Characterization of sets of finite local and non-local perimeter via non-local heat equation
-
Andrea Kubin
Abstract
In this paper we provide a characterization of sets with finite local and non-local perimeter via a Γ-convergence result. As an application, we give a short proof of the isoperimetric inequality, both in the local and in the non-local cases.
Introduction
In this article we analyze the asymptotic behavior of the energy
as
The function
Theorem 0.1.
Let
In the above theorem,
History of the problem and motivation
Without aiming at completeness, we now briefly recall related results from the literature.
Our result is inspired on an idea of De Giorgi (see [11]), where the author used the heat kernel to define the perimeter of a set.
Later, Ledoux in [24] build upon this idea to provide a new proof of the isoperimetric inequality using the heat kernel semigroup, in both the Euclidean and Gaussian setting. The result in [11] was generalized in the context of Riemannian manifolds in [27] (see also [8] and [15] for Carnot groups), where the authors provided a generalization of the asymptotic behavior of the heat semigroup shown in [24] and also a characterization for sets of finite perimeter via the heat semigroup. This line of research has been developed in several directions.
We mention the contribution in [20], where the authors consider the case of Riemannian manifolds with unbounded Ricci curvature, and [5], where the authors prove a characterization of Sobolev space using, as energy, the s-Gagliardo seminorms, also providing pointwise convergence result as the parameter
When
In this paper we extend the result of De Giorgi in [11]. In fact, we also provide a characterization of sets of finite local and non-local perimeter via the fractional heat semigroup.
Outline of the proof
We now describe the strategy used to prove Theorem 0.1. Since we aim to prove a Γ-convergence result, we need to prove: compactness, the
In Section 2, we prove compactness properties.
Here we encounter the first difficulty, as the first non-trivial order in the power expansion of
After establishing good compactness properties, we prove
In Section 3
we study the case
We continue in Section 4, where we address the problem with
The case
Finally, in Section 5 we use our result to prove the isoperimetric inequality, both in the local and non-local framework.
The proof relies on Hardy rearrangement inequality (the approximating kernels are
We conclude this introduction with a brief heuristic interpretation of our result. By Bochner subordination formula, one way to obtain the fractional laplacian of a function is
where
Hence, the fractional perimeter of a set E can be computed by evolving the characteristic function of E via the heat equation and then computing the s-derivative (with respect to time) at time 0 of the energy of the evolving solution at time
for
1 Notation and preliminary results
1.1 Sets of finite perimeter
We recall some notation and basic results of geometric measure theory from [3].
We denote with
This means that a set E is a finite perimeter if
Theorem 1.1 (Compactness in
BV
).
Let
Then, there exist a subsequence
For every
1.2 Non-local perimeter
The non-local perimeter of a set E is defined via the fractional laplacian of the function
The Fourier transform is invertible
Let
where
with
For all
and we denote with
If
and we observe that
Theorem 1.2.
Let
There exist a subsequence
1.3 Fractional heat equation
Here we collect some known facts about fractional heat equation and refer to [17] for a detailed discussion.
Given
We denote by
where
and
for
When
and the scaling property
Finally, we also need the estimate for the gradient of the fractional heat kernel
1.4 Technical results
In order to prove compactness when
Lemma 1.3 ([18, Lemma 15]).
Let
Lemma 1.3 it is proved in [18, Lemma 15] for
Finally, in order to compute the constant in the Γ-limit for
Lemma 1.4.
Let
In particular,
where
Before proving Lemma 1.4, we recall the definition of Euler’s beta function
The beta and the gamma function are connected by the identity
Proof of Lemma 1.4.
We observe that the function
hence (1.10). Formula (1.11) follows by (1.10) with
2 Compactness
In this section we study compactness properties of sequences
where
This compactness properties suggests the candidate Γ-limit: for
The main theorem of this section is the following.
Theorem 2.1.
Let
There exists
As already observed in the introduction, for
2.1 Proof of Theorem 2.1 when
s
∈
(
0
,
1
2
)
∪
(
1
2
,
1
)
Compactness for
2
s
<
1
.
Let
are not increasing. The proof of this fact is quite immediate, and we skip it. Observe that
where in the last equality we have used
Recalling the definition of
Moreover, since
Therefore,
Compactness for
2
s
>
1
.
