Abstract
We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of p-Laplace type when \(\tfrac{2n}{n+2}< p\le 2\). The result is based on a reverse Hölder inequality in intrinsic cylinders combining p-intrinsic and (p, q)-intrinsic geometries. A singular scaling deficits affects the range of q.
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1 Introduction
This paper discusses the local higher integrability of the spatial gradient of weak solutions \(u=u(z)=u(x,t)\) to parabolic double-phase systems with the prototype
in \(\Omega _T=\Omega \times (0,T)\), where \(\Omega \) is a bounded domain in \({\mathbb {R}}^n\), \(n\ge 2\), and \(T>0\). The coefficient function \(a\in C^{\alpha ,\alpha /2}(\Omega _T)\) is non-negative and Hölder continuous. The higher integrability result in Theorem 2.2 was obtained when \(p\ge 2 \) in [11] and here we extend this result to singular the parameter range. More precisely, in this paper we assume that
When \(p=\tfrac{2n}{n+2}\) we have \(\mu = 0\), while at \(p=2\) the range of q is the same as in [11]. Note that \(\tfrac{\mu }{p}\) is the usual scaling deficit appearing in singular p-parabolic problems, cf. [7, Section VIII]. An upper bound for q in terms of \(p, \alpha \) and n appears naturally in regularity properties of double-phase problems. Otherwise the solution may not be regular already in the elliptic case, see [8].
The method for showing the higher integrability result in this paper originates from [13] where the result was shown for parabolic p-Laplace systems. There a reverse Hölder inequality was shown in p-intrinsic cylinders as in (2.1). See also [2] for the gradient higher integrability of p(x, t)-Laplace systems. On the other hand, in [10] the same result was shown for the Orlicz setting including parabolic (p, q)-Laplace problems (corresponding to \(a(z)=a_0\) for some constant \(a_0>0\)) in (p, q)-intrinsic cylinders. For the double-phase model \(\inf a(z)\) may be zero and a priori neither the p-term nor the q-term dominates. Instead, the behaviour of the system varies locally between two distinct phases based on which of the terms is dominating. To incorporate this into the argument, we divide into cases at every point \(z_0\) by comparing \(a(z_0)\) to the level of the gradient. If \(a(z_0)\) is sufficiently small, we show a reverse Hölder inequality in a p-intrinsic cylinder. In the complementary case, it follows that a(z) is comparable to \(a(z_0)\) in a sufficiently large neighborhood of \(z_0\) and the reverse Hölder inequality can be shown in a (p, q)-intrinsic cylinder.
To construct the intrinsic cylinders, we use a stopping time argument to find a p-intrinsic cylinder at every point in a suitable upper level set. Moreover, we obtain a decay estimate for the radius of a p-intrinsic cylinder in terms of the level. This estimate, stated in Lemma 3.1, gives the comparability of a(z) around (p, q)-intrinsic cylinders, see the property (p,q-2). Lemma 3.1 is also used in the p-intrinsic case to transform terms involving q into terms of a p-Laplace system, for example in the proof of Lemma 3.7. This argument gives the range of q in (1.1), see Remark 3.2. Note that (1.1) allows for the situation that \(q>2\) while \(p<2\). However, this case does not have to be considered separately and the division to p- and (p, q)-intrinsic cylinders is sufficient.
Stationary double-phase problems have been studied extensively in [1, 3,4,5,6]. Note that the double-phase model in these papers is not included in the (p, q)-problems studied for instance in [14]. For parabolic double-phase problems existence has been studied in [15] and [12] while many regularity questions remain open.
