Abstract
In this paper we establish a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems. The proof is based on a new intrinsic scaling that involves both the solution and its spatial gradient. It allows to compensate for the different scaling of the system in |u| and |Du|. The result covers the range of parameters \(p>\frac{2n}{n+2}\) and \(0<q\le 1\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and results
In this paper, we consider weak solutions \(u:\Omega _T\rightarrow \mathbb {R}^N\) of doubly non-linear parabolic equations, resp. systems, in a space-time cylinder \(\Omega _T{:}{=} \Omega \times (0,T)\), where \(\Omega \subset \mathbb {R}^n\) is a bounded open domain with \(n\ge 2\), \(N\ge 1\), and \(T>0\). The model equation is given by
with \(p>1\) and \(q>0\). For \(q=1\) we have the parabolic p-Laplace, for \(p=2\) we have the porous medium equation and for \(q+1=p\) we recover the homogeneous doubly nonlinear equation
sometimes called Trudinger’s equation. The model equation (1.1) is a special case of the general doubly nonlinear parabolic equation
with a Carathéodory vector-field \(\mathbf {A}:\Omega _T\times \mathbb {R}^N\times \mathbb {R}^{Nn}\rightarrow \mathbb {R}^{Nn}\) that satisfies the p-growth and ellipticity conditions
for a.e. \((x,t)\in \Omega _T\) and any \((u,\xi )\in \mathbb {R}^N\times \mathbb {R}^{Nn}\), with constants \(0<\nu \le L<\infty \). Equation (1.1) is of both physical and mathematical interest. It occurs, among other things, in filtration processes of gases or liquids in porous media. It also plays an important role in the study of non-Newtonian fluids; see [16] and [28] and references therein. It is also used to describe the dynamics of glaciers [19] and of shallow water flows [1]; see also [24] and [25]. The special case \(p = 3/2\) in Trudinger’s equation (1.2) (in one space dimension) is used as a model for a friction-dominated flow in a gas network; see [2, 18]. The equation (1.1) has a different behavior when \(q+1<p\) and \(q+1\ge p\). In the first region perturbations propagate with finite speed and moving boundaries exist. Therefore, this region is termed the slow diffusion case. In the second range, perturbations propagate with infinite speed and extinction may occur in finite time. This range is called fast diffusion case, and Trudinger’s Eq. (1.2) represents the limiting case between the slow and fast diffusion regions.
Before stating our main result, we specify our notion of weak solution to (1.3).
Definition 1.1
Assume that the Carathéodory vector field \({\mathbf {A}}:\Omega _T\times \mathbb {R}^N\times \mathbb {R}^{Nn}\rightarrow \mathbb {R}^{Nn}\) satisfies (1.4), and that \(F\in L^p_{\mathrm{loc}} (\Omega _T,\mathbb {R}^{Nn})\). We identify a measurable map \(u:\Omega _T\rightarrow \mathbb {R}^{N}\) in the class
as a weak solution to the doubly non-linear parabolic Eq. (1.3) if and only if the identity
holds, for any testing function \(\varphi \in C_0^\infty (\Omega _T,\mathbb {R}^N)\). \(\square \)
The purpose of this paper is to establish a local higher integrability result for the spatial gradient of weak solutions to doubly non-linear parabolic equations and systems of the type (1.3).
