Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
-
Bogi Kim
und
Abhrojyoti Sen
Abstract
This article is devoted to exploring the Lipschitz truncation method for parabolic multi-phase problems. The method is based on Whitney decomposition and covering lemmas with a delicate comparison scheme of appropriate alternatives to distinguish phases, as introduced by the first and second authors in [B. Kim and J. Oh, Higher integrability for weak solutions to parabolic multi-phase equations, J. Differential Equations 409 2024, 223–298].
1 Introduction
In this paper, we prove energy estimates for weak solutions to parabolic multi-phase problems of type
where
Energy estimates are very important for proving the existence and regularity results for partial differential equations. We prove the energy estimates using the Lipschitz truncation method for parabolic multi-phase problems. Here, the Lipschitz truncation is a method of redefining a given function so that it keeps its values on a specific “good set”, while redefining it in “bad sets” using a partition of unity related to a Whitney covering argument. Acerbi and Fusco [1, 2] have introduced the Lipschitz truncation for elliptic problems. Furthermore, Kinnunen and Lewis [30] and Kim, Kinnunen and Särkiö [28] have studied related methods for the parabolic p-Laplace system and for the parabolic double phase systems, respectively.
In this paper, we deal with a parabolic multi-phase system
where Carathéodory vector fields
and
where the function
for
We assume that the modulating coefficients
and
Here,
for all
Definition 1.1.
A function
and
is a weak solution to (1.1) if
for every
The parabolic multi-phase problems derive from elliptic double phase problems. The elliptic double phase problems of type
was first introduced in [35, 36, 37, 38]. These problems originate from the Lavrentiev phenomenon and the homogenization of strongly anisotropic materials. According to [13, 21], the conditions
and
are sharp for obtaining regularity results of weak solutions. In fact, when (1.7) or (1.8) holds, the gradient of a weak solution u is Hölder continuous, see [6, 14, 19, 13]. Moreover, the Calderón–Zygmund estimates have been discussed in [3, 17, 15]. Also, Baroni, Colombo and Mingione [5] have investigated the Harnack’s inequality. In addition, other regularity results for elliptic double phase problems have been discussed in [31, 32, 10, 9, 8, 11, 23, 22, 25]. The regularity results for elliptic multi-phase problems given by
with
On the other hand, the regularity for parabolic double phase problems
with
In this paper, our goal is to prove the energy estimates of the weak solution to (1.1). For this, we denote parabolic cylinders
with
Our main theorem of this paper is as follows.
Theorem 1.2.
Let u be a weak solution to (1.1). Then, for
Remark 1.3.
In [24], the energy estimate presented in their Lemma 3.1 is derived under the assumption that
Remark 1.4.
We would like to mention that the existence and uniqueness results for (1.1) with Dirichlet boundary condition can be obtained by closely following the proofs of [28, Theorem 2.6 and Theorem 2.7]. Therefore, we choose not to pursue such aspects in this paper.
To prove Theorem 1.2, we will construct the Whitney decomposition by dividing into p-,
2 Whitney decomposition and covering lemmas
In this section, we construct a family of Whitney decomposition and show that the family covers the bad set
2.1 Auxiliary definitions
Let
where
Let us define the strong maximal function for
where the supremum is taken over cubes
Lemma 2.1.
For
Without loss of generality we can assume that the modulating coefficient functions
and let
Let
Chebyshev’s inequality implies that
Now, we write
Note that for each
for
for
Using this distance, we construct a family of open subsets
Here, the intrinsic cylinders denoted as follows:
for any
Let
If Q is one of the four intrinsic cylinders defined above, we denote it as follows:
The integral average on a ball
We intend to use a Vitali-type argument to find a countable collection of points
and
Moreover, we denote
2.2 Some preliminary estimates
We start with some basic estimates.
Lemma 2.2.
Let
Proof.
We claim that
Now using
Raising the power to
Now using
which is a contradiction. This completes the proof of
and hence
It follows that
and this proves
Lemma 2.3.
Let
Proof.
First, we note that
Raising power
Using
Moreover, the proof of the first statement follows from the previous lemma. This completes the proof. ∎
Lemma 2.4.
Let
Proof.
Since
Moreover, following the previous lemma, we get
Also, the first statement follows from Lemma 2.2. This completes the proof. ∎
Lemma 2.5.
Let
Proof.
From Lemma 2.2, we get
Now, note that it is enough to prove
which is a contradiction. ∎
Lemma 2.6.
Let
Proof.
From Lemma 2.4, we have
for any
which gives a contradiction. This completes the proof. ∎
Lemma 2.7.
Let
Proof.
It is enough to prove the first statement. The second statement of the lemma follows from Lemma 2.2. We claim
which is a contradiction. ∎
Lemma 2.8.
Let
Proof.
The proof can be completed analogously to the argument presented in the proof of Lemma 2.2. ∎
Lemma 2.9.
