to be used in the proof of Theorem 2.1,let us recall the geometric properties of smooth boundaries which we are going to use. We refer e.g. to [15,Section I.2] for a detailed discussion on the subject.

Proposition 2.3. — Let A ⊂ R^N be a uniformly C^1,1-regular set.

Then, there areε, δ >0 such that the maps

i) ∂A×[0, δ]→A^δ, (p, d)→p+d·ν(p) ii) ∂A×[0, ε]→Aε, (p, d)→p−d·ν(p),

whereν(p)is the outward unit normal to∂Aatp, areC^1,1-diffeomorphisms.

Moreover, for every η >0 there is a locally finite coveringV = (Vi) of

∂AandC^1,1 diffeomorphisms

ψ_i:D_i→V_i, D_i open subset ofR^N−1 such that

|Dν(p)|η p∈Vi

|Dψi(ξ)−Ri|η ξ∈Di (2.3) for suitable linear mapsR_i. For everyV_i andψ_i define

Ψi(ξ, ) :=ψi(ξ) +ν(ψi(ξ)), ξ∈Di, ∈(−ε, δ).

Short-time heat flow and functions of bounded variation inR

Denoting by U_i the open set U_i:= Ψ_i

D_i×(−ε, δ) ,

the familyU = (Ui)turns out to be a covering ofAε∪A^δ, and

|DΨi−Li|< η (2.4)

for every i, whereLi are orthogonal maps such that|LieN−ν|η.

Proof of Theorem 2.1. — Recall that we denote byp_N the Gauss-Weierstrass kernel. We divide the proof in three steps.

Step1 - We first consider the case when A={x∈R^N : x_N <0},so that we have

A^δ =R^N−1×(0, δ), Aε=R^N−1×(−ε,0)

andϕindependent ofx_N. Denotingx= (x, x_N) (wherex = (x₁, . . . , x_N−1)∈ R^N−1) observe thatp_N(x, y, t) =p_N−1(x, y, t)p₁(x_N, y_N, t) and takeϕ= ϕ(x) inCc(R^N−1). Then we have

π t

A^δ

Aε

p_N(x, y, t)ϕ(x)dy dx=

= π

t

A^δ

Aε

p_N−1(x, y, t)p₁(x_N, y_N, t)ϕ(x)dy dx

= π

t

R^N−1

R^N−1

pN−1(x, y, t)ϕ(x) _δ

0

₀

−εp₁(x_N, y_N, t)dy_Ndx_Ndy dx

=

R^N−1

ϕ(x)dx π

t δ

0

0

−εp1(xN, yN, t)dyNdxN. Let us now consider only _π

t

_δ

0 dxN

₀

−εp1(xN, yN, t)dyN in the last line above: taking the new variablez= (y_N −x_N)/√

2twe obtain π

t _δ

0

₀

−ε

p₁(x_N, y_N, t)dy_Ndx_N =

= π

t δ

0

^−xN√ 2t

−ε−√xN 2t

e^−z2/2 2√

πt

√2t dz dxN

= 1

√2t δ

0

R

e^−z2/2χ(−ε,0)(√

2tz+xN)dz dxN

= 1

√2t ₀

−∞

e^−z2/2 _δ

0

χ_(−ε,0)(√

2tz+x_N)dx_Ndz.

By Lemma 2.2,

t→0lim

√1 2t

δ 0

χ(−ε,0)(√

2tz+xN)dxN =−z, and then,by the monotone convergence theorem we conclude that

tlim→0

π t

A^δ

Aε

pN(x, y, t)ϕ(x)dy dx=

R^N−1

ϕ(x)dx. (2.5) Step2 - Take now a functionϕ∈Cc(R^N) and denote byAthe same set as in Step1. Notice that

ω(τ) := sup

(x,xN)∈R^N−1×[−τ,τ]

|ϕ(x, xN)−ϕ(x,0)| →0 asτ→0. (2.6) We can then write

π t

A^δ

Aε

p_N(x, y, t)ϕ(x)dy dx=

= π

t

A^δ

Aε

pN(x, y, t)

ϕ(x,0) + (ϕ(x)−ϕ(x,0)) dy dx .