Theorem 1.2
Let \(p>\frac{2n}{n+2}\), \(0<q\le 1\), \(\sigma >p\), and \(F\in L^\sigma _{\mathrm{loc}}(\Omega _T,\mathbb {R}^N)\). Then, there exist \(\varepsilon _o=\varepsilon _o(n,p,q,\nu ,L)\in (0,1]\) and \(c=c(n,p,q,\nu ,L)\ge 1\) such that whenever u is a weak solution of (1.3) in the sense of Definition 1.1, then there holds
where \(\varepsilon _1{:}{=}\min \{\varepsilon _o,\frac{\sigma }{p}-1\}\). Moreover, for every \(\varepsilon \in (0,\varepsilon _1]\) and every cylinder \(Q_{\varrho }{:}{=}B_\varrho (x_o)\times (t_o-\varrho ^{1+q}, t_o+\varrho ^{1+q})\Subset \Omega _T\), we have the quantitative local higher integrability estimate
with \(p^\sharp {:}{=}\max \{p,q+1\}\) and the scaling deficit
Theorem 1.2 ensures in the range \((p,q)\in \big (\frac{2n}{n+2},\infty )\times (0,1]\) that weak solutions of (1.3) belong to a slightly better Sobolev space than the natural energy space and therefore obey a self-improving property of integrability. The improvement in integrability of the spatial gradient, as stated in (1.6), is the direct consequence of the quantitative reverse Hölder type estimate (1.7). The range of (p, q) covered by Theorem 1.2 is composed of three parts that are illustrated in the diagram in Fig. 1 below. The part that lies above the line \(q=p-1\) (the red triangle) belongs to the fast diffusion range, while the parts below the line (the green and blue region) are contained in the slow diffusion range. The green and the blue region differ in the properties of the diffusion part of the differential equation (1.3), which becomes degenerate for \(p\ge 2\) (the green region) and singular for \(p<2\) (the blue region). By this we mean that the modulus of ellipticity degenerates for small values of |Du| in the green region, while it becomes singular in the blue region. Because of the differences described above, each of the three regions requires slightly different techniques. The common feature of the three colored regions is that we can work with cylinders as in (2.2) that are scaled in time, while the cases with \(q>1\) require a scaling in the spatial directions, as it has been used for the singular porous medium equation in [7, 12]. This is the reason why we restrict ourselves to the cases in which \(0<q\le 1\).
The lower bound \(p>\frac{2n}{n+2}\) in Theorem 1.2 already emerges in the case of the parabolic p-Laplace system; cf. [17]. It is needed in the proof of the Sobolev-Poincaré inequality, see Lemma 4.4. Theorem 1.2 contains, among others, as limiting cases the parabolic p-Laplace system \(q=1\), \(p> \frac{2n}{n+2}\), which was originally treated in [17], the slow diffusion range for the porous medium equation \(0<q\le 1\), \(p=2\), cf. [6, 11], and Trudinger’s equation, i.e. the case when \(q=p-1\), in the range \(\frac{2n}{n+2}<p\le 2\); cf. [5]. We should point out that in these special cases inequality (1.7) reproduces the reverse Hölder type inequalities established in [5, 6, 17].
The key to the proof of our main result is a suitable intrinsic geometry, a concept which was originally introduced by DiBenedetto and Friedman [9]; see also the monographs [10, 27]. Meanwhile, variants of this idea have been successfully used to prove higher integrability for the parabolic p-Laplace system [17] and more recently for porous medium type equations [11, 12], for porous medium type systems [6, 7], and for Trudinger’s equation [5]. Our idea is to consider space-time cylinders \(Q_{\varrho ,s}(z_o){:}{=}B_\varrho (x_o)\times (t_o-s,t_o+s)\), with \(z_o=(x_o,t_o)\), such that the quotient \(\frac{s}{\varrho ^{1+q}}\) satisfies
with
Thus, the scaling in time direction depends on both, the solution and its spatial gradient. It allows us to compensate for the different scaling of |u| and |Du| in the system. To our knowledge, this is the first time such a geometry is used in the context of gradient estimates for doubly nonlinear equations. We note that a related construction of cylinders, which also depend on two parameters, was used in a different context in [26]. On the cylinders described above, we are able to prove Sobolev–Poincaré and reverse Hölder type inequalities. The construction of the cylinders is quite complex, since the domain of integration on the right-hand side of (1.9) also depends on the parameters \(\lambda \) and \(\theta \). This is achieved in the final part of the paper. For any \(\lambda \) and any radius \(\varrho \) we find \(\theta =\theta (\lambda , \varrho )\) such that (1.9)\(_2\) is satisfied. However, since \(\theta \) does not depend monotonically on \(\varrho \) we have to modify our choice at the cost that the identity in (1.9)\(_2\) has to be replaced by ’\(\ge \)’. Such cylinders are called \(\theta \)-sub-intrinsic. In the course of the construction we modify an argument from [11]; see also [5,6,7, 23]. Next, we fix a parameter \(\lambda \) and consider the super-level set \(\{|Du|>\lambda \}\). For any point in this set, we find by a stopping time argument a radius \(\varrho \), which in turn fixes \(\theta (\lambda ,\varrho )\), such that the associated cylinder satisfies (1.9)\(_1\). By construction the cylinder is also \(\theta \)-sub-intrinsic. On these cylinders we can apply the previously proved reverse Hölder inequality. However, some complications occur, since they are only \(\theta \)-sub-intrinsic. The final part of the proof is quite standard. Once the reverse Hölder inequalities are established, we conclude gradient estimates on super-level sets. Integration with respect to the levels \(\lambda \) then yields the quantitative higher integrability estimate.
