Abstract
Building upon the recent results in [M. Focardi and E. Spadaro,
On the measure and the structure of the free boundary of the lower-dimensional obstacle problem,
Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the
free boundary for the solutions to the fractional obstacle problem in
local finiteness of the
classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most
Instead, if
1 Introduction
Quasi-geostrophic flow models [10], anomalous diffusion in disordered media [4] and American options with jump processes [11] are some instances of constrained variational problems involving free boundaries for thin obstacle problems.
In this paper we analyze the fractional obstacle problem with
exponent
on the cone
where
for
The most challenging regularity issues are then that of w itself and that of its free boundary
To investigate the fine properties of the solution w of (1.1)
the groundbreaking paper by Caffarelli and Silvestre [5] introduces
an equivalent local counterpart for the fractional obstacle problem in terms
of the so called a-harmonic extension argument.
Indeed, it is inspired by the case
In particular, note that u is the unique minimizer of the Dirichlet energy
on the class
One then is interested into the regularity of the
free boundary
The locality of the operator
in (3.1) is the main advantage of the new formulation
to perform the analysis of
Optimal interior regularity for u has been established Caffarelli, Salsa and Silvestre
in [9, Theorem 6.7 and Corollary 6.8] for any
termed in the existing literature as the subset of regular, singular and nonregular/nonsingular points, respectively.
These sets are defined via the infinitesimal behavior of appropriate rescalings of the
solution itself. More precisely, for
then the family of functions
According to the regularity of φ different results are known in literature:
Singular points: for φ analytic and
It is also worth mentioning the paper by Barrios, Figalli and Ros-Oton [3],
in which the authors study the fractional obstacle problem (1.1) with
nonzero obstacle φ having compact support and satisfying suitable
concavity assumptions. Under these assumptions, they are able to fully characterize
the free boundary, showing that
For ease of expositions we start with the simpler case in which the
obstacle function φ is analytic, actually the slightly milder assumption
(1.5) below suffices (see Section 3 for related results in
the case
Theorem 1.1.
Let u be a solution to the fractional obstacle problem
(1.2) with obstacle function
Then:
The free boundary
(1.6)where
The free boundary
(1.7)
Moreover, there exists a subset
The analysis is more involved in case φ is not analytic,
since one cannot in principle avoid contact points of infinite order
between the solution and the obstacle, and the free boundary
can be locally an arbitrary compact set
where
Theorem 1.2.
Let u be a solution to the fractional obstacle problem
(1.2) with obstacle function
This note extends the results of [18] to the case of nonconstant obstacles.
It is clear by the examples of arbitrary compact sets as contact sets of suitable
solutions of the problem, that in general the free boundary for nonconstant smooth
obstacles does not possess any structure and that the key ingredient for the analysis
of the free boundary is the analyticity of the obstacles as shown in Theorem 1.1.
Nevertheless, for a distinguished of the free boundary, characterized as those points of
finite order of contact, e.g. the points
2 Analytic obstacles
In this section we deal with analytic obstacles. We report first on some results related to the Caffarelli–Silvestre a-harmonic extension argument that will be instrumental to reduce the analytic-type fractional obstacle problem to the lower-dimensional obstacle problem. We provide then the proof of Theorem 1.1.
2.1 Extension results
We start off with a lemma in which it is proved that there exists a canonical a-harmonic extension
of a polynomial in the class of polynomials (see [21, Lemma 5.2]).
We denote by
Lemma 2.1.
For every
Proof.
Let
with
Remark 2.2.
In particular,
We provide next the main result that reduces locally the analytic case to the zero obstacle case (cf. [21, Lemma 5.1]).
Lemma 2.3.
Let
Proof.
For every
2.2 Proof of Theorem 1.1
Theorem 1.1 follows straightforwardly
from [18, Theorems 1.1–1.3].
As explained in the introduction
3
C
k
+
1
obstacles
In this section we deal with the more demanding case of
for
For any
where
and that
for all unit vectors
Let then
where
Set
and
Recalling that
First, from the regularity assumption on φ, from Lemma 2.1 and from
estimate (3.2) we infer that
In turn, this yields that the distribution
The following result resumes the regularity theory developed in [9, Proposition 4.3].
Theorem 3.1.
Let u be a solution to the fractional obstacle problem (3.1)
in
where
In particular, the function u is analytic in
In particular,
Furthermore, for
where in the second equality we have used that
In particular, since the last addend in (3.13) only depends on the boundary values of
among all functions
Remark 3.2.
We record here some bounds that shall be employed extensively
in what follows. By using the linearity and continuity of the extension
operator
for some constant
for some constant
3.1 A frequency-type function
Building upon the approach developed in [18], we consider
a quantity strictly related to Almgren’s frequency function (see [12])
and instrumental for developing the free boundary analysis in the subsequent
sections. Let
then given the solution u to (3.1), a point
where
and
Here
For later convenience, we introduce also the notation
and
In particular, note that for all
by Cauchy–Schwarz inequality.
Remark 3.3.
In case
Remark 3.4.
