Short-time Heat Flow and Functions of Bounded Variation in R^{N}

M. Miranda Jr^{∗} D. Pallara^{∗} F. Paronetto^{∗} M. Preunkert^{†}

Abstract We prove a characterisation of sets with finite perimeter and BV functions in terms of the short time behaviour of the heat semigroup in R^{N}. For sets with smooth boundary a more precise result is shown.

Résumé On prouve une caractérisation des ensembles avec perimètre fini et des fonctions à variation bornée en termes du comportément du semi-groupe de la chaleur dans R^{N} au voisinage de t = 0. On prouve aussi un résultat plus précis pour les ensembles avec frontière assez régulière.

Mathematics subject classification (2000): 35K05, 47D06, 49Q15 Keywords: Heat semigroup, Functions of bounded variation.

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 and BV functions. De Giorgi’s original definition of the perimeter of a (measurable) set E ⊂ R^{N} was based on the heat semigroup (T (t))_{t≥0} in R^{N}, because of its regularising effects, and can be phrased as follows:

P (E) = lim_{t→0} ∥∇_{x}T (t)χ_{E}∥_{L^{1}(R^{N})},

where χ_{E} denotes the characteristic function of E. We denote by p_{N} : R^{N} ×R^{N} ×R_{+} → R the Gauss-Weierstrass kernel, defined by

p_{N}(x, y, t) := \frac{1}{(4πt)^{N/2}} e^{− \frac{|x−y|^{2}}{4t}} ,

so that T (t)u(x) = ∫_{R^{N}} p_{N}(x, y, t)u(y)dy for every u ∈ L^{1}(R^{N}).

∗Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy.

e-mail: michele.miranda@unile.it, diego.pallara@unile.it, fabio.paronetto@unile.it

†Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany.

e-mail: mapr@fa.uni-tuebingen.de

In [8] M. Ledoux investigated in a different perspective some connections between the heat semigroup (T (t))_{t≥0} on L^{2}(R^{N}) and the isoperimetric inequality, observing that the L^{2}-inequality

∥T (t)χ_{E}∥_{L^{2}(R^{N})} ≤ ∥T (t)χ_{B}∥_{L^{2}(R^{N})}  t ≥ 0  (1.1)

for all sets E with smooth boundary with the same volume |E| as the ball B implies the isoperimetric inequality. By the self-adjointness of the operators T (t) and

∥T (t)χ_{E}∥_{L^{2}(R^{N})}^{2} = ⟨T (t)χ_{E}, T (t)χ_{E}⟩ = ⟨T (2t)χ_{E}, χ_{E}⟩,  (1.2)

the behaviour of ⟨T (t)χ_{E}, χ_{E}⟩ is related to the L^{2}-norm of T (t)χ_{E}, where we use the notation ⟨f, g⟩ = ∫_{R^{N}} fgdx whenever 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 = χ_{E} and g = g^{∗} = p_{N}(·, ·, t) in (1.3), so that f^{∗} = h^{∗} = χ_{B}, the inequality (1.1) follows immediately:

∥T (t)χ_{E}∥_{L^{2}(R^{N})}^{2} = ⟨T (2t)χ_{E}, χ_{E}⟩
= ∫_{R^{N} ×R^{N}} χ_{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)χ_{B}, χ_{B}⟩ = ∥T (t)χ_{B}∥_{L^{2}(R^{N})}^{2}

In [8] one important point has been the formula

lim_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{B}, χ_{B^{c}}⟩ = P (B),  (1.4)

where B is a ball, and the inequality

\sqrt{\frac{π}{t}} ⟨T (t)χ_{E}, χ_{E^{c}}⟩ ≤ P (E)  for every t ≥ 0,  (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). If E and B have the same volume, from the elementary relation

|E| = ⟨T (t)χ_{E}, χ_{E}⟩ + ⟨T (t)χ_{E}, χ_{E^{c}}⟩ for every t ≥ 0

and (1.2) it follows that the L^{2}-inequality (1.1) is equivalent to

⟨T (t)χ_{E}, χ_{E^{c}}⟩ ≥ ⟨T (t)χ_{B}, χ_{B^{c}}⟩ for every t ≥ 0,

and the semigroup inequality (1.1) implies the isoperimetric inequality for Caccioppoli sets in R^{N}. In connection with these results, it seems to be interesting 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 a localised 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 of u, 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 behaviour of the heat semigroup for general BV functions. Recalling that u ∈ BV (R^{N}) if u ∈ L^{1}(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}} u div gdx : g ∈ [C_{c}^{1}(R^{N})]^{N}, ∥g∥_{L^{∞}(R^{N})} ≤ 1},

we show that the equality

|Du|(R^{N}) = lim_{t→0} \frac{\sqrt{π}}{2\sqrt{t}} ∫_{R^{N} ×R^{N}} |u(x) − u(y)|p_{N}(x, y, t)dxdy

holds. Since P (E) = |Dχ_{E}|(R^{N}) if u = 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

(L^{1}(R^{N}), D(Δ))_{1/2,∞} = \{u ∈ L^{1}(R^{N}) : sup_{0<t<1} \frac{1}{\sqrt{t}}∥T (t)u − u∥_{L^{1}(R^{N})} < +∞\}, (1.6)

we have that BV (R^{N}) ⊂ (L^{1}(R^{N}), D(Δ))_{1/2,∞} and, from a known characterisation of the above interpolation space, an embedding theorem for BV into a Besov space follows.