Range of p and q
At this point, a few words to place our result in the history of the problem of higher integrability are in order. In the stationary elliptic case, the higher integrability was first observed by Elcrat & Meyers [20], see also the monographs [13, Chapter 11, Theorem 1.2] and [15, Sect. 6.5] and the references therein. The first higher integrability result for parabolic systems goes back to Giaquinta & Struwe [14, Theorem 2.1]. For parabolic systems with p-growth whose main prototype is the parabolic p-Laplace system, the higher integrability of the gradient of weak solutions was established by Kinnunen & Lewis [17] in the range \(p>\frac{2n}{n+2}\). This lower bound is natural and also appears in other contexts in the regularity theory of parabolic p-Laplace systems; cf. the monograph [10]. In the meantime, the result has been generalized in several directions, such as global results, higher order parabolic systems with p-growth, and parabolic systems with p(x, t)-growth; cf. [3, 4, 8, 21]. The corresponding problem for the porous medium equation, i.e. (1.1) with \(p=2\), proved to be more complicated and remained open for a long time, even in the scalar case for non-negative solutions. In addition to the obvious anisotropic behavior of the equation with respect to scalar multiplication of solutions, it is also not possible to add constants to a solution without losing the property of being a solution. This difficulty has recently been overcome by Gianazza & Schwarzacher [11] who proved in the slow diffusion range \(0<q<1\) that non-negative weak solutions of porous medium type equations admit the self-improving property of higher integrability of the gradient. The main novelty in their proof is the use of a new intrinsic scaling. Instead of scaling cylinders with respect to |Du| as in the case of the parabolic p-Laplace (cf. [10] and the references therein), they work with cylinders which are intrinsically scaled with respect to |u|. The proof, however, uses the method of expansion of positivity and therefore can not be extended to signed solutions, to the case of porous medium type systems, and to the fast diffusion range. A simpler and more flexible proof that does not rely on the expansion of positivity and that covers both signed solutions and vector-valued solutions, can be found in [6]. The singular case of the porous medium equation has been treated independently in [12] for non-negative solutions and in [7] for vector-valued solutions. Next, in [5] the higher integrability is established for Trudinger’s equation. In this equation aspects of both the porous medium equation and the parabolic p-Laplace equation play a role. Therefore the intrinsic scaling has to take into account the degeneracy of the system both with respect to the gradient variable and with respect to the solution itself. In [5] the higher integrability is established for exponents p in the somewhat unexpected range \(\max \{\frac{2n}{n+2},1\}<p<\frac{2n}{(n-2)_+}\). The lower bound also appears for the parabolic p-Laplace system [17], while the upper bound corresponds exactly to the lower bound for the porous medium equation in the fast diffusion range. In the scalar case of non-negative solutions of Trudinger’s equation with \(F=0\), the upper bound could be eliminated in [22]. However, the techniques used in [22] are strictly limited to the scalar case.