If u is a solution to the fractional obstacle problem (3.1),
then for every
and
Thus,
In particular, this shows that the frequency function is scaling invariant, in the sequel we will use this property repeatedly.
3.2 Almost monotonicity of
I
u
x
0
at distinguished points
In this subsection we prove the quasi-monotonicity of
Lemma 3.5.
Let u be a solution to the fractional obstacle problem (3.1) in
Remark 3.6.
With an abuse of notation, the integration in the last addends in (3.19) and (3.21) is meant with respect
to the reference measure
Proof.
To show (3.19), (3.20) and (3.21),
we assume without loss of generality that
For (3.19) we consider the vector field
so that
Therefore, (3.19) follows from the divergence theorem by taking into account that V is compactly supported.
Next, (3.20) is a consequence of the direct computation
Finally, to prove (3.21) we consider the compactly supported vector field
Moreover, conditions (3.10)-(3.12) and Lemma 2.1 imply that
Therefore, we infer that
and we conclude (3.21) by direct differentiation since
As a consequence we derive a first monotonicity formula for
Corollary 3.7.
Let u be a solution to the fractional obstacle problem (3.1).
Then, for all
In particular, if
Moreover, for all
Proof.
The proof of (3.22) (and hence of (3.23) and (3.24)) follows from the differential equation in (3.20). The proof of (3.25) is a simple consequence of a dyadic integration argument:
where in the last inequality we used that
We establish next an auxiliary lemma containing useful bounds for some quantities
related to the
Lemma 3.8.
Let u be a solution to the fractional obstacle problem (3.1) in
Proof.
By the co-area formula for Lipschitz functions we check that
and
Therefore, an integration by parts gives
By (3.8), as
with
Next we show an explicit expression for the radial derivative
of
Proposition 3.9.
Let u be a solution to the fractional obstacle problem (3.1) in
for
and
Proof.
It is not restrictive to assume
where
From this we conclude (3.32) straightforwardly.
For (3.33), we estimate separately each term appearing in the integral
defining
with
and
with
Estimate (3.33) turns out to be useful to analyze the subsets of points
with
Remark 3.10.
Note that
Proposition 3.11.
For every
Proof.
Since
therefore, for
In addition, from (3.19) and (3.2) we get for all
Therefore, from (3.33), if
Hence, from (3.18), (3.32) and (3.2) we find
and the monotonicity of
The proof is complete. ∎
Remark 3.12.
The monotonicity for the truncated Almgren’s frequency function
proved in [9] and [21] is essentially equivalent to Proposition 3.11.
We derive next an additive quasi-monotonicity formula for the frequency.
Corollary 3.13.
For every
3.3 Lower bound on the frequency and compactness
We first show that the frequency of a solution u to
(3.1) at points in
Lemma 3.14.
For every
Proof.
In view of (3.26) and since
Inequality (3.45) is a straightforward consequence of estimate (3.43) and
the latter estimate provided that
For the free boundary analysis developed in [18] it is mandatory to consider the critical set of a solution. In the current framework, the natural substitute for the critical set is given by
Notice that
We can then give the following compactness result.
For
Note that
with obstacle function
among all functions
Corollary 3.15.
Let
Proof.
By taking into account inequality (3.43) in Proposition 3.11, we get for l large,
In particular, we infer that
among all functions
By taking into account that
Therefore, one can easily show that the sequence
defined by
if
Items (3.50)–(3.52) are then a straightforward consequence of Theorem 3.1 and (3.53) (cf. the arguments in [9, Lemma 6.2]). ∎
A sharp lower bound on the frequency then follows.
Corollary 3.16.
Let
Proof.
Note that
4 Main estimates on the frequency
In this section we prove the principal estimates on the frequency
that we are going to exploit in the sequel. We start with an elementary lemma. Recall that all obstacles functions φ
are assumed to satisfy the normalization condition
Lemma 4.1.
Let
Remark 4.2.
Note that as a byproduct of the first estimate in (4.1)
in Lemma 4.1 frequencies at the scale ϱ
are well-defined at every point
Proof.
In order to prove (4.1), we argue by contradiction:
we can assume that there exist
In particular, by almost monotonicity of the frequency function (cf. Proposition 3.11)
and the lower bound on the frequency (cf. Corollary 3.16) we infer that
To prove the first set of inequalities in (4.1), we compute
Moreover, by estimate (3.15) in Remark 3.2 and since
Therefore, recalling that
Since
Moreover, by (4.3) and (4.5), by the strong
Being the left-hand side finite, necessarily
Hence,
The second set of inequalities in (4.1) is proven by the same argument. Indeed, assuming that
we have
and since by (3.16) in Remark 3.2 and by (3.39),
we get (recall
By the strong convergence of
Thus, by analyticity
The latter equality contradicts
that follows from strong
Finally, (4.2) follows straightforwardly from (4.1) for
We introduce the following notation for the radial variation of
(modified) frequency at a point
Note that
Lemma 4.3.
Let
Proof.
Without loss of generality we prove the result for
We now use the integral estimate (whose elementary proof is left to the readers)
In the last inequality we have used that, if
Hence, from (4) we conclude that
where we used that
4.1 Oscillation estimate of the frequency
The following lemma shows how the spatial oscillation of the frequency in two nearby points at a given scale is in turn controlled by the radial variations at comparable scales.