Notation We denote by H^{k} the k-dimensional Hausdorff measure (which coincides with the classical measure for k-dimensional smooth submanifolds). For every measure μ and measurable set E, we denote by μ E the restriction measure, i.e., μ E(B) = μ(E ∩ B) for all measurable B. For E measurable and t ∈ [0, 1] we denote by E^{t} the set of points of density t, i.e., we set

x ∈ E^{t} ⟺ lim_{ϱ→0} \frac{|E ∩ B_{ϱ}(x)|}{|B_{ϱ}|} = t.  (1.7)

Recall that the essential boundary of E is ∂^{∗}E = R^{N} \ (E^{0} ∪ E^{1}), and the reduced boundary FE is defined as follows. For E ⊂ R^{N} such that χ_{E} ∈ BV_{loc}(R^{N}), i.e., E has locally finite perimeter, x ∈ supp|Dχ_{E}| belongs to FE if the limit

ν_{E}(x) := lim_{ϱ→0} \frac{Dχ_{E}(B_{ϱ}(x))}{|Dχ_{E}|(B_{ϱ}(x))}

exists in R^{N} and satisfies |ν_{E}(x)| = 1. The function ν_{E} : E → S^{N−1} is called the generalised inner normal to E, 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 plane to FE at x, i.e.,

lim_{ϱ→0} \frac{1}{ϱ^{N−1}} ∫_{E} ϕ(\frac{x − y}{ϱ} )dy = ∫_{π_{x}} ϕ(y)dH^{N−1}(y)  ∀ ϕ ∈ C_{c}(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 χ_{F} is given by the R^{N}-valued measure

Dχ_{F} = ν_{F} H^{N−1} FF.  (1.10)

2 Diffusion for regular sets

In this section we study the short-time behaviour of T (t)χ_{A} for sets A with smooth boundary. By smooth we 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 that A ⊂ R^{N} is uniformly C^{1,1}- regular if there are ϱ, L > 0 such that for every p ∈ ∂A the set ∂A ∪ B_{ϱ}(p) is the graph of a C^{1,1} function ψ with ∥∇ψ∥_{∞} ≤ L. Setting

A^{δ} := {x ∈ A^{c} : dist(x, ∂A) ≤ δ},  A_{ε} := {x ∈ A : dist(x, ∂A) ≤ ε}  (2.1)

for δ, ε > 0, we prove the following theorem.

Theorem 2.1 Let A ⊂ R^{N} be uniformly C^{1,1}-regular. Let A_{ε} and A^{δ} be an inner and outer tubular neighborhood of ∂A defined in (2.1). Then for every continuous φ : R^{N} → R with compact support the equality

lim_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{A_{ε}}, φχ_{A^{δ}}⟩ = ∫_{∂A} φ dH^{N−1}

holds.

Equality (1.4) for uniformly C^{1,1}-regular sets follows easily from Theorem 2.1 (see Theorem 2.4). Our proof of Theorem 2.1 is based on a preliminary one-dimensional computation and then on a partition of unity argument, 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 computation below.

Lemma 2.2 Fix δ, ε > 0, and for t > 0 define the functions f_{t} by

f_{t}(z) := \frac{1}{\sqrt{t}} ∫_{0}^{δ} χ_{\{−ε≤r+\sqrt{2t} z≤0\}}(r) dr,  z ≤ 0.  (2.2)

Then f_{t} ≥ f_{s} for every 0 < t < s and

lim_{t→0} f_{t}(z) = −\sqrt{2} z, z ≤ 0.

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

r ∈ I_{t} = [0, δ] ∩ [−ε − \sqrt{2t} z, −\sqrt{2t} z],

and that the interval I_{t} is

I_{t} = \{ \begin{matrix}[ − ε − \sqrt{2t}z, δ] & if z ≤ min\{−ε/\sqrt{2t}, −δ/\sqrt{2t}\} , \\ [0, δ] & if z ∈ [−ε/\sqrt{2t}, −δ/\sqrt{2t}] , \\ [ − ε − \sqrt{2t}z, −\sqrt{2t}z] & if z ∈ [−δ/\sqrt{2t}, −ε/\sqrt{2t}] , \\ [0, −\sqrt{2t}z] & if z ≥ max\{−ε/\sqrt{2t}, −δ/\sqrt{2t}\} . \end{matrix}

Notice that only one between the second and third possibility for I_{t} may occur, according to the relative size of ε and δ. Consequently, since

f_{t}(z) = \frac{1}{\sqrt{t}}|I_{t}| ,

we have

f_{t}(z) = \{ \begin{matrix}(δ + ε)/\sqrt{t} + \sqrt{2}z & −∞ < z ≤ min\{−ε/\sqrt{2t}, −δ/\sqrt{2t}\} , \\ min\{ε, δ\}/\sqrt{t} & min\{−ε/\sqrt{2t}, −δ/\sqrt{2t}\} ≤ z ≤ max\{−ε/\sqrt{2t}, −δ/\sqrt{2t}\} , \\ −\sqrt{2} z & max\{−ε/\sqrt{2t}, −δ/\sqrt{2t}\} ≤ z ≤ 0 , \end{matrix}

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 are ε, δ > 0 such that the maps

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

where ν(p) is the outward unit normal to ∂A at p, are C^{1,1}-diffeomorphisms.

Moreover, for every η > 0 there is a locally finite covering V = (V_{i}) of ∂A and C^{1,1} diffeomorphisms

ψ_{i} : D_{i} → V_{i} , D_{i} open subset of R^{N−1}

such that

\begin{matrix}|Dν(p)| ≤ η & p ∈ V_{i} \\ |Dψ_{i}(ξ) − R_{i}| ≤ η & ξ ∈ D_{i} \end{matrix}  (2.3)

for suitable linear maps R_{i}. For every V_{i} and ψ_{i} define

Ψ_{i}(ξ, ϱ) := ψ_{i}(ξ) + ϱν(ψ_{i}(ξ)) , ξ ∈ D_{i} , ϱ ∈ (−ε, δ).