By Step1 we have that limt→0

π t

A^δ

Aε

pN(x, y, t)ϕ(x,0)dy dx=

R^N−1

ϕ(x,0)dx . It remains to prove that

t→0lim π

t

A^δ

Aε

p_N(x, y, t)[ϕ(x)−ϕ(x,0)]dy dx= 0. (2.7) As done in Step 1 we obtain,denoting by K the projection of supp ϕon {x∈R^N|x_N = 0}

π

t

A^δ

Aε

p_N(x, y, t)[ϕ(x)−ϕ(x,0)]dy dx

π t

K

δ 0

0

−ε

|ϕ(x)−ϕ(x,0)|p₁(x_N, y_N, t)dy_Ndx_Ndx

= π

t

K

_δ

0

R

|ϕ(x)−ϕ(x,0)|χ_(−ε,0)(y_N)e^{−|xN−yN|2/4t}

(4πt)^1/2 dy_Ndx_Ndx

= 1

√2t

K

_δ

0

₀

−∞|ϕ(x)−ϕ(x,0)|χ_(−ε,0)(x_N +√

2tz)e^−z2/2dzdx_Ndx

Short-time heat flow and functions of bounded variation inR

where in the last equality we have set z= (y_N −x_N)/√ 2t.

Now fixσ >0: then we can findz₀<0 for which

_z0

−∞ze^−z2/2dz< σ . (2.8) Going on estimating,from above we derive,for−√

2tz0δ,

π t

A^δ

Aε

p_N(x, y, t)[ϕ(x)−ϕ(x,0)]dy dx

= 1

√2t

K

_z0

−∞

_δ

0

|ϕ(x)−ϕ(x,0)|χ_(−ε,0)(x_N +√

2tz)e^−z2/2dx_Ndzdx + + 1

√2t

K

0 z0

δ 0

|ϕ(x)−ϕ(x,0)|χ_(−ε,0)(xN+√

2tz)e^−z2/2dxNdzdx

ω(δ)

K

_z0

−∞

f_t(z)

√2 e^−z2/2dzdx +

K

₀

z0

ω(−√

2tz₀)f_t(z)

√2 e^−z2/2dzdx where ω is defined in (2.6) and ft is as in Lemma 2.2. By Lemma 2.2 we know that ft(z)→ −√

2z andftfsfor 0< t < s. Then we infer,also by (2.8),

t→0lim⁺ω(δ)

K

_z0

−∞

f_t(z)

√2 e^−z2/2dzdx =−ω(δ)|K| _z0

−∞

ze^−z2/2dzdx< ω(δ)|K|σ . Since |^ft√(z)₂ e^−z2/2||ze^−z2/2|andω(−√

2tz0) converges uniformly to 0 as t→0+,we finally obtain (2.7).

Step 3 - Now we consider a uniformly C^1,1-regular set A,fix a function ϕ∈C_c(R^N) and use the notation in Proposition 2.3. Assume at first that the support of ϕ is contained in a fixed U_i (this hypothesis can easily be removed by a partition of unity argument). Since for every i∈ N, x∈U_i the function p_N(x, y, t)/√

t goes to 0 as t → 0 for every y ∈ A_ε\U_i,by dominated convergence we have

t→0lim π

t

A^δ

dx

Aε

p_N(x, y, t)ϕ(x)dy

= lim

t→0

π t

A^δ∩Ui

dx

Aε∩Ui

p_N(x, y, t)ϕ(x)dy.

Since the indexiis fixed,we may drop this index everywhere,and write the kernel p_N(x, y, t),x∈A^δ ∩U and y ∈A_ε∩U,using the new variables in D×[0, δ] andD×[−ε,0],i.e.,we may write

y= Ψ(z, r) z∈D , r∈[0, δ], x= Ψ(ξ, ) ξ∈D , ∈[−ε,0].

so that

ψ(z) =ψ(ξ) +Dψ(ξ)·(z−ξ) +o(|z−ξ|), ν(ψ(z)) =ν(ψ(ξ)) +Dν(ψ(ξ))·

Dψ(ξ)·(z−ξ)

+o(|z−ξ|). (2.9) Then,using these equalities,we obtain

A^δ

Aε

pN(x, y, t)ϕ(x)dy dx=

D×(0,δ)

D×(−ε,0)p˜_N

(z, r),(ξ, ), t

ϕ(ψ(z) +rν(ψ(z)))dz dr dξ d where we have defined

˜ pN

(z, r),(ξ, ), t

=pN

Ψ(z, r),Ψ(ξ, ), t)|DΨ(z, r)||DΨ(ξ, )|. By (2.4),we have

|DΨ(ξ, )|= 1 +O(η) inD×(0, δ),

|DΨ(z, r)|= 1 +O(η) inD×(−ε,0) (2.10) and consequently

|DΨ⁻¹(x)|= 1 +O(η) and |Dψ⁻¹(x)|= 1 +O(η) in U . (2.11) Then,writing

Ψ(z, r) = Ψ(ξ, )+[DΨ−L]((z, r)−(ξ, ))+L((z, r)−(ξ, ))+o(|(z, r)−(ξ, )|) for the orthogonal mapLgiven by Proposition 2.3,we can compute,using (2.4),