Let
2.3 Vitali covering and their properties
In this subsection, we want to choose a countable collection of intrinsic cylinders from
First, we claim that
and
This is a contradiction, and we conclude
Let
Since
Then, in the same way as [28, Subsection 3.3], we have a countable subcollection
Since
We claim that
Let
Moreover, by (2.17) and the standard Vitali covering argument, we obtain that
Now, we prove the inclusion in (2.18) in the time direction, by considering the sixteen cases depicted in Table 1.
The combinations of
|
|
|
|
|
|
|
|
(1-1) | (1-2) | (1-3) | (1-4) |
|
|
(2-1) | (2-2) | (2-3) | (2-4) |
|
|
(3-1) | (3-2) | (3-3) | (3-4) |
|
|
(4-1) | (4-2) | (4-3) | (4-4) |
Case
and hence
Cases
Case
From
Since (1.5) implies
Thus, we conclude
and hence
Cases
Case
Then (2.20) implies
and hence
Cases
Cases
Thus, we have established a countable covering family
Lemma 2.10.
We have
Proof.
Since, by the definition of
the second inequality is satisfied. Moreover, from the triangle inequality we obtain that, for any
Since
Next, we prove the property (v) in Lemma 2.18.
Lemma 2.11.
We have
Proof.
It is enough to prove that
for
Let
To complete the proof, we consider 16 cases in Table 1 with
Case
Case
where the last inequality follows from
and hence, by (2.22), we have the conclusion.
Case
Case
Case
and, similarly,
Thus, we obtain
By this lemma, we get
for all
Lemma 2.12.
For any
We show from the previous two lemmas that the measures of the neighboring cylinders are comparable.
Lemma 2.13.
There exists c depending on n and K such that
Proof.
We divide it into sixteen cases as in Lemma 2.11.
Case
for all
Case
Case
for all
Case
Now, we establish the inclusion property of
Lemma 2.14.
Let
Proof.
Since
and hence
Lemma 2.15.
Let
Proof.
We may assume that
Since
Thus, we have
and hence
Lemma 2.16.
Let
Proof.
The proof can be obtained by using
Through the above three lemmas, we obtain the condition
where
Lemma 2.17.
There exists a constant c depending only on n and K such that
Proof.
Since the cylinders
and hence
We summarize the above results below.
Lemma 2.18.
Let K be as in (2.1) and
There exists a constant
If
If
For any
2.4 Partition of unity
The following lemma demonstrates the construction of a partition of unity subordinate to Whitney decomposition
Lemma 2.19.
There exists a partition of unity
There exists a constant
There exists a constant
for any
Proof.
In the previous subsection, we obtain the Whitney decomposition
and
Since
is well-defined and satisfies
Thus, the collection
for any
3 Construction of test function via Lipschitz truncation
The main goal of this section is to construct a Lipschitz function
3.1 Definition of test function
Take
Let
Now for
as a suitable candidate for test function to be used to prove energy estimates. For
where
Similarly, we define
where
3.2 Preliminary lemmas
In this subsection, we discuss some preparatory machinery to prove Lemma 3.5. We denote a family of parameters as
First, we recall the following lemma from [7, Lemma 8.1].
Lemma 3.1.
Let
where
Lemma 3.2.
Let
Moreover, there exists a constant
Proof.
Let
and that proves (3.5). By Lemma 2.18 (iv), we have
Case I:
Hence, we get
and this proves (3.6) for this case.
Case II:
By Lemma 2.18 (iv), there exists a
Hence, from (3.7), we have
Since
and finally, we have
which proves (3.6) for this case.
Case III:
By Lemma 2.18 (iv), there exists a
Hence, from (3.8), we have
Since
and finally, we have
which proves (3.6) for this case.
Case IV:
Now we prove a parabolic Poincaré-type result.
Lemma 3.3.
Let
There exists a constant
(3.9)If in addition
(3.10)
In addition, the above estimates hold with
Proof.
For
Furthermore, let
By taking a test function
By the definition of f, (1.2) and (1.3), we have
First, we get from Lemma 3.1 that
and
Note that
Therefore, we have
Next, we obtain from the one-dimensional Lebesgue differentiation theorem that
Combining the above inequalities, we conclude that
where c depends only on
Now, we estimate the left-hand sides of (3.9) and (3.10). First, we prove (3.9). By the standard Poincaré inequality, we have
(3.12)
It follows from the definition of f, Lemma 3.1 and (3.1) that
Moreover, we obtain
(3.14)
To estimate the first term, we use the fact that
(3.15)
Combining (3.11), (3.12), (3.13), (3.14) and (3.15), we have the conclusion.
Finally, we prove (3.10). As in (3.12), we obtain
Then we have
Since
Thus, combining (3.11), (3.13), (3.15), (3.16), (3.17) and (3.18), we have the conclusion. ∎
Next, we recall the boundary version of the Poincaré inequality from [29, Theorem 6.22].
Lemma 3.4.
Let
3.3 Poincaré-type inequality for the test function
In this subsection, we prove the Poincaré-type inequality for
Lemma 3.5.
Let
If
(3.19)(3.20)If
(3.21)(3.22)
Proof.