In this case the proof
is similar
In fact, setting again
where
in the second inequality we have used the gradient kernel estimate (1.9), while
in the last inequality we have used the assumption (2.1). Thus, there exists
2.2 Proof of Theorem 2.1 when
s
=
1
2
This subsection is devoted to the proof of Theorem 2.1 when
Lemma 2.2.
Let
Proof.
By translational invariance, it is enough to prove the claim only for
i.e., (2.3). ∎
Lemma 2.3.
For every
Proof.
We can assume without loss of generality that
The following result is a localized isoperimetric inequality for the non-local energy (0.1) when
Lemma 2.4.
Let
Proof.
Fix
Let ρ and
We claim that there exists
Indeed, if
where in the last step we have used (2.5) for
where in the last inequality we used (2.5). If
Hence (2.6) holds also in case
We prove that (2.6) implies (2.4). We observe that
and hence
Therefore by (2.6) and Lemma 1.3, with ε replaced by
and hence recalling the very definition of
We are now in a position to prove Theorem 2.1 when
Proof.
We divide the proof into three steps.
Throughout the proof, we write
Step 1.
Let
(2.7)
For every
We claim that there exists a constant
where M is the constant in (2.1). Indeed, we have
where the first inequality follows by (2.7), the second from Lemma 2.3, the third from (2.3) and the last one follows directly by (2.1).
Step 2. For every
holds for i large enough.
We omit the dependence on i by setting
By Lemma 2.4, there exist
For every
i.e., (2.9).
Step 3. Here we conclude the proof of the compactness result. Fix
By step 2, we have that
3 Γ-convergence for
s
∈
(
0
,
1
2
)
The main result of this section is the following.
Theorem 3.1.
Let
(Lower bound) Let
(3.1)(Upper bound) For every
(3.2)Here
3.1 Proof of Theorem 3.1 (i)
Recall that
(otherwise (3.1) is trivial). To prove (3.1), we observe
(3.3)
where in the last step we used the Plancherel theorem and
We recall that
We observe that the function
where
The Fourier transform of the function
(3.5)
where
(3.6)
where in the last step we used that
3.2 Proof of Theorem 3.1 (ii)
Let
where we used that
and the fact that
At this point, we just need to use that the Fourier transform is an isometry and
and therefore
Since
Finally, to conclude we just need to observe that
3.3 Characterization of sets of finite
2
s
-fractional perimeter
As a byproduct of our Γ-convergence analysis, we prove that a set
Theorem 3.2.
Let
4 Γ-convergence for
s
∈
[
1
2
,
1
)
The main result of this section is the following Γ-convergence theorem.
Theorem 4.1.
Let
(Lower bound) Let
(Upper bound) For every
Here
The argument to prove Theorem 4.1 is completely different from the used to show
Theorem 3.1, as it relies on geometric measure theory. We start computing the limit as
4.1 Estimates on cubes
Throughout this subsection, we set
The main goal of this subsection is to prove the following result.
Proposition 4.2.
Let
To begin, we consider the case when
Lemma 4.3.
For any
where
Proof.
Using Fubini Theorem, we have
We observe that
where we used the change of variable
and we have set
A straightforward computation gives
(4.7)
with
and
which is exactly (4.4). ∎
We now estimate the integral on cubes from below.
Lemma 4.4.
For any
where
Proof.
To prove this lemma, we just need to show that the error passing from a small cube to the hyperplane
to observe that, by using the change of variable
Using the integration in polar coordinate, we obtain
(4.8)
Remark 4.5.
Thanks to Lemma 1.4 we get
We now proceed performing the same computation for
Lemma 4.6.
Let
Proof.
We start by observing that
We claim that
For all
Since the Fourier transform is an isometry, we have
where with a little abuse of notation we denoted the Fourier transform in
From the above formula and by
we obtain (4.10).
Now, using Fubini Theorem and (4.10) with
(4.11)
We use basic trigonometry to get
Thus, using the above formula, we have that (4.11) becomes
We observe that
where in the last equality we have used a change of variable
It is quite easy to show that h is bounded and
which gives (4.9). ∎
The next lemma is an obvious consequence of the previous one.