Ψ(z, r)−Ψ(ξ, )²

=[DΨ−L]((z, r)−(ξ, )) +L((z, r)−(ξ, )) +o(|(z, r)−(ξ, )|)²

=|(z, r)−(ξ, )|²

1 +O(η) and consequentlypN

Ψ(z, r),Ψ(ξ, ), t) = _(4πt)¹N/2 exp

−|(z−ξ, r−)|² 4t

1 +O(η)

by which finally

˜ pN

Ψ(z, r),Ψ(ξ, ), t) = 1

(4πt)^N/2 exp

−|(z−ξ, r−)|² 4t

1 +O(η) . Then from the above equality and (2.10) we derive

π t

A^δ

Aε

pN(x, y, t)ϕ(x)dy dx=

=

1 +O(η) π

t

D×(0,δ)

D×(−ε,0)pN

(z, r),(ξ, ), t

ϕ(Ψ(ξ, ))dξ d dz dr .

Short-time heat flow and functions of bounded variation inR

Taking the limit ast→0,by Step2 and (2.11) we obtain limt→0

π t

A^δ

dx

Aε

p_N(x, y, t)ϕ(x)dy

=

1 +O(η)

D

ϕ(Ψ(ξ,0))dξ

=

1 +O(η)

D

ϕ(ψ(ξ))dξ

=

1 +O(η)

∂A∩Uϕ(x)|Dψ⁻¹(x)|dH^N−1(x)

=

1 +O(η)

∂A∩U

ϕ(x)dH^N−1(x) which concludes the proof.

Of course,for sets with smooth boundary,equality (1.4) can be deduced from Theorem 2.1.

Theorem 2.4. — LetA⊂R^N be a compact subset withC^1,1-boundary

∂A. Then

tlim→0

π

t T(t)χA, χA^c =P(A) holds.

Proof. — From the elementary relations

T(t)χ_Aε, χ_Aδ +T(t)χ_Aε, χ_Ac\A^δ = T(t)χ_Aε, χ_A^c , T(t)χAε, χA^c +T(t)χA\Aε, χA^c = T(t)χA, χA^c

it follows that

T(t)χAε, χ_A^δ T(t)χAε, χA^c , T(t)χ_Aε, χ_A^c T(t)χ_A, χ_A^c and therefore,together with (1.5),

π

t T(t)χ_Aε, χ_Aδ π

t T(t)χ_A, χ_A^c P(A).

Then by this last inequality and Theorem 2.1 it follows that P(A) = lim

t→0

π

t T(t)χAε, χ_A^δ lim inf

t→0

π

t T(t)χA, χA^c

lim sup

t→0

π

t T(t)χ_A, χ_A^c P(A).

Remark 2.5. — Since for compact subsetsA⊂R^N with smooth bound- ary∂Athe perimeterP(A) and the (N−1)-dimensional Hausdorff measure H^N−1(∂A) coincide (cf [2,Proposition 3.62]),we also have

limt→0

π

t T(t)χ_A, χ_A^c =H^N−1(∂A).

3. Diffusion for Caccioppoli sets

In this section we show that a measurable setE⊂R^N has finite perime- ter if and only if

π/tT(t)χ_E, χ_E^c is bounded. To begin with,let us study the short-time behaviour ofT(t) with respect to two arbitrary sets of finite perimeter. Recall that forEwith finite perimeterν_Edenotes the generalised (or measure-theoretic) inner unit normal to the reduced boundary.

Theorem 3.1. — LetE, F ⊂R^N be sets of finite perimeter. Then the following equality holds:

tlim→0

π

tχE−T(t)χE, χF =

FE∩FF

νE(x)·νF(x)dH^N−1(x). (3.1)

Proof. — Since

T(t)χ_E−χ_E= _t

0

∆T(s)χ_Eds, we have

T(t)χ_E−χ_E, χ_F = _t

0

∆T(s)χ_E, χ_F ds.

Moreover,by (1.10),integrating by parts we obtain

∆T(s)χE, χF =

R^N

∆T(s)χE(x)χF(x)dx

= −

R^N

∇T(s)χE(x)·dDχF(x)

= −

FF

∇T(s)χE(x)·νF(x)dH^N−1(x).