2 Notation and main result
2.1 Notation
We denote a point in \({\mathbb {R}}^{n+1}\) as \(z=(x,t)\), where \(x\in {\mathbb {R}}^n\) and \(t\in {\mathbb {R}}\). A ball with center \(x_0\in {\mathbb {R}}^n\) and radius \(\rho >0\) is denoted as
Parabolic cylinders with center \(z_0=(x_0,t_0)\) and quadratic scaling in time are denoted as
where
We use the following notation for the double-phase functional. With the non-negative coefficient function \(a(\cdot )\) being fixed, we define a function \(H(z,s):\Omega _T\times {\mathbb {R}}^+\longrightarrow {\mathbb {R}}^+\) as
We use two types of intrinsic cylinders. For \(\lambda \ge 1\) and \(\rho > 0\), a p-intrinsic cylinder centered at \(z_0=(x_0,t_0)\) is
and a (p, q)-intrinsic cylinders centered at \(z_0=(x_0,t_0)\) is
Note that \(\tfrac{\lambda ^p}{H(z_0,\lambda )}\rho ^2=\tfrac{\lambda ^2}{H(z_0,\lambda )}(\lambda ^\frac{p-2}{2}\rho )^2\) and thus \(G_\rho ^\lambda (z_0)\) is the standard intrinsic cylinder for (p, q)-Laplace system. For \(c>0\), we write
We also consider parabolic cylinders with arbitrary scaling in time and denote
The \((n+1)\)-dimensional Lebesgue measure of a set \(E\subset {\mathbb {R}}^{n+1}\) is denoted as |E|. For \(f\in L^1(\Omega _T,{\mathbb {R}}^N)\) and a measurable set \(E\subset \Omega _T\) with \(0<|E|<\infty \), we denote the integral average of f over E as
2.2 Main result
We consider weak solutions to the parabolic double-phase system
in \(\Omega _T=\Omega \times (0,T)\), where \(\Omega \) is a bounded domain in \({\mathbb {R}}^n\), \(n\ge 2\), and \(T>0\). Here \({\mathcal {A}}(z,\nabla u):\Omega _T\times {\mathbb {R}}^{Nn}\longrightarrow {\mathbb {R}}^{Nn}\) with \(N\ge 1\) is a Carathéodory vector field satisfying the following structure assumptions: there exist constants \(0<\nu \le L<\infty \) such that
for almost every \(z\in \Omega _T\) and every \(\xi \in {\mathbb {R}}^{Nn}\). The source term \(F:\Omega _T\longrightarrow {\mathbb {R}}^{Nn}\) satisfies
We assume that \(a \ge 0\) and \(a\in C^{\alpha ,\frac{\alpha }{2}}(\Omega _T)\) for some \(\alpha \in (0,1]\). Here \(a\in C^{\alpha ,\frac{\alpha }{2}}(\Omega _T)\) means that \(a\in L^{\infty }(\Omega _T)\) and there exists a constant \([a]_\alpha =[a]_{\alpha ,\alpha /2;\Omega _T}<\infty \), such that
for every \((x,y)\in \Omega \) and \((t,s)\in (0,T)\).
Definition 2.1
A map \(u:\Omega _T\longrightarrow {\mathbb {R}}^N\) satisfying
and
is a weak solution to (2.3), if
for every \(\varphi \in C_0^\infty (\Omega _T,{\mathbb {R}}^N)\).
The main result of this paper is the following higher integrability estimate for the gradient of a weak solution to (2.3). The constants depend on
Theorem 2.2
Let u be a weak solution to (2.3). There exist constants \(0<\epsilon _0=\epsilon _0( data )\) and \(c=c( data ,\Vert a\Vert _{L^\infty (\Omega _T)})\ge 1\), such that
for every \(Q_{2r}(z_0)\subset \Omega _T\) and \(\epsilon \in (0,\epsilon _0)\).
2.3 Auxiliary lemmas
We start with two estimates derived from the weak formulation of (2.3). A priori Definition 2.1 does not guarantee that u can be used as a test function in the weak formulation and thus we do not immediately obtain the following Caccioppoli inequality. A Lipschitz truncation method could be used as in the degenerate case [12], but we omit the proof since it is beyond the scope of this paper.
Lemma 2.3
Let u be a weak solution to (2.3). There exists a constant \(c=c(n,p,q,\nu ,L)\), such that
for every \(Q_{R,\ell }(z_0)\subset \Omega _T\), with \(R,\ell >0\), \(r\in [R/2,R)\) and \(\tau \in [\ell /2^2,\ell )\).
The following parabolic Poincaré inequality can be shown in the same way as in [11].
Lemma 2.4
Let u be a weak solution to (2.3). There exists a constant \(c=c(n,N,m,L)\), such that
for every \(Q_{R,\ell }(z_0\subset \Omega _T\) with \(R,\ell >0\), \(m\in (1,q]\) and \(\theta \in (1/m,1]\).
Finally, we have two technical lemmas. The first lemma is a Gagliardo–Nirenberg inequality and the second one is a standard iteration lemma, see [9, Lemma 8.3].
Lemma 2.5
Let \(B_{\rho }(x_0)\subset {\mathbb {R}}^n\), \(\sigma ,s,r\in [1,\infty )\) and \(\vartheta \in (0,1)\) such that
Then there exists a constant \(c=c(n,\sigma )\), such that
for every \(v\in W^{1,s}(B_{\rho }(x_0))\).