Denoting by U_{i} the open set

U_{i} := Ψ_{i}(D_{i} × (−ε, δ)),

the family U = (U_{i}) turns out to be a covering of A_{ε} ∪ A^{δ}, and

|DΨ_{i} − L_{i}| < η  (2.4)

for every i, where L_{i} are orthogonal maps such that |L_{i}e_{N} − ν| ≤ η.

Proof of Theorem 2.1. Recall that we denote by p_{N} the Gauss-Weierstrass kernel. We divide the proof in three steps. Step 1 - We first consider the case when A = {x ∈ R^{N} : x_{N} < 0}, so that we have

A^{δ} = R^{N−1} × (0, δ) , A_{ε} = R^{N−1} × (−ε, 0)

and φ independent of x_{N}. Denoting x = (x′, x_{N}) (where x′ = (x_{1}, …, x_{N−1}) ∈ R^{N−1}) observe that p_{N}(x, y, t) = p_{N−1}(x′, y′, t)p_{1}(x_{N}, y_{N}, t) and take φ = φ(x′) in C_{c}(R^{N−1}). Then we have

\sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)φ(x) dy dx =

= \sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N−1}(x′, y′, t)p_{1}(x_{N}, y_{N}, t)φ(x′) dy dx

= \sqrt{\frac{π}{t}} ∫_{R^{N−1}} ∫_{R^{N−1}} p_{N−1}(x′, y′, t)φ(x′) ∫_{0}^{δ} ∫_{−ε}^{0} p_{1}(x_{N}, y_{N}, t) dy_{N} dx_{N} dy′ dx′

= ( ∫_{R^{N−1}} φ(x′)dx′)\sqrt{\frac{π}{t}} ∫_{0}^{δ} ∫_{−ε}^{0} p_{1}(x_{N}, y_{N}, t) dy_{N} dx_{N}.

Let us now consider only \sqrt{\frac{π}{t}} ∫_{0}^{δ} dx_{N} ∫_{−ε}^{0} p_{1}(x_{N}, y_{N}, t) dy_{N} in the last line above: taking the new variable z = (y_{N} − x_{N})/\sqrt{2t} we obtain

\sqrt{\frac{π}{t}} ∫_{0}^{δ} ∫_{−ε}^{0} p_{1}(x_{N}, y_{N}, t) dy_{N} dx_{N} =

= \sqrt{\frac{π}{t}} ∫_{0}^{δ} ∫ \frac{−x_{N}}{\sqrt{2t}} ; \frac{−ε−x_{N}}{\sqrt{2t}} \frac{e^{−z^{2}/2}}{2\sqrt{πt}} \sqrt{2t} dz dx_{N}

= \frac{1}{\sqrt{2t}} ∫_{0}^{δ} ∫_{R} e^{−z^{2}/2}χ_{(−ε,0)}(\sqrt{2t}z + x_{N}) dz dx_{N}

= \frac{1}{\sqrt{2t}} ∫_{−∞}^{0} e^{−z^{2}/2} ∫_{0}^{δ} χ_{(−ε,0)}(\sqrt{2t}z + x_{N}) dx_{N} dz.

By Lemma 2.2,

lim_{t→0} \frac{1}{\sqrt{2t}} ∫_{0}^{δ} χ_{(−ε,0)}(\sqrt{2t}z + x_{N}) dx_{N} = −z,

and then, by the monotone convergence theorem we conclude that

lim_{t→0} \sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)φ(x) dy dx = ∫_{R^{N−1}} φ(x′)dx′.  (2.5)

Step 2 - Take now a function φ ∈ C_{c}(R^{N}) and denote by A the same set as in Step 1. Notice that

ω(τ) := sup_{(x′,x_{N})∈R^{N−1}×[−τ,τ]} |φ(x′, x_{N}) − φ(x′, 0)| → 0 as τ → 0.  (2.6)

We can then write

\sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)φ(x) dy dx =

= \sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)[φ(x′, 0) + (φ(x) − φ(x′, 0))] dy dx .

By Step 1 we have that

lim_{t→0} \sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)φ(x′, 0) dy dx = ∫_{R^{N−1}} φ(x′, 0)dx′ .

It remains to prove that

lim_{t→0} \sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} 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 spt φ on {x ∈ R^{N} | x_{N} = 0}

|\sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)[φ(x) − φ(x′, 0)] dy dx| ≤

≤ \sqrt{\frac{π}{t}} ∫_{K} ∫_{0}^{δ} ∫_{−ε}^{0} |φ(x) − φ(x′, 0)|p_{1}(x_{N}, y_{N}, t)dy_{N}dx_{N}dx′

= \sqrt{\frac{π}{t}} ∫_{K} ∫_{0}^{δ} ∫_{R} |φ(x) − φ(x′, 0)|χ_{(−ε,0)}(y_{N})\frac{e^{−|x_{N} −y_{N} |^{2}/4t}}{(4πt)^{1/2}} dy_{N}dx_{N}dx′

= \frac{1}{\sqrt{2t}} ∫_{K} ∫_{0}^{δ} ∫_{−∞}^{0} |φ(x) − φ(x′, 0)|χ_{(−ε,0)}(x_{N} + \sqrt{2t}z)e^{−z^{2}/2}dzdx_{N}dx′

where in the last equality we have set z = (y_{N} − x_{N})/\sqrt{2t}. Now fix σ > 0: then we can find z_{0} < 0 for which

| ∫_{−∞}^{z_{0}} ze^{−z^{2}/2}dz| < σ .  (2.8)

Going on estimating, from above we derive, for −\sqrt{2t}z_{0} ≤ δ,

|\sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)[φ(x) − φ(x′, 0)] dy dx| ≤

= \frac{1}{\sqrt{2t}} ∫_{K} ∫_{−∞}^{z_{0}} ∫_{0}^{δ} |φ(x) − φ(x′, 0)|χ_{(−ε,0)}(x_{N} + \sqrt{2t}z)e^{−z^{2}/2}dx_{N}dzdx′ +