Proposition 4.4.
Let
for every
Proof.
We start off noting that by Remark 4.2 and the
choice
To ease the readability of the proof we divide it in several substeps.
Step 1.
With fixed
and consider the map
Set
Recalling the very definition of
because by linearity (the details of the elementary computations are left to the readers)
and
Moreover, from the very definition of
Thus, from (4.11) and the latter equality, by direct calculation it follows that
and thus we may conclude that
Moreover, note also that
Step 2. Thanks to the previous formulas, for all
In addition, integrating by parts gives
Then, by formula (3.19) together with (4.1) and (4.16), we have that
In what follows we estimate separately the two terms
Step 3. We start off with
We estimate separately the factors of the integrand defining
with
Using inequalities (3.15)–(3.16) in Remark 3.2, we estimate the last two addends as follows:
for some constant
For the second factor, we note that for
To estimate the last three addends, we have used the very definition of
By collecting (4.17) and (4.18), using Hölder
inequality we conclude that there exists
Clearly, we have that
In the second inequality, we have used that
In addition, thanks to (3.25) and
By collecting (4.19)–(4.21), we conclude that for some
where, in the last inequality, we have used Lemma 4.1
and that
Finally, in view of the very definition of the spatial oscillation of the frequency and Corollary 3.13, we deduce for some constant depending on A that
Step 4. We estimate next
Then, arguing as in (4.4), thanks to
estimate (3.15) in Remark 3.2,
we get as
In addition, (3.8) and (4.13) yield
Therefore, by applying repeatedly Lemma 4.1,
by taking into account (3.43) and by choosing
since
The conclusion in (4.10) follows at once from estimates (4.22) and (4.1). ∎
5 Proof of the main result
5.1 Mean-flatness
Here we show a control of the Jones’ β-number by the oscillation of the frequency.
Given a Radon measure μ in
where the infimum is taken among all affine
If
and
Then
where
Proposition 5.1.
Let
Proof.
The proof is a variant of the [18, Proposition 4.2], which in turn follows closely the original arguments by Naber and Valtorta in [28, 29], therefore we only highlight the main differences.
Without loss of generality assume that
Since
From the very definitions of
(5.4)
where
Next we estimate the two sides of (5.4).
For estimating the left-hand side of (5.4), we can show by compactness that
Here we use the same contradiction argument in [18, Proposition 4.2] using the compactness given by Corollary 3.15.
For what concerns the right-hand side of (5.4) we proceed as follows. By the triangular inequality we have
The addends
For
Note that the extra term with respect to [18, Proposition 4.2] arises as a consequence of the additional error term in Proposition 4.4.
To estimate
Then we use estimates (3.15) and (3.16) in Remark 3.2
to deduce that for all
Therefore, integrating the last estimate we conclude that
We can now collect estimates (5.5)–(5.9) and use Corollary 3.13 to get
Finally, by assumption
5.2 Rigidity of homogeneous solutions
In this subsection we extend the results on the rigidity of almost homogeneous solutions established in [18].
We denote by
and set
As observed in [18], the maximal dimension of the spine
of a function in
We next introduce the notion of almost homogeneous solutions.
Given
Definition 5.2.
Let
The following lemma justifies this terminology and it is the analog of [18, Lemma 5.5].
Lemma 5.3.
Let
for some homogeneous solution
Proof.
The proof follows by a contradiction argument similar to [18, Lemma 5.5].
Assume that for some
and satisfying the bounds
Consider
in turn implying that
and thus we would conclude that
A rigidity property of the type shown in [18, Proposition 5.6] holds for the nonzero obstacle problem.
Proposition 5.4.
Let
Either for every point
(5.13)or there exists a linear subspace
(5.14)
Proof.
The proof proceeds by contradiction and follows the strategy
developed in [18, Proposition 5.6].
Let
there exists
(5.15)for every linear subspace
(5.16)
We consider the rescaled functions
If
In addition, by (3.10) and being
Hence, we conclude that
which is a contradiction to the constancy of the frequency
at critical points of the homogeneous solution
Moreover, (3.43)
can be employed in the last two equalities as
Instead, if
Again, note that (3.43) can be employed since
5.3 Proof of Theorem 1.2
We start off noting that it suffices to prove that
Then, recalling the definition of
where
Hence, we may establish the result for
Note that as
solves the fractional obstacle problem (3.1) in
with
We choose
In such a case the proof, of the main results can be obtained by following verbatim [18, Sections 6–8].
Indeed, [18, Proposition 6.1], that leads both to the local finiteness of the Minkowski content of
Finally, the classification of blowup limits is exactly that stated in [18, Theorem 1.3], and proved in [18, Section 8], in view of Lemma 3.14 and Corollary 3.15.
Funding statement: The authors have been partially funded by the ERC-STG Grant n. 759229 HiCoS “Higher Co-dimension Singularities: Minimal Surfaces and the Thin Obstacle Problem” and by GNAMPA of INdAM.
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