We start by proving (3.20). Note that we assume
Case I:
Since
Plugging (3.23) in the above estimate, we get
Since
Using Hölder’s inequality and applying (3.6), we further estimate
where the last inequality follows from Lemma 2.18 (ix). Plugging the above estimate in (3.24), we obtain
Case II:
Note that
The last part of the above calculation follows from Case I.
Case III:
The estimates of
Case IV:
From Lemma 2.18 (x), we have
where the last inequality follows from (3.6). This completes the proof of (3.20).
Now we prove (3.22). If
and using
which completes the proof of (3.22).
Next, we focus on the estimates involving Steklov averages, i.e., (3.19) and (3.21). If
We note that
This completes the proof of (3.19). To prove (3.21) in the case
Now using Lemma 3.1 and (3.5), we estimate the right-hand side of the above expression as
which gives the proof of (3.21). ∎
Corollary 3.6.
We have the following estimates on
- (3.25)
- (3.26)
- (3.27)
- (3.28)
Proof.
We start by proving (i). Indeed, we obtain
If
If
The proof of (ii) follows from (3.19) and (3.21) as above.
Now, let us prove (iv) and the proof of (iii) is similar. First, we assume
where the last inequality follows from (3.20).
On the other hand, let
The other case
3.4 Bounds on Lipschitz truncation and its derivatives
In this subsection, we show that
Lemma 3.7.
We have
Proof.
Fix
Moreover, we have from Lemma 2.19 that
Therefore, we conclude from Lemma 2.18 (xi) and Lemma 2.19 that
On the other hand, for each
As in (3.29), we have
and hence we have the conclusion. ∎
Lemma 3.8.
For any
Furthermore, we have
and
Proof.
Fix
and
Note that Lemma 2.19 implies
Thus, we get from Lemma 2.18 (xi), Lemma 2.19 and Corollary 3.6 that
and
Next, by the above arguments, we get
and
Therefore, by Lemma 2.18 (xi), Lemma 2.19 and Corollary 3.6, we have the conclusion. ∎
Lemma 3.9.
Let
for almost every
Proof.
We start with proving (i):
Now using Corollary 3.6 (ii) and (3.32), we can estimate the last term of the above inequality to obtain
The proof of (ii) can be obtained similarly. To prove (iii), first let
Now let us consider
Case I:
Case II:
Case III:
Case IV:
This completes the proof. ∎
3.5 Lipschitz regularity of
v
h
Λ
In this subsection, we show that
where
Lemma 3.10 (Campanato characterization).
Assume that
for every
Proof.
Since
This completes the proof. ∎
We remark that the above conclusion holds for every
Lemma 3.11.
There exists a constant
for every
Proof.
We first note that, since
Applying Lemma 3.10, we get
(3.37)
Note that
We fix the cube
Case 1:
Here,
Lemma 3.8 implies the uniform estimate
for all
To prove (3.38), let
Thus, we conclude
Case 2:
We denote
On the other hand, if
Recalling that in this case
Case 3:
We want to show that the radii
and hence
Thus, we have
Note that
As in Case 2, we can estimate the second term on the right-hand side. To estimate the first term, we obtain from the definition of
Since
Combining the previous inequalities with (3.25), we obtain
Since
Thus, the proof is completed. ∎
Corollary 3.12.
Let
Proof.
Since
This completes the proof. ∎
3.6 Some more properties of Lipschitz truncation
In the following proposition, we collect some more properties of Lipschitz truncation.
Proposition 3.13.
Let
Proof.
To show (i), we first note from Definition 1.1 and the definition of Steklov averages that
Moreover, the definition of
The proof of (ii) is obvious from Lemma 3.9 (iii). Indeed, we have
The proof of (iii) follows from the definitions (3.3) and (3.4). In fact, since
Since
The proof of (v) follows from
4 Proof of energy estimate
In this section, we prove the energy estimate in Theorem 1.2 using the Lipschitz truncation
4.1 Proof of Theorem 1.2
We closely follow the proof of [28, Subsection 5.1]. For
Note that, by Proposition 3.13 (i),
Now we estimate the each term above by dividing the integral domain into
Estimate of
I
.
By integration by parts and the product rule, we obtain
First, we consider
Note that integration by parts implies
Since
Letting
Since the Lipschitz truncation is done only in the bad set
Since
For the same reason, we also have
Now, we estimate
Then Hölder’s inequality implies
Since
Thus, we get
and hence we conclude
Estimate of
II
.
As in [28], we obtain
Since the integral within the good set does not contain Λ, letting
where
where c depends only on
Moreover, it is easy to show
Estimate of
III
.
The estimate for
Applying (1.3) and Young’s inequality, we get
for some
Combining all the estimates for
Finally, we complete the proof by letting
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: RS-2023-00217116
Award Identifier / Grant number: RS-2025-00555316
Funding statement: Bogi Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. RS-2023-00217116). Jehan Oh is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. RS-2025-00555316). Abhrojyoti Sen is supported by research grants from the Alexander von Humboldt Foundation for postdocs.
Acknowledgements
The authors would like to express their sincere gratitude to the anonymous referee who provided valuable comments and suggestions on the earlier version, which have greatly improved the quality and clarity of the manuscript.
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Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