+\frac{1}{\sqrt{2t}} ∫_{K} ∫_{z_{0}}^{0} ∫_{0}^{δ} |φ(x) − φ(x′, 0)|χ_{(−ε,0)}(x_{N} + \sqrt{2t}z)e^{−z^{2}/2}dx_{N}dzdx′

≤ ω(δ) ∫_{K} ∫_{−∞}^{z_{0}} \frac{f_{t}(z)}{\sqrt{2}}e^{−z^{2}/2}dzdx′ + ∫_{K} ∫_{z_{0}}^{0} ω(−\sqrt{2t}z_{0})\frac{f_{t}(z)}{\sqrt{2}}e^{−z^{2}/2}dzdx′

where ω is defined in (2.6) and f_{t} is as in Lemma 2.2. By Lemma 2.2 we know that f_{t}(z) → −\sqrt{2}z and f_{t} ≤ f_{s} for 0 < t < s. Then we infer, also by (2.8),

lim_{t→0^{+}} ω(δ) ∫_{K} ∫_{−∞}^{z_{0}} \frac{f_{t}(z)}{\sqrt{2}}e^{−z^{2}/2}dzdx′ = −ω(δ)|K| ∫_{−∞}^{z_{0}} ze^{−z^{2}/2}dzdx′ < ω(δ)|K|σ .

Since |\frac{f_{t}(z)}{\sqrt{2}}e^{−z^{2}/2}| ≤ |ze^{−z^{2}/2}| and ω(−\sqrt{2t}z_{0}) 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)/\sqrt{t} goes to 0 as t → 0 for every y ∈ A_{ε} \ U_{i}, by dominated convergence we have

lim_{t→0} \sqrt{\frac{π}{t}} ∫_{A^{δ}} dx ∫_{A_{ε}} p_{N}(x, y, t)φ(x) dy = lim_{t→0} \sqrt{\frac{π}{t}} ∫_{A^{δ}∩U_{i}} dx ∫_{A_{ε}∩U_{i}} p_{N}(x, y, t)φ(x) dy.

Since the index i is 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] and D × [−ε, 0], i.e., we may write

\begin{matrix}y = Ψ(z, r) & z ∈ D , r ∈ [0, δ] , \\ x = Ψ(ξ, ϱ) & ξ ∈ D , ϱ ∈ [−ε, 0] . \end{matrix}

so that

\begin{matrix}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|) . \end{matrix}  (2.9)

Then, using these equalities, we obtain

∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)φ(x) dy dx =
∫_{D×(0,δ)} ∫_{D×(−ε,0)} p̃_{N}((z, r), (ξ, ϱ), t)φ(ψ(z) + rν(ψ(z)))dz dr dξ dϱ

where we have defined

p̃_{N}((z, r), (ξ, ϱ), t) = p_{N}(Ψ(z, r), Ψ(ξ, ϱ), t)|DΨ(z, r)||DΨ(ξ, ϱ)| .

By (2.4), we have

\begin{matrix}|DΨ(ξ, ϱ)| = 1 + O(η)  in D × (0, δ) , \\ |DΨ(z, r)| = 1 + O(η)  in D × (−ε, 0) \end{matrix}  (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, ϱ) + [DΨ − L]((z, r) − (ξ, ϱ)) + L((z, r) − (ξ, ϱ)) + o(|(z, r) − (ξ, ϱ)|)

for the orthogonal map L given by Proposition 2.3, we can compute, using (2.4),

|Ψ(z, r) − Ψ(ξ, ϱ)|^{2} = |[DΨ − L]((z, r) − (ξ, ϱ)) + L((z, r) − (ξ, ϱ)) + o(|(z, r) − (ξ, ϱ)|)|^{2}
= |(z, r) − (ξ, ϱ)|^{2}(1 + O(η))

and consequently p_{N}(Ψ(z, r), Ψ(ξ, ϱ), t) = \frac{1}{(4πt)^{N/2}} exp \{−\frac{|(z − ξ, r − ϱ)|^{2}}{4t} \}(1 + O(η)) by

which finally

p̃_{N}(Ψ(z, r), Ψ(ξ, ϱ), t) = \frac{1}{(4πt)^{N/2}} exp \{−\frac{|(z − ξ, r − ϱ)|^{2}}{4t} \}(1 + O(η)) .

Then from the above equality and (2.10) we derive

\sqrt{\frac{π}{t}} ∫_{A^{δ}} ∫_{A_{ε}} p_{N}(x, y, t)φ(x) dy dx =

= (1 + O(η))\sqrt{\frac{π}{t}} ∫_{D×(0,δ)} ∫_{D×(−ε,0)} p_{N}((z, r), (ξ, ϱ), t)φ(Ψ(ξ, ϱ))dξ dϱ dz dr .

Taking the limit as t → 0, by Step 2 and (2.11) we obtain

lim_{t→0} \sqrt{\frac{π}{t}} ∫_{A^{δ}} dx ∫_{A_{ε}} 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∩U} φ(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 Let A ⊂ R^{N} be a compact subset with C^{1,1}-boundary ∂A. Then

lim_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{A}, χ_{A^{c}}⟩ = P (A)

holds. Proof. From the elementary relations

⟨T (t)χ_{A_{ε}}, χ_{A^{δ}}⟩ + ⟨T (t)χ_{A_{ε}}, χ_{A^{c}\A^{δ}}⟩ = ⟨T (t)χ_{A_{ε}}, χ_{A^{c}}⟩,
⟨T (t)χ_{A_{ε}}, χ_{A^{c}}⟩ + ⟨T (t)χ_{A\A_{ε}}, χ_{A^{c}}⟩ = ⟨T (t)χ_{A}, χ_{A^{c}}⟩

it follows that

⟨T (t)χ_{A_{ε}}, χ_{A^{δ}}⟩ ≤ ⟨T (t)χ_{A_{ε}}, χ_{A^{c}}⟩,
⟨T (t)χ_{A_{ε}}, χ_{A^{c}}⟩ ≤ ⟨T (t)χ_{A}, χ_{A^{c}}⟩

and therefore, together with (1.5),

\sqrt{\frac{π}{t}} ⟨T (t)χ_{A_{ε}}, χ_{A^{δ}}⟩ ≤ \sqrt{\frac{π}{t}} ⟨T (t)χ_{A}, χ_{A^{c}}⟩ ≤ P (A).