Lemma 2.6
Let \(0<r<R<\infty \) and \(h:[r,R]\longrightarrow {\mathbb {R}}\) be a non-negative and bounded function. Suppose there exist \(\vartheta \in (0,1)\), \(A,B\ge 0\) and \(\gamma >0\) such that
Then there exists a constant \(c=c(\vartheta ,\gamma )\), such that
3 Reverse Hölder inequality
In this section we provide a reverse Hölder inequality for u, a weak solution to (2.3). The reverse Hölder inequality is used to show the higher integrability result in the next section. We consider the p-intrinsic and (p, q)-intrinsic cases in separate subsections. In both cases we show parabolic Sobolev–Poincaré inequalities and a series of estimates leading to the reverse Hölder inequality.
Throughout this section, let \(z_0=(x_0,t_0)\in \Omega _T\), with \(x_0\in \Omega \) and \(t_0\in (0,T)\), be a Lebesgue point of \(|\nabla u(z)|^p+a(z)|\nabla u(z)|^q\) satisfying
for some \(\Lambda >1+\Vert a\Vert _{L^\infty (\Omega _T)}\). Note that \(H(z_0,s)\) is strictly increasing and continuous with
Therefore, by the intermediate value theorem for continuous functions, there exists \(\lambda =\lambda (z_0)>1\), such that
We also use the constants
and
In the p-intrinsic case we consider a cylinder \(Q_\rho ^\lambda (z_0)\) defined as in (2.1) and assume the following:
-
(p-1)
p-intrinsic case: \(K\lambda ^{p}\ge a(z_0)\lambda ^q\).
-
(p-2)
Stopping time argument for a p-intrinsic cylinder:
- (p-i)
-
(p-ii)
for every \(s\in (\rho ,2\kappa \rho ]\).
In the (p, q)-intrinsic case we consider a cylinder \(G_\rho ^\lambda (z_0)\) defined as in (2.2) and assume the following:
-
(p,q-1)
(p, q)-intrinsic case: \(K\lambda ^{p}< a(z_0)\lambda ^q\).
-
(p,q-2)
\(\tfrac{a(z_0)}{2}\le a(z)\le 2a(z_0)\) for every \(z\in G_{4\rho }^\lambda (z_0).\)
-
(p,q-3)
Stopping time argument for a (p, q)-intrinsic cylinder:
- (p,q-i)
-
(p,q-ii)
for every \(s\in (\rho ,2\kappa \rho ]\).
The fact that these two cases are complementary will be shown in Sect. 4.1.
The following decay estimate will be used in this and the next section. Note that the estimate holds without assumption (p-1).
Lemma 3.1
Assumption (p-i) implies
where \(\mu >0\) is defined in (1.1).
Proof
It follows from (p-i) and (3.1) that
Therefore, we have by (1.1) that \(\rho ^\alpha \le M_1^\frac{\alpha }{n+2}\lambda ^{-\frac{\alpha \mu }{n+2}}\le \tfrac{K}{40[a]_\alpha }\lambda ^{-\frac{\alpha \mu }{n+2}}\). Also \(\rho ^\alpha \lambda ^q\le \tfrac{K}{40[a]_\alpha }\lambda ^p\) follows from (1.1). \(\square \)
Remark 3.2
The range of q is determined to satisfy the second inequality of (3.2) and this is where the intrinsic deficit appears in the range of q. Although it is not mentioned in [11], the same argument holds for the degenerate case.
3.1 The p-intrinsic case
In this subsection we show a reverse Hölder inequality in the p-intrinsic cylinder \(Q_\rho ^\lambda (z_0)\) satisfying (p-1), (p-2) and \(Q_{2\kappa \rho }^\lambda (z_0)\subset \Omega _T\). The scaling deficit \(\mu \) defined in (1.1) plays a role throughout the argument. In particular, note that \(0<p-1-\tfrac{\alpha \mu }{n+2}<1\). We begin by estimating the last term in Lemma 2.4.
Lemma 3.3
For \(s\in [2\rho ,4\rho ]\) and \(\theta \in ((q-1)/p,1]\), there exists a constant \(c=c(n,p,q,\alpha ,L,[a]_{\alpha },M_1)\), such that
Proof
By (2.4) there exists a constant \(c=c([a]_{\alpha })\), such that
We apply (p-1) to estimate the second term on the right-hand side of (3.3) and obtain
In order to estimate the last term on the right-hand side of (3.3), note that by (3.2) we have
As \(q-1<p\) by \(\alpha \le 1 \) and (1.1), it follows from Hölder’s inequality, (3.4) and (p-ii) that
where \(c=c(n,p,\alpha ,M_1)\) and \(\theta \in ((q-1)/p,1]\). It follows from (p-ii), \(\lambda \ge 1\) and (1.1) that
where \(c=c(n,p,q,\alpha )\). We conclude that
where \(c=c(n,p,q,\alpha ,M_1)\). Similarly, replacing \(|\nabla u|\) by |F| in the above argument, we have
This completes the proof. \(\square \)
Next, we provide a p-intrinsic parabolic Poincaré inequality.