Notice that,if we define for everyx∈ FE ands >0 the measures dµs,x=L^N

E−x

√s

,

Short-time heat flow and functions of bounded variation inR

we have

∇T(s)χ_E(x) =

E

∇x



e^−|x−y|

2 4s

(4πs)^N/2



dy=−

E

(x−y) 2s

e^−|x−y|

2 4s

(4πs)^N/2dy

= 1

2√ s

E−x

√s

e^−|z|2/4 (4π)^N/2zdz.

= 1

2√ s

R^N

e^−|z|2/4

(4π)^N/2zdµs,x(z).

Moreover,setting,for every x∈ FE, HνE(x)=

z∈R^N :z·νE(x)0 ,

the existence of the approximate tangent plane forx∈ FE,see (1.8),implies that the measures µ_s,x are locally weakly^∗ convergent as s → 0 to the measure

dµ_x=L^N H_νE(x), and also

s→0lim

R^N

z·ν_F(x)e^−|z|2/4

(4π)^N/2dµ_s,x(z) =

H_νE(x)

z·ν_F(x)e^−|z|2/4 (4π)^N/2dz.

Summing up,we can write π

tχ_E−T(t)χ_E, χ_F =

FFg(x, t)dH^N−1(x), whereg:FF×(0,+∞) is given by

g(x, t) = π

t _t

0

1 2√ s

R^N

e^−|z|2/4

(4π)^N/2z·ν_F(x)dµ_s,x(z)ds, and by (1.9) we have

t→0lim⁺g(x, t) =





√π (4π)^N/2

H_νE(x)

z·ν_F(x)e^−|z|2/4dz forx∈ FE∩ FF

0 forx∈

E⁰∪E¹

∩ FF,

whereE⁰, E¹are defined according to (1.7). Since

|g(x, t)|

√π (4π)^N/2

R^N

|z|e^−|z|2/4dz=cN,

we can apply Lebesgue dominated convergence theorem and obtain

∃ lim

t→0

π

tχE−T(t)χE, χF

=

√π (4π)^N/2

FE∩FF

H_νE(x)

z·ν_F(x)e^−|z|2/4dzdH^N−1(x)

=

√π (4π)^N/2

FE∩FF

H_νE(x)

(νE(x)·νF(x))(z·νE(x))e^−|z|2/4dzdH^N−1(x)

=

FE∩FFν_E(x)·ν_F(x)dH^N−1(x),

becauseν_F(x) = (ν_E(x)·ν_F(x))ν_E(x) forH^N−1-a.e. x∈ FE∩ FF and

√π (4π)^N/2

H_νE(x)

z·νE(x)e^−|z|2/4dz= 2(4π)^(N−1)/2 ∀x∈ FE.

Remark 3.2. — Notice that if |F\E| = 0 in the preceding statement, thenνE(x) =νF(x) forH^N−1-a.e. x∈ FE∩ FF,hence the equality

t→0lim π

tχE−T(t)χE, χF =H^N−1(FE∩ FF) (3.2) holds.

As a special case,we may takeE=F in the above theorem,and obtain the following result,which generalises formula (1.4).

Theorem 3.3. — Let E ⊂ R^N be a set of finite perimeter; then the following equality holds

tlim→0

π

tT(t)χE, χE^c =P(E). (3.3) Proof. — Since T(t)χEL¹(R^N) =|E| for allt0,insertingF =E in (3.2) we obtain

χE−T(t)χE, χE =χE−T(t)χE,1−χE^c =T(t)χE, χE^c

and the assertion follows.

Let us now prove the reverse implication of Theorem 3.3.

Short-time heat flow and functions of bounded variation inR

Theorem 3.4. — LetE ⊂R^N be a set such that eitherE or E^c has finite measure, and

lim inf

t→0⁺

T(t)χ√E, χE^c

t <+∞.

ThenE has finite perimeter.

Proof. — Assume that|E|<+∞. We can write

√1

tT(t)χ_E, χ_E^c = 1 (4π)^N/2√

t

R^N

R^N

χ_E^c(x)χ_E(x+√

ty)e^−|y|2/4dydx

= 1

(4π)^N/2√ t

R^N

e^−|y|2/4

R^N

χ_E−^√_ty(x)−χ_E(x)χ_E−^√_ty(x)

dxdy

= 1

(4π)^N/2√ t

R^N

e^−|y|2/4

|E| − |E∩(E−√ ty)|

dy

= 1

2(4π)^N/2√ t

R^N

e^−|y|2/4|E(E−√ ty)|dy

= 1

2(4π)^N/2

R^N

|y|e^−|y|2/4|E(E−√ ty)|

√t|y| dy,

whereEF= (E∪F)\(E∩F) . Then,if we define

|D_νχ_E|= lim inf

t→0⁺

|E(E−tν)|

t ,

from the previous estimate we get that

R^N

|y|e^−|y|2/4|D_y/|y|χE|dylim inf

t→0⁺

R^N

|y|e^−|y|2/4|E(E−√

√ ty)|

t|y| dy <+∞.