Then by this last equality and Theorem 2.1 it follows that

P (A) = lim_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{A_{ε}}, χ_{A^{δ}}⟩ ≤ lim inf_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{A}, χ_{A^{c}}⟩
≤ lim sup_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{A}, χ_{A^{c}}⟩ ≤ P (A).

Remark 2.5 Since for compact subsets A ⊂ R^{N} with smooth boundary ∂A the perimeter P (A) and the (N −1)-dimensional Hausdorff measure H^{N−1}(∂A) coincide (cf [2, Proposition 3.62]), we also have

lim_{t→0} \sqrt{\frac{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 set E ⊂ R^{N} has finite perimeter if and only if \sqrt{π/t}⟨T (t)χ_{E}, χ_{E^{c}}⟩ is bounded. To begin with, let us study the short-time behaviour of T (t) with respect to two arbitrary sets of finite perimeter. Recall that for E with finite perimeter ν_{E} denotes the generalised (or measure-theoretic) inner unit normal to the reduced boundary.

Theorem 3.1 Let E, F ⊂ R^{N} be sets of finite perimeter. Then the following equality holds:

lim_{t→0} \sqrt{\frac{π}{t}} ⟨χ_{E} − T (t)χ_{E}, χ_{F} ⟩ = ∫_{FE∩FF} ν_{E}(x) · ν_{F} (x)dH^{N−1}(x).  (3.1)

Proof. Since

T (t)χ_{E} − χ_{E} = ∫_{0}^{t} ΔT (s)χ_{E}ds,

we have

⟨T (t)χ_{E} − χ_{E}, χ_{F} ⟩ = ∫_{0}^{t} ⟨ΔT (s)χ_{E}, χ_{F} ⟩ds.

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

⟨ΔT (s)χ_{E}, χ_{F} ⟩ = ∫_{R^{N}} ΔT (s)χ_{E}(x)χ_{F} (x)dx = − ∫_{R^{N}} ∇T (s)χ_{E}(x) · dDχ_{F} (x)
= − ∫_{FF} ∇T (s)χ_{E}(x) · ν_{F} (x)dH^{N−1}(x).

Notice that, if we define for every x ∈ FE and s > 0 the measures

dμ_{s,x} = L^{N} (\frac{E − x}{\sqrt{s}}) ,

we have

∇T (s)χ_{E}(x) = ∫_{E} ∇_{x} (\frac{e^{− \frac{|x−y|^{2}}{4s}}}{(4πs)^{N/2}} ) dy = − ∫_{E} \frac{(x − y)}{2s} \frac{e^{− \frac{|x−y|^{2}}{4s}}}{(4πs)^{N/2}} dy

= \frac{1}{2\sqrt{s}} ∫_{\frac{E−x}{\sqrt{s}}} \frac{e^{−|z|^{2}/4}}{(4π)^{N/2}} zdz.

= \frac{1}{2\sqrt{s}} ∫_{R^{N}} \frac{e^{−|z|^{2}/4}}{(4π)^{N/2}} zdμ_{s,x}(z).

Moreover, setting, for every x ∈ FE,

H_{ν_{E}(x)} = {z ∈ R^{N} : z · ν_{E}(x) ≥ 0} ,

the existence of the approximate tangent plane for x ∈ 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_{ν_{E}(x)},

and also

lim_{s→0} ∫_{R^{N}} z · ν_{F} (x)\frac{e^{−|z|^{2}/4}}{(4π)^{N/2}} dμ_{s,x}(z) = ∫_{H_{ν_{E} (x)}} z · ν_{F} (x)\frac{e^{−|z|^{2}/4}}{(4π)^{N/2}} dz.

Summing up, we can write

\sqrt{\frac{π}{t}} ⟨χ_{E} − T (t)χ_{E}, χ_{F} ⟩ = ∫_{FF} g(x, t)dH^{N−1}(x),

where g : FF × (0, +∞) is given by

g(x, t) = \sqrt{\frac{π}{t}} ∫_{0}^{t} \frac{1}{2\sqrt{s}} ∫_{R^{N}} \frac{e^{−|z|^{2}/4}}{(4π)^{N/2}} z · ν_{F} (x)dμ_{s,x}(z)ds,

and by (1.9) we have

lim_{t→0^{+}} g(x, t) = \{ \begin{matrix}\frac{\sqrt{π}}{(4π)^{N/2}} ∫_{H_{ν_{E} (x)}} z · ν_{F} (x)e^{−|z|^{2}/4}dz & for x ∈ FE ∩ FF \\ 0 & for x ∈ (E^{0} ∪ E^{1}) ∩ FF, \end{matrix}

where E^{0}, E^{1} are defined according to (1.7). Since

|g(x, t)| ≤ \frac{\sqrt{π}}{(4π)^{N/2}} ∫_{R^{N}} |z|e^{−|z|^{2}/4}dz = c_{N},

we can apply Lebesgue dominated convergence theorem and obtain

∃ lim_{t→0} \sqrt{\frac{π}{t}} ⟨χ_{E} − T (t)χ_{E}, χ_{F} ⟩ = \frac{\sqrt{π}}{(4π)^{N/2}} ∫_{FE∩FF} ∫_{H_{ν_{E} (x)}} z · ν_{F} (x)e^{−|z|^{2}/4}dzdH^{N−1}(x)