Lemma 3.4
For \(s\in [2\rho ,4\rho ]\) and \(\theta \in ((q-1)/p,1]\), there exists a constant \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\), such that
Proof
By Lemmas 2.4 and 3.3, there exists a constant \(c=c(n,N,p,q, \alpha ,L,[a]_{\alpha },M_1)\), such that
To estimate the second term on the right-hand side of (3.5), we use Hölder’s inequality and (p-ii) to obtain
where \(c=c(n,p)\). Similarly, the third term on the right-hand side of (3.5) is estimated by
where \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\). This finishes the proof. \(\square \)
Lemma 3.5
For \(s\in [2\rho ,4\rho ]\) and \(\theta \in ((q-1)/p,1]\), there exists a constant \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\), such that
Proof
By Lemmas 2.4 and 3.3, there exists a constant \(c=c(n,N,p,q, \alpha ,L,[a]_{\alpha },M_1)\), such that
By (p-1) and (p-ii), we obtain for the second term on the right-hand side of (3.6) that
where \(c=c(n,p,\alpha ,[a]_{\alpha },M_1)\). Similarly, the third and the fourth terms on the right-hand side of (3.6) can be estimated by
and
The conclusion follows from Hölder’s inequality. \(\square \)
In the following lemma we estimate the quadratic term
Lemma 3.6
There exists a constant \(c=c( data )\), such that
Proof
Let \(2\rho \le \rho _1<\rho _2\le 4\rho \). By Lemma 2.3, there exists a constant \(c=c(n,p,q,\nu ,L)\), such that
We estimate the first term on the right-hand side of (3.7). From Lemma 3.4 and (p-ii), we obtain
where \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\). On the other hand, we observe that
By Lemma 3.5 and (p-ii), we have
where \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\). For the other term in (3.9), we obtain from Lemma 2.5 with \(\sigma =q\), \(s=p\), \(r=2\) and \(\vartheta =\tfrac{p}{q}\), that
where \(c=c(n,q)\). We have by (3.8) and (p-ii) and (3.2) that
where the last inequality follows from (1.1) and \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\). We conclude that
where \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\).
Next, we estimate the second term on the right-hand side of (3.7). Using Lemma 2.5 with \(\sigma =2, s=p, r=2, \vartheta =1/2\), and then (3.8) and (p-ii), we have
where \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\). Observe that by \(p> \tfrac{2n}{n+2}\) it was possible to use these parameters in Lemma 2.5 as
For the last term on the right-hand side of (3.7) we obtain by (p-ii) that
Combining the estimates, we conclude from (3.7) that
where \(c=c(n,N,p,q,\alpha ,L,[a]_{\alpha },M_1)\). Finally, we apply Young’s inequality twice, with conjugate pairs (2, 2) and \((\tfrac{2}{q-p},\tfrac{2}{2-q+p})\), to obtain
The proof is concluded by an application of Lemma 2.6. \(\square \)
Next, we prove an estimate for the first term on the right-hand side of the energy estimate in Lemma 2.3 by using Lemma 2.5.
Lemma 3.7
There exist constants \(c=c( data )\) and \(\theta _0=\theta _0(n,p,q)\in (0,1)\), such that for any \(\theta \in (\theta _0,1)\) we have
Proof
By (2.4) we obtain
We begin with the first term on the right-hand side of (3.10). The condition in Lemma 2.5 with \(\sigma =p\), \(s=\theta p\), \(r=2\) and \(\vartheta = \theta \) is satisfied for \(\theta \in (n/(n+2),1)\), and we obtain
where \(c=c(n,p)\).