Lemma 4.7.
For
where
Finally, we prove the crucial estimate to prove the
Lemma 4.8.
For
where
Proof.
To prove this lemma, we just need to show that the error passing from a small cube to the hyperplane
and observe that, by (1.8) and using a change of variable, we have
Using (1.7), we have
(4.13)
The proof of (4.12) is now direct consequence of Lemma 4.6 and inequality (4.13). ∎
Proof of Proposition 4.2.
Formula
(4.3) is now a direct consequence of Lemma 4.3 and Lemma 4.4
when
We are now ready to prove our Γ-convergence theorem.
4.2 Proof of Theorem 4.1 (i)
Proposition 4.9.
Let
Then, for any sequence
where the constant
Proof.
We set
Since the set E is of finite perimeter, for
Now, given a cube
and
We claim that, setting
Without loss of generality, we assume
For all
Then we infer
(4.18)
Now, we distinguish between the case
Case
Up to extract a further subsequence, we assume that
Using that
where the constant
Case
Up to extract a further subsequence, we can assume
with
We are now in a position to prove Theorem 4.1 (i).
Proof of Theorem 4.1 (i).
To conclude the proof, we need to show that
4.3 Proof of Theorem 4.1 (ii)
Proof of Theorem 4.1 (ii).
We start by proving that for a polyhedron Π it holds
For any
if
Moreover, since we are interested in the limit as
For
We now have to distinguish among many cases.
Case one:
and therefore
(4.21)
Case two:
and define
Clearly,
and
Integrating the above inequality on the set
(4.22)
Now we need to distinguish between the case
Therefore, (4.22) becomes
While, for
where
Therefore, in any case we found that for
Case three:
For the first integral, arguing as in (4.21), we have
Therefore, for each
(4.23)
Summing (4.23) for
Collecting (4.21), (4.22), (4.24), (4.23) and sending
The argument for a generic set E of finite perimeter follows by density of the polyhedral sets in the class of the sets with finite perimeter. ∎
4.4 Characterization of sets of finite perimeter
As a consequence of our Γ-convergence analysis, we prove that a set
Theorem 4.10.
Let
If
If E is a set of finite perimeter, then
(4.25)
Proof.
To prove (i), we just need to apply Theorem 2.1 to the constant sequence
We prove (ii). First, we observe that a simple approximation argument provides
Let
(4.27)
By (4.27) and the decay estimate for
Hence, we have that (4.25) is true for all
Let now
Hence, using the very definition of
(4.28)
where in the last step we have used the L’Hôpital’s rule and the fact that
5 Application: The local and the non-local isoperimetric inequality
As a consequence of the Γ-convergence result, we have a short proof of the isoperimetric inequality.
Recall that
where
where
and by density of the polyhedral sets in the class of the sets with finite perimeter we recover the isoperimetric inequality for all sets of finite perimeter.
For
which is the fractional isoperimetric inequality.
Funding statement: Andrea Kubin was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and by the Academy of Finland grant 314227. Domenico Angelo La Manna is a member of GNAMPA and has been partially supported by PRIN2022E9CF89.
Acknowledgements
We thank L. Gennaioli and G. Stefani for spotting a mistake in the first draft of this paper.
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Articles in the same Issue
- Frontmatter
- On the asymptotic behavior of a diffraction problem with a thin layer
- Degenerate parabolic p-Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions
- Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV
- A Sobolev gradient flow for the area-normalised Dirichlet energy of H 1 maps
- An elliptic approximation for phase separation in a fractured material
- The L 1-relaxed area of the graph of the vortex map: Optimal upper bound
- Compactness of Palais–Smale sequences with controlled Morse index for a Liouville-type functional
- Characterization of sets of finite local and non-local perimeter via non-local heat equation
- On nonexpansiveness of metric projection operators on Wasserstein spaces
- A continuous model of transportation in the Heisenberg group
- Universality of renormalisable mappings in two dimensions: the case of polar convex integrands
- Convergence of a heterogeneous Allen–Cahn equation to weighted mean curvature flow
- Harnack inequality for degenerate elliptic equations with matrix weights
- Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory
- The Capacitary John–Nirenberg Inequality Revisited