Noticing that

R^N

|y|e^−|y|2/4|D_y/|y|χE|dy=CN

S^N−1

|DνχE|dν,

we have proved that

S^N−1

|DνχE|dν <+∞.

This implies that the function ν → |DνχE| is finite for a.e. ν ∈ S^N−1; in particular,there exist M > 0 and an orthonormal system of coordinates ν1, . . . , νN of Lebesgue points of|DνχE|such that

|DνiχE|M, ∀i= 1, . . . , N.

Without loss of generality,we can assume thatν_i=e_i; now, ifφ∈C_c¹(R^N), the function

φt(x) = φ(x+tei)−φ(x) t

is uniformly convergent to∂iφ(x). This implies that

R^N

χE(x)∂iφ(x)dx= lim

t→0⁺

R^N

χE(x)φt(x)dx.

But

R^N

χE(x)φt(x)dx=

R^N

χE(x−tei)−χE(x)

t φ(x)dx,

hence

R^N

χ_E(x)φ_t(x)dx

φ∞|E(E+te_i)|

t .

From this it follows that

R^N

χE(x)∂iφ(x)dx

φ_∞lim inf

t→0⁺

|E(E+tei)|

t

= φ∞|D_iχ_E|Mφ∞. In the end,we have proved that

R^N

χ_E(x)divφ(x)dxN Mφ∞, ∀φ∈C_c¹(R^N), and thenχ_E∈BV(R^N).

Remark 3.5. — Given E with finite perimeter and setting f(x, t) = T(t)χ_E(x)−χ_E(x),we have

R^Nf dx= 0 for all t0,hence

R^Nf⁺dx=

R^Nf⁻dx. Since f⁺ = (T(t)χ_E−χ_E)χ_E^c we obtain

T(t)χE−χEL¹(R^N)= 2T(t)χE, χE^c (3.4) and therefore from equality (3.3) we deduce

P(E) = lim

t→0

√π 2√

tT(t)χ_E−χ_EL¹(R^N). (3.5) Moreover,since by [12,Proposition 8] ¹₂

π/tT(t)χE−χEL¹(R^N)P(E) for allt >0,the limit in (3.5) is in fact a supremum.

Short-time heat flow and functions of bounded variation inR

4. Diffusion forBV functions

In this section we apply the results of the preceding one to generalBV functions and prove a characterisation of BV functions which generalises Theorems 3.3 and 3.4. Let us recall the coarea formula,which is repeat- edly used in the sequel (see e.g. [2,Theorem 3.40] for details). For every u∈L¹(R^N) the following equality holds:

|Du|(R^N) =

R

P({u > τ})dτ, (4.1) and both sides are finite if and only ifu∈BV(R^N).

Theorem 4.1. — Letu∈BV(R^N); then the following equality holds:

|Du|(R^N) = lim

t→0

√π 2√

t

R^N×R^N

|u(x)−u(y)|p_N(x, y, t)dxdy.

Proof. — From (3.4),(3.5) and the coarea formula (4.1),setting Eτ ={u > τ}and noting that,for almost everyx, y∈R^N

R

|χEτ(x)−χEτ(y)|dτ =|u(x)−u(y)|, (4.2) we get by Remark 3.5

|Du|(R^N) =

R

P(E_τ)dτ

R

√π 2√

t

R^N

|T(t)χ_Eτ −χ_Eτ|dxdτ

√π 2√

t

R

R^N×R^N

|χ_Eτ(y)−χ_Eτ(x)|p_N(x, y, t)dxdydτ

=

√π 2√

t

R^N×R^N

|u(x)−u(y)|p_N(x, y, t)dxdy.

For the reverse inequality,from the Fatou Lemma and from the fact that χ_Eτ(x)χ_Eτ(y)= 0 iff τ <min{u(x), u(y)},

we have that

|Du|(R^N) =

R

P(E_τ)dτ =

R t→0lim

√π

√t

R^N

T(t)χ_Eτ(x)χ_Eτ^c(x)dxdτ lim inf

t→0

√π

√t

R

R^N×R^N

(χ_Eτ(y)−χ_Eτ(y)χ_Eτ(x))p_N(x, y, t)dxdydτ

= lim inf

t→0

√π

√t

R^N×R^N

(u(y)−min{u(x), u(y)})p_N(x, y, t)dxdy.