For the second term on the right-hand side of (3.10), we apply Lemma 2.5 with \(\sigma =q\), \(s=\theta q\), \(\vartheta = \theta \) and \(r=2\). Again the condition of the lemma holds for \(\theta \in (n/(n+2),1)\). We obtain
where \(c=c(n,q)\). By using (p-1), we have
Then we consider the last term on the right-hand side of (3.10). The assumptions in Lemma 2.5 with \(\sigma =q\), \(s=\theta p\), \(r = 2\) and \(\vartheta =\theta p/q\) are satisfied for \(\theta \in (nq/((n+2)p),1)\), and we obtain
where \(c=c(n,q)\). Note that
and that from (3.2) we obtain \(\rho \le c(M_1,n)\lambda ^\frac{-\mu }{n+2}\). Therefore
where \(c=c(n,p,q,\alpha ,{{\,\textrm{diam}\,}}(\Omega ),M_1,M_2)\). Observe that the last inequality follows from (1.1), as
The claim follows by combining the estimates above. \(\square \)
Now we are ready to prove the reverse Hölder inequality in the p-intrinsic case.
Lemma 3.8
There exist constants \(c=c( data )\) and \(\theta _0=\theta _0(n,p,q)\in (0,1)\), such that for any \(\theta \in (\theta _0,1)\) we have
Proof
Lemma 2.3 implies that
where \(c=c(n,p,q,\nu ,L)\). To estimate the first term on the right-hand side of (3.11), we apply Lemmas 3.6 and 3.7 to conclude that there exist \(\theta _0=\theta _0(n,p,q)\in (0,1)\) and \(c=c( data )\), such that for any \(\theta \in (\theta _0,1)\) we have
By Lemmas 3.4 and 3.5 we obtain
Note that \(p-1-\tfrac{\alpha \mu }{n+2} > 0\) by (1.1). Letting
we obtain from (3.12) by (p-ii) that
To estimate the second term on the right-hand side of (3.11), we apply Lemma 2.5 with \(\sigma = 2, s= \theta p, \vartheta = \tfrac{1}{2}\) and \( r=2\), where \(\theta \in (2n/((n+2)p),1)\). This and Lemma 3.6 gives
where \(c=c( data )\). Applying Lemma 3.4 and (p-ii) to the right-hand side implies
Combining the estimates for the terms in (3.11) and applying (p-ii) gives
By applying Young’s inequality, we obtain
and using (p-i) to absorb \(\tfrac{1}{2}\lambda ^p\) into the left hand side we conclude that
This completes the proof. \(\square \)
We finish this subsection with a corollary of the previous lemma which is used in the proof of higher integrability. The distribution sets are denoted as
and
Lemma 3.9
There exist constants \(c=c( data )\) and \(\theta _0=\theta _0(n,p,q)\in (0,1)\), such that for any \(\theta \in (\theta _0,1)\) we have
Proof
The condition (p-ii) implies that
By representing \(Q_{2\rho }^\lambda (z_0)\) as a union of \(Q_{2\rho }^\lambda (z_0)\cap \Psi ((4c)^{-1/\theta }\lambda ^p)\) and \(Q_{2\rho }^\lambda (z_0)\setminus \Psi ((4c)^{-1/\theta }\lambda ^p)\) , we have
for any \(c > 0\). A similar argument gives
It follows from Lemma 3.8 that
By recalling (p-2), we obtain
Thus, we have
We note that
where we applied (p-1). The estimate above implies that
Therefore, by replacing \(2K(4c)^{1/\theta _0}\) with c, (3.15) can be written as
This completes the proof. \(\square \)
3.2 The (p, q)-intrinsic case
In this subsection we show a reverse Hölder inequality in the (p, q)-intrinsic cylinder \(G_\rho ^\lambda (z_0)\) satisfying (p,q-1), (p,q-2), (p,q-3) and \(G_{2\kappa \rho }^\lambda (z_0)\subset \Omega _T\). Note that (p,q-2) and (p,q-ii) imply
It follows that
We start with a (p, q)-intrinsic parabolic Poincaré inequality.
Lemma 3.10
For \(s\in [2\rho ,4\rho ]\) and \(\theta \in ((q-1)/p,1]\), there exists a constant \(c=c(n,N,p,q,L)\), such that
Proof
Note that
Therefore, by Lemma 2.4 and (p,q-2), there exists a constant \(c=c(n,N,p,q,L)\), such that
To estimate the second term on the right-hand side of (3.17), we note that
By (3.16) and Hölder’s inequality, and the same argument for the term with \(H(z_0,|F|)\), we have
where \(c=c(p,q)\). We conclude that
where \(c=c(p,q)\). In order to estimate the first term on the right-hand side of (3.18), we apply Hölder’s inequality and (3.16) to get
for any \(\theta \in ((q-1)/p,1]\) with \(c=c(n,p)\). Similarly, we have for any \(\theta \in ((q-1)/q,1]\) that
where \(c=c(n,p)\). Combining the above inequalities, we conclude that
which completes the proof. \(\square \)
Note that by replacing \(H(z_0,s)^\theta \) with \(s^{\theta p}\) in the proof of Lemma 3.10, we also get the following result. All necessary calculations are already contained in the proof of the previous lemma.