= \frac{\sqrt{π}}{(4π)^{N/2}} ∫_{FE∩FF} ∫_{H_{ν_{E} (x)}}(ν_{E}(x) · ν_{F} (x))(z · ν_{E}(x))e^{−|z|^{2}/4}dzdH^{N−1}(x)

= ∫_{FE∩FF} ν_{E}(x) · ν_{F} (x)dH^{N−1}(x),

because ν_{F} (x) = (ν_{E}(x) · ν_{F} (x))ν_{E}(x) for H^{N−1}-a.e. x ∈ FE ∩ FF and

\frac{\sqrt{π}}{(4π)^{N/2}} ∫_{H_{ν_{E} (x)}} z · ν_{E}(x)e^{−|z|^{2}/4}dz = 2(4π)^{(N−1)/2} ∀ x ∈ FE.

Remark 3.2 Notice that if |F \ E| = 0 in the preceding statement, then ν_{E}(x) = ν_{F} (x) for H^{N−1}-a.e. x ∈ FE ∩ FF, hence the equality

lim_{t→0} \sqrt{\frac{π}{t}} ⟨χ_{E} − T (t)χ_{E}, χ_{F} ⟩ = H^{N−1}(FE ∩ FF)  (3.2)

holds.

As a special case, we may take E = 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

lim_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{E}, χ_{E^{c}}⟩ = P (E).  (3.3)

Proof. Since ∥T (t)χ_{E}∥_{L^{1}(R^{N})} = |E| for all t ≥ 0, inserting F = E in (3.2) we obtain

⟨χ_{E} − T (t)χ_{E}, χ_{E}⟩ = ⟨χ_{E} − T (t)χ_{E}, 1 − χ_{E^{c}}⟩ = ⟨T (t)χ_{E}, χ_{E^{c}}⟩

and the assertion follows.

Let us now prove the reverse implication of Theorem 3.3. Theorem 3.4 Let E ⊂ R^{N} be a set such that either E or E^{c} has finite measure, and

lim inf_{t→0^{+}} \frac{⟨T (t)χ_{E}, χ_{E^{c}}⟩}{\sqrt{t}} < +∞.

Then E has finite perimeter. Proof. Assume that |E| < +∞. We can write

\frac{1}{\sqrt{t}}⟨T (t)χ_{E}, χ_{E^{c}}⟩ = \frac{1}{(4π)^{N/2}\sqrt{t}} ∫_{R^{N}} ∫_{R^{N}} χ_{E^{c}}(x)χ_{E}(x + \sqrt{t}y)e^{−|y|^{2}/4}dydx
= \frac{1}{(4π)^{N/2}\sqrt{t}} ∫_{R^{N}} e^{−|y|^{2}/4} ∫_{R^{N}} (χ_{E−\sqrt{t}y}(x) − χ_{E}(x)χ_{E−\sqrt{t}y}(x)) dxdy
= \frac{1}{(4π)^{N/2}\sqrt{t}} ∫_{R^{N}} e^{−|y|^{2}/4} (|E| − |E ∩ (E − \sqrt{t}y)|) dy
= \frac{1}{2(4π)^{N/2}\sqrt{t}} ∫_{R^{N}} e^{−|y|^{2}/4}|E△(E − \sqrt{t}y)|dy
= \frac{1}{2(4π)^{N/2}} ∫_{R^{N}} |y|e^{−|y|^{2}/4} \frac{|E△(E − \sqrt{t}y)|}{\sqrt{t}|y|}dy,

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

|D_{ν}χ_{E}| = lim inf_{t→0^{+}} \frac{|E△(E − tν)|}{t},

from the previous estimate we get that

∫_{R^{N}} |y|e^{−|y|^{2}/4}|D_{y/|y|}χ_{E}|dy ≤ lim inf_{t→0^{+}} ∫_{R^{N}} |y|e^{−|y|^{2}/4} \frac{|E△(E − \sqrt{t}y)|}{\sqrt{t}|y|}dy < +∞.

Noticing that

∫_{R^{N}} |y|e^{−|y|^{2}/4}|D_{y/|y|}χ_{E}|dy = C_{N} ∫_{S^{N−1}} |D_{ν}χ_{E}|dν,

we have proved that

∫_{S^{N−1}} |D_{ν}χ_{E}|dν < +∞.

This implies that the function ν ↦ |D_{ν}χ_{E}| is finite for a.e. ν ∈ S^{N−1}; in particular, there exist M > 0 and an orthonormal system of coordinates ν_{1}, …, ν_{N} of Lebesgue points of |D_{ν}χ_{E}| such that

|D_{ν_{i}}χ_{E}| ≤ M, ∀i = 1, …, N.

Without loss of generality, we can assume that ν_{i} = e_{i}; now, if ϕ ∈ C_{c}^{1}(R^{N}), the function

ϕ_{t}(x) = \frac{ϕ(x + te_{i}) − ϕ(x)}{t}

is uniformly convergent to ∂_{i}ϕ(x). This implies that

∫_{R^{N}} χ_{E}(x)∂_{i}ϕ(x)dx = lim_{t→0^{+}} ∫_{R^{N}} χ_{E}(x)ϕ_{t}(x)dx.

But

∫_{R^{N}} χ_{E}(x)ϕ_{t}(x)dx = ∫_{R^{N}} \frac{χ_{E}(x − te_{i}) − χ_{E}(x)}{t}ϕ(x)dx,

hence

|∫_{R^{N}} χ_{E}(x)ϕ_{t}(x)dx| ≤ ∥ϕ∥_{∞} \frac{|E△(E + te_{i})|}{t}.

From this it follows that

|∫_{R^{N}} χ_{E}(x)∂_{i}ϕ(x)dx| ≤ ∥ϕ∥_{∞} lim inf_{t→0^{+}} \frac{|E△(E + te_{i})|}{t}
= ∥ϕ∥_{∞}|D_{i}χ_{E}| ≤ M∥ϕ∥_{∞}.

In the end, we have proved that

∫_{R^{N}} χ_{E}(x)div ϕ(x)dx ≤ NM∥ϕ∥_{∞}, ∀ϕ ∈ C_{c}^{1}(R^{N}),

and then χ_{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^{N}} fdx = 0 for all t ≥ 0, hence ∫_{R^{N}} f^{+}dx = ∫_{R^{N}} f^{−}dx. Since f^{+} = (T (t)χ_{E}−χ_{E})χ_{E^{c}} we obtain

∥T (t)χ_{E} − χ_{E}∥_{L^{1}(R^{N})} = 2⟨T (t)χ_{E}, χ_{E^{c}}⟩  (3.4)

and therefore from equality (3.3) we deduce

P (E) = lim_{t→0} \frac{\sqrt{π}}{2\sqrt{t}}∥T (t)χ_{E} − χ_{E}∥_{L^{1}(R^{N})}.  (3.5)

Moreover, since by [12, Proposition 8] \frac{1}{2} \sqrt{π/t}∥T (t)χ_{E} − χ_{E}∥_{L^{1}(R^{N})} ≤ P (E) for all t > 0, the limit in (3.5) is in fact a supremum.

4 Diffusion for BV functions

In this section we apply the results of the preceding one to general BV functions and prove a characterisation of BV functions which generalises Theorems 3.3 and 3.4. Let us recall the coarea formula, which is repeatedly used in the sequel (see e.g. [2, Theorem 3.40] for details). For every u ∈ L^{1}(R^{N}) the following equality holds:

|Du|(R^{N}) = ∫_{R} P ({u > τ})dτ,  (4.1)

and both sides are finite if and only if u ∈ BV (R^{N}).

Theorem 4.1 Let u ∈ BV (R^{N}); then the following equality holds:

|Du|(R^{N}) = lim_{t→0} \frac{\sqrt{π}}{2\sqrt{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 E_{τ} = {u > τ} and noting that, for almost every x, y ∈ R^{N}

∫_{R} |χ_{E_{τ}} (x) − χ_{E_{τ}} (y)|dτ = |u(x) − u(y)|,  (4.2)

we get by Remark 3.5

|Du|(R^{N}) = ∫_{R} P (E_{τ})dτ ≥ ∫_{R} \frac{\sqrt{π}}{2\sqrt{t}} ∫_{R^{N}} |T (t)χ_{E_{τ}} − χ_{E_{τ}} |dxdτ
≥ \frac{\sqrt{π}}{2\sqrt{t}} ∫_{R} ∫_{R^{N} ×R^{N}} |χ_{Eτ} (y) − χ_{E_{τ}} (x)|p_{N}(x, y, t)dxdydτ
= \frac{\sqrt{π}}{2\sqrt{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_{E_{τ}} (x)χ_{E_{τ}} (y) ≠ 0 iff τ < min[u(x), u(y)],

we have that

|Du|(R^{N}) = ∫_{R} P (E_{τ})dτ = ∫_{R} lim_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{t} \end{matrix} ∫_{R^{N}} T (t)χ_{E_{τ}} (x)χ_{E_{τ}^{c}} (x)dxdτ
≤ lim inf_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{t} \end{matrix} ∫_{R} ∫_{R^{N} ×R^{N}} (χ_{Eτ} (y) − χ_{E_{τ}} (y)χ_{E_{τ}} (x))p_{N}(x, y, t)dxdydτ
= lim inf_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{t} \end{matrix} ∫_{R^{N} ×R^{N}} (u(y) − min\{u(x), u(y)\})p_{N}(x, y, t)dxdy.

But then the assertion follows, since min\{u(x), u(y)\} = \frac{1}{2}[u(x) + u(y) − |u(x) − u(y)|].

The following result is the reverse implication of Theorem 4.1 and is indeed a corollary of Theorem 3.4.

Proposition 4.2 Let u ∈ L^{1}(R^{N}) be such that

lim inf_{t→0} \frac{1}{\sqrt{t}} ∫_{R^{N} ×R^{N}} |u(x) − u(y)| p_{N}(x, y, t)dxdy < +∞;  (4.3)

then u ∈ BV (R^{N}).

Proof. Using (4.2) and setting again E_{τ} = {u > τ}, we can write

∫_{R^{N} ×R^{N}} |u(x) − u(y)| p_{N}(x, y, t)dxdy = ∫_{R} ∫_{R^{N} ×R^{N}} |χ_{E_{τ}} (x) − χ_{E_{τ}} (y)| p_{N}(x, y, t)dxdydτ .

Since ⟨T (t)χ_{E}, χ_{E^{c}}⟩ ≥ 0 (see Remark 3.5) by the above equality and 4.3 we get

0 ≤ ∫_{R} lim inf_{t→0} \frac{1}{\sqrt{t}}⟨T (t)χ_{E_{τ}} , χ_{E_{τ}^{c}} ⟩dτ ≤ lim inf_{t→0} ∫_{R} \frac{1}{\sqrt{t}}⟨T (t)χ_{E_{τ}} , χ_{E_{τ}^{c}} ⟩dτ
≤ lim inf_{t→0} \frac{1}{\sqrt{t}} ∫_{R^{N} ×R^{N}} ∫_{R} |χ_{E_{τ}} (x) − χ_{E_{τ}} (y)| p_{N}(x, y, t)dxdydτ < +∞ .