Lemma 3.11
For \(s\in [2\rho ,4\rho ]\) and \(\theta \in ((q-1)/p,1]\), there exists a constant \(c=c(n,N,p,q,L)\), such that
As in the previous subsection, we estimate the term
Lemma 3.12
There exists a constant \(c=c(n,N,p,q,\nu ,L)\), such that
Proof
Let \(2\rho \le \rho _1<\rho _2\le 4\rho \). By Lemma 2.3, there exists a constant \(c=c(n,p,q,\nu ,L)\), such that
For the first term on the right-hand side of (3.19), we apply Lemma 3.10, together with (p,q-2) and (p,q-ii), to obtain
where \(c=c(n,N,p,q,L)\).
For the second term on the right-hand side of (3.19) we obtain by Lemma 2.5, as in the proof of Lemma 3.6, that
where \(c=c(n,N,p)\). Using Lemma 3.11 and (3.16), we obtain
where \(c=c(n,N,p,q,L)\). Combining the estimates and applying (p,q-ii) for the last term on the right-hand side of (3.19), we get
The claim follows by applying Young’s inequality and Lemma 2.6 as in the proof of Lemma 3.6. \(\square \)
Lemma 3.13
There exists a constant \(c=c(n,p,q)\), such that for any \(\theta \in (n/(n+2),1)\) we have
Proof
We obtain from (p,q-2) that
As in the proof of Lemma 3.7, by Lemma 2.5 there exists a constant \(c=c(n,p,q)\), such that for any \(\theta \in (n/(n+2),1)\) we have
and
We conclude that
This completes the proof. \(\square \)
Now we are ready to show the reverse Hölder inequality in (p, q)-intrinsic cylinders.
Lemma 3.14
There exist constants \(c=c(n,N,p,q,\nu ,L)\) and \(\theta _0=\theta _0(n,p,q)\in (0,1)\), such that for any \(\theta \in (\theta _0,1)\) we have
Moreover, we have
where \(\Psi (\Lambda )\) and \(\Phi (\Lambda )\) are defined in (3.13) and (3.14).
Proof
Lemma 2.3 gives
Using Lemmas 3.13, 3.10 and 3.12 for the first term on the right-hand side of (3.20), we obtain
As in the proof of Lemma 3.8, we obtain from Lemmas 3.6 and 3.12 that
We conclude from Lemma 3.11 that
Therefore, we have for the second term on the right-hand side of (3.20) that
Note that by Hölder’s inequality
Using a similar argument for |F|, we conclude from (3.21) that
Collecting the estimates for the terms in (3.20) and applying Young’s inequality and (p,q-2), we obtain
We use (p,q-i) to absorb \(\tfrac{1}{2}\Lambda \) into the left hand side. This completes the proof of the first statement.
To show the second statement, observe that as in the proof of Lemma 3.9, we obtain from the first statement that
It follows from (p,q-3) that
and we have
This completes the proof.
\(\square \)
4 Proof of the main result
In this section we complete the proof of Theorem 2.2. In the first subsection, we use a stopping time argument to construct intrinsic cylinders which are either p-intrinsic, as in (p-1)-(p-2), or (p, q)-intrinsic, as in (p,q-1)-(p,q-3). In the second subsection, we construct a Vitali type covering for this collection of intrinsic cylinders. Also here the decay estimate of Lemma 3.1 is needed to show the covering property of the intrinsic cylinders. In the last subsection, we complete the proof of the gradient estimate by applying Fubini’s theorem together with Lemma 2.6.
4.1 Stopping time argument
Let
Moreover, recalling the definition of \(M_1\) in (3.1), let
Recalling the notation in (3.13) and (3.14), for \(\rho \in [r,2r]\) we denote
and
Next, we apply a stopping time argument. Let \(r\le r_1<r_2\le 2r\) and
where \(\kappa \) is as in (4.2). For every \( w\in \Psi (\Lambda ,r_1)\), let \(\lambda _w>0\) be such that
We claim that
For a contradiction, assume that the inequality above does not hold. Then
which is a contradiction with (4.3). Therefore, (4.5) is true and we have for every \(s\in [(r_2-r_1)/(2\kappa ),r_2-r_1)\) that
By (4.4) we have \( w\in \Psi (\lambda _w^p,r_1)\). Therefore, by the Lebesgue differentiation theorem there exists \(\rho _{ w}\in (0,(r_2-r_1)/(2\kappa ))\), such that
and
for every \(s\in (\rho _{ w},r_2-r_1)\). This shows that at each point \( w\in \Psi (\Lambda ,r_1)\) we have a p-intrinsic cylinder satisfying (p-2).