In particular, by Theorem 3.4, for a.e. τ ∈ R the set E_{τ} has finite perimeter and the limit lim_{t→0} \frac{1}{\sqrt{t}}⟨T (t)χ_{E_{τ}} , χ_{E_{τ}^{c}} ⟩ exists. Then by the coarea formula and Theorem 3.3,

|Du|(R^{N}) = ∫_{R} P (E_{τ})dτ = ∫_{R} lim_{t→0} \sqrt{\frac{π}{t}} ⟨T (t)χ_{E_{τ}} , χ_{E_{τ}^{c}} ⟩dτ
≤ lim inf_{t→0} \sqrt{\frac{π}{t}} ∫_{R^{N} ×R^{N}} |u(x) − u(y)| p_{N}(x, y, t)dxdy < +∞

that is, u ∈ BV (R^{N}).

Let us add a further result concerning BV functions, which in some sense complements Theorem 3.1. In order to state it, let us recall some properties of u ∈ BV (R^{N}). Setting

u^{∨}(x) = inf {t ∈ [−∞, +∞] : lim_{ϱ→0} ϱ^{−N}|{u > t} ∩ B_{ϱ}(x)| = 0} ,
u^{∧}(x) = sup {t ∈ [−∞, +∞] : lim_{ϱ→0} ϱ^{−N}|{u < t} ∩ B_{ϱ}(x)| = 0} .

and S_{u} = {x ∈ R^{N} : u^{∧}(x) < u^{∨}(x)}, a Borel unit vector field ν_{u} is defined H^{N−1}- a.e. in S_{u} in such a way that the measure Du, restricted to S_{u}, can be represented as Du S_{u} = (u^{∨} − u^{∧})ν_{u}H^{N−1} S_{u}, see e.g. [2, Theorem 3.78].

Theorem 4.3 Let u, v ∈ BV (R^{N}) ∩ L^{2}(R^{N}), or u, v ∈ BV (R^{N}), v ∈ L^{∞}(R^{N}). Then

lim_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{t} \end{matrix} ⟨u − T (t)u, v⟩ = ∫_{S_{u}∩S_{v}}(u^{∨} − u^{∧})(v^{∨} − v^{∧})ν_{u} · ν_{v}dH^{N−1},

where E_{τ} = {u > τ} and F_{σ} = {v > σ}. The previous identity makes also sense for any u, v ∈ BV_{loc}(R^{N}) without the assumption to be L^{2}, possibly having on both sides infinite value.

Proof. Notice that

x ∈ FE_{τ} ⇒ τ ∈ [u^{∧}(x), u^{∨}(x)].

Then if x is a Lebesgue point either for u or for v, we have either

∫_{R} χ_{FE_{τ}} (x)dτ = 0  or  ∫_{R} χ_{FF_{σ}}(x)dσ = 0.

Moreover, since for H^{N−1}-a.e. x ∈ S_{v} and σ ∈ (v^{∧}(x), v^{∨}(x)) there holds x ∈ FF_{σ}, we get

∫_{R×R} ∫_{FE_{τ} ∩FF_{σ}} ν_{E_{τ}} · ν_{F_{σ}}dH^{N−1}dτdσ = ∫_{R} ∫_{FE_{τ}} ∫_{R} ν_{E_{τ}} · ν_{F_{σ}}χ_{FF_{σ}}dσdH^{N−1}dτ
= ∫_{R} ∫_{FE_{τ} ∩S_{v}}(v^{∨} − v^{∧})ν_{E_{τ}} · ν_{v}dH^{N−1}dτ
= ∫_{S_{v}} ∫_{R}(v^{∨} − v^{∧})ν_{E_{τ}} · ν_{v}χ_{FE_{τ}} dτdH^{N−1}
= ∫_{S_{u}∩S_{v}}(u^{∨} − u^{∧})(v^{∨} − v^{∧})ν_{u} · ν_{v}dH^{N−1}.

Remark 4.4 Immediate consequences of (1.2) and Theorem 4.3 is the following equality:

lim_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix} (⟨u, v⟩ − ⟨T (t)u, T (t)v⟩) = ∫_{S_{u}∩S_{v}}(u^{∨} − u^{∧})(v^{∨} − v^{∧})ν_{u} · ν_{v}dH^{N−1}.

In fact,

lim_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix} (⟨u, v⟩ − ⟨T (t)u, T (t)v⟩) = lim_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix} (⟨u, v⟩ − ⟨T (2t)u, v⟩)
= lim_{t→0} \begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix} (⟨u − T (2t)u, v⟩) .

Remark 4.5 The interpolation space (L^{1}(R^{N}), D(Δ))_{1/2,∞} defined in (1.6) coincides with the Besov space B_{1∞}^{1/2}(R^{N}) (see [13, Theorem 4^{∗}]). From Theorem 4.1 it follows that the BV -norm is stronger than the B_{1∞}^{1/2}(R^{N}) norm, because clearly

∥T (t)u − u∥_{L^{1}(R^{N})} ≤ ∫_{R^{N} ×R^{N}} |u(x) − u(y)|p_{N}(x, y, t)dxdy

for every t > 0. On the other hand, Remark 3.5 shows that in the above estimate the equality holds if u = χ_{E} and P (E) is finite. Let us point out an example showing that (L^{1}(R^{N}), D(Δ))_{1/2,∞} is larger than BV (R^{N}). Taking N = 1 and using the following norm, which is equivalent to the natural norm in the above interpolation space (see [14, Theorem 1.13.6.1])

∥u∥ = sup_{0<t<1} \frac{∥(T_{A}(t) − I)^{2}u∥_{L^{1}(R)}}{t} < ∞,

where in our case Au = u′ and T_{A}(t) is the translation semigroup T_{A}(t)u(x) = u(x + t), it is easily seen that the function u(x) = log |x| · χ_{(−1,1)}(x) belongs to L^{1}(R) and to (L^{1}(R), D(Δ))_{1/2,∞}, but u does not belong to BV (R).

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