Next, we assume that
and show that in this case there exists a (p, q)-intrinsic cylinder satisfying (p,q-2) and (p,q-3). For every \(s\in [\rho _{ w},r_2-r_1)\), we have by (2.1), (2.2), (4.4) and (4.6) that
Recall that \(w\in \Psi (\Lambda ,r_1)\). Again by the Lebesgue differentiation theorem, we find \(\varsigma _{w}\in (0,\rho _{w}]\) such that
and
for every \(s\in (\varsigma _{w},r_2-r_1)\).
To show (p,q-2), we claim that
Assume for contradiction that the opposite holds. By (4.7) and the negation of (4.8), we have
As (4.6) holds true also in this case, Lemma 3.1 gives
This is a contradiction and therefore (4.8) is true. It follows from (4.8), that
and
Therefore, when (4.7) is true
As \(\varsigma _w \le \rho _w\), we have shown the properties (p,q-1)-(p,q-3).
4.2 Vitali type covering argument
For each \( w\in \Psi (\Lambda ,r_1)\), we consider
We prove a Vitali type covering lemma for this collection of intrinsic cylinders. We denote
Recall that \(l_{ w}\in (0,R)\) for every \( w\in \Psi (\Lambda ,r_1)\), where \(R=(r_2-r_1)/\kappa \) and \(\kappa \) is as in (4.2). Let
We construct subcollections \({\mathcal {G}}_j\subset {\mathcal {F}}_j\), \(j\in {\mathbb {N}}\), recursively as follows. Let \({\mathcal {G}}_1\) be a maximal disjoint collection of cylinders in \({\mathcal {F}}_1\). Observe that for each \(U(w)\in {\mathcal {F}}_j\) we have
which implies that the collection is finite. Suppose that we have selected \({\mathcal {G}}_1,\ldots ,{\mathcal {G}}_{k-1}\) with \(k\ge 2\), and let
be a maximal collection of pairwise disjoint cylinders. It follows that
is a countable subcollection of pairwise disjoint cylinders in \({\mathcal {F}}\). We claim that for each \(U( w)\in {\mathcal {F}}\), there exists \(U( v)\in {\mathcal {G}}\) such that
For every \(U( w)\in {\mathcal {F}}\), there exists \(j \in {\mathbb {N}}\) such that \(U( w)\in {\mathcal {F}}_j\). By the construction of \({\mathcal {G}}_j\), there exists a cylinder \(U( v)\in \cup _{i=1}^j {\mathcal {G}}_i\) for which the first condition in (4.10) holds true. Moreover, since \(l_{ w}\le \tfrac{R}{2^{j-1}}\) and \(l_{ v} \ge \tfrac{R}{2^j}\), we have
In the remaining of this subsection, we prove the second claim in (4.10). First, we show the comparability of \(\lambda _w\) and \(\lambda _v\) using the following observations. For \(i\in \{v,w\}\), there exist \(2\rho _i\ge l_i>0\) and \(\lambda _i>0\) such that
and
It follows from (4.13) and (3.2) that
Moreover, the first condition in (4.10) and (4.11) imply that \(Q_{l_{w}}(w) \cap Q_{l_{v}}(v) \ne \emptyset \) and
Therefore, we have by (2.4) that
Now we show the comparability of \(\lambda _w\) and \(\lambda _v\). First, we claim that if \(\lambda _w \le \lambda _v\), then
For a contradiction, assume that (4.16) does not hold. By (4.12) and (4.15), we have
From the negation of (4.16) and (4.14), we obtain
Negation of (4.16) and the above estimates lead to the contradiction
and thus (4.16) holds.
On the other hand, if \(\lambda _v\le \lambda _w\), we claim that
Again, assume for contradiction that the opposite holds. It follows from (4.14) that
which is a contradiction. We conclude that
We show that the second claim in (4.10) holds in all four possible cases that may occur:
-
(i)
\(U(v)=Q_{l_v}^{\lambda _v}(v)\) and \(U(w)=Q_{l_w}^{\lambda _w}(w)\),
-
(ii)
\(U(v)=G_{l_v}^{\lambda _v}(v)\) and \( U(w)=G_{l_w}^{\lambda _w}(w)\),
-
(iii)
\(U(v)=G_{l_v}^{\lambda _v}(v)\) and \( U(w)=Q^{\lambda _w}_{l_w}(w)\) or
-
(iv)
\(U(v)=Q_{l_v}^{\lambda _v}(v)\) and \(U(w)=G_{l_w}^{\lambda _w}(w)\).
Proof for the spatial inclusion is the same in all the cases. We denote \(v=(x_v,t_v)\) and \(w=(x_w,t_w)\), where \(x_v,x_w\in {\mathbb {R}}^n\) and \(t_v,t_w\in {\mathbb {R}}\), and recall the notation in (2.1) and (2.2). For any \(\xi \in B_{l_w}^{\lambda _w}(w)\) we have by (4.11) and (4.17), that
and therefore \(B_{l_w}^{\lambda _w}(w) \subset 6KB_{l_v}^{\lambda _v}(v)\).
We show the inclusion in time direction in the four possible cases separately. In case (i), we have by (4.11) for any \(\tau \in I_{l_w}(t_w)\) that
and therefore \(I_{l_w}(t_w) \subset 9I_{l_v}(t_v)\).
In case (ii), we have by (4.11) and (4.17) for any \(\tau \in J^{\lambda _w}_{l_w}(t_w)\) that
and therefore \(J^{\lambda _w}_{l_w}(t_w)\subset 9KJ^{\lambda _v}_{l_v}(t_v)\).
In case (iii) we have \(K\lambda _w^p\ge a(w)\lambda _w^q\), which along with (4.17) and (4.12) gives
Therefore, we have for any \(\tau \in I_{\rho _w}(t_w)\) that
Together with the spatial inclusion this implies \(Q_{l_w}^{\lambda _w}(w)\subset 6KG_{l_v}^{\lambda _v}(v)\).
Finally, in case (iv) we have by (4.11) and (4.12) for any \(\tau \in J_{l_w}^{\lambda _w}(t_w)\) that
and therefore \(J_{l_w}^{\lambda _w}(t_w) \subset 9KI_{l_v}(t_v)\). We have covered every case and conclude that (4.10) holds.
4.3 Final proof of the gradient estimate
We write the countable pairwise disjoint collection \({\mathcal {G}}\) defined in (4.9) as \({\mathcal {G}}=\cup _{j=1}^\infty U_j\), where \(U_j=U( w_j)\) with \( w_j \in \Psi (\Lambda ,r_1)\). By Lemma 3.9 and Lemma 3.14, there exist \(c=c( data )\) and \(\theta _0=\theta _0(n,p,q)\in (0,1)\), such that
for every \(j\in {\mathbb {N}}\) with \(\theta = (\theta _0+1)/2\). By summing over j and applying the fact that the cylinders in \({\mathcal {G}}\) are pairwise disjoint, we obtain
Moreover, since
we conclude from (4.18) that
For \(k\in {\mathbb {N}}\), let
and
It is easy to see that if \(\Lambda >k\), then \(\Psi _k(\Lambda ,\rho )=\emptyset \), and if \(\Lambda \le k\), then \(\Psi _k(\Lambda ,\rho )=\Psi (\Lambda ,\rho )\). Therefore, we deduce from (4.19) that
Recalling (4.3), we denote
Then for any \(\Lambda >\Lambda _1\), we obtain
Let \(\epsilon \in (0,1)\) to be chosen later. We multiply the inequality above by \(\Lambda ^{\epsilon -1}\) and integrate each term over \((\Lambda _1,\infty )\), which implies
We apply Fubini’s theorem to estimate \(\textrm{I}\) and obtain
Since
we have
Similarly, by Fubini’s theorem, we have
and
By combining the estimates above, we obtain
We choose \(\epsilon _0=\epsilon _0( data )\in (0,1)\) so that for any \(\epsilon \in (0,\epsilon _0)\),
Then, by applying Lemma 2.6 we get
The claim follows by letting \(k\longrightarrow \infty \) and recalling (4.1).
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Acknowledgements
The authors would like to thank Kristian Moring for helpful discussions about this problem. The second author is supported by a doctoral training grant from Vilho, Yrjö and Kalle Väisälä Foundation.
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Kim, W., Särkiö, L. Gradient higher integrability for singular parabolic double-phase systems. Nonlinear Differ. Equ. Appl. 31, 40 (2024). https://doi.org/10.1007/s00030-024-00928-5
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DOI: https://doi.org/10.1007/s00030-024-00928-5