2 Preliminaries
2.1 Notations
Throughout the paper, we use space-time cylinders of the form
with center \(z_o=(x_o,t_o)\in \mathbb {R}^n\times \mathbb {R}\), radius \(\varrho >0\) and scaling parameters \(\lambda ,\theta >0\). Here
where
For the case that \(\theta =1=\lambda \), we simply omit the scaling parameters in our notation and instead of \(Q_\varrho ^{(1,1)}(z_o)\) write
For a mapping \(u\in L^1\big (0,T;L^1(\Omega ,\mathbb {R}^N)\big )\) and a given measurable set \(A\subset \Omega \) with positive Lebesgue measure the slice-wise mean \((u)_{A}:(0,T)\rightarrow \mathbb {R}^N\) of u on A is defined by
Note that if \(u\in C^0\big ([0,T];L^{1+q}(\Omega ,\mathbb {R}^N)\big )\) the slicewise means are defined for any \(t\in [0,T]\). If the set A is a ball \(B_\varrho (x_o)\), then we abbreviate \((u)_{x_o;\varrho }(t){:}{=}(u)_{B_\varrho (x_o)}(t)\). Similarly, for a given measurable set \(E\subset \Omega \times (0,T)\) of positive Lebesgue measure the mean value \((u)_{E}\in \mathbb {R}^N\) of u on E is defined by
If \(E= Q_\varrho ^{(\lambda ,\theta )}(z_o)\), we use the short hand notation \((u)^{(\lambda ,\theta )}_{z_o;\varrho }{:}{=}(u)_{Q_\varrho ^{(\lambda ,\theta )}(z_o)}\).
For a power of a vector \(u\in \mathbb {R}^N\), we use the notation
for \(\alpha >0\), which we interpret as 
in the case \(u=0\). Finally, we define the boundary term
for any \(u,a\in \mathbb {R}^N\).
2.2 Auxiliary material
In order to “re-absorb” certain quatities, we will use the following iteration lemma, which can be retrieved by a simple change of variable from [15, Lemma 6.1].
Lemma 2.1
Let \(0<\vartheta <1\), \(A,C\ge 0\) and \(\alpha ,\beta > 0\). Then there exists a universal constant \(c = c(\alpha ,\beta ,\vartheta )\) such that there holds: For any non-negative bounded function satisfying
we have
The next lemma can be verified using the arguments from [15, Lemma 8.3].
Lemma 2.2
For any \(\alpha >0\), there exists a constant \(c=c(\alpha )\) such that, for all \(a,b\in \mathbb {R}^N\), \(N\in {\mathbb {N}}\), we have
As an easy consequence of the preceding lemma, we obtain
Lemma 2.3
For any \(\alpha \ge 1\), there exists a constant \(c=c(\alpha )\) such that, for all \(a,b\in \mathbb {R}^N\), \(N\in {\mathbb {N}}\), we have
The following estimate, which is known as the quasi-minimality of the mean value, can be retrieved from [5, Lemma 3.5].
Lemma 2.4
Let \(p\ge 1\) and \(\alpha \ge \frac{1}{p}\). Then, there exists a constant \(c=c(\alpha ,p)\) such that whenever \(A\subseteq B\subset \mathbb {R}^k\), \(k\in {\mathbb {N}}\), are two bounded domains of positive measure, then for any function \(u \in L^{\alpha p}(B,\mathbb {R}^N)\) and any constant \(a\in \mathbb {R}^N\), we have
The proof of the following lemma can be found in [6, Lemma 2.3 (i)] for the case \(0<q\le 1\) and in [5, Lemma 3.4] for parameters \(q>1\).
Lemma 2.5
For any \(q>0\) there exists a constant \(c=c(q)\) such that for any \(u,v\in \mathbb {R}^N\), \(N\in {\mathbb {N}}\), the following estimates hold true:
Finally, we recall Gagliardo-Nirenberg’s inequality, which can be formulated in the following way.
Lemma 2.6
Let \(1\le p,q,r<\infty \) and \(\theta \in (0,1)\) such that \( - \frac{n}{p} \le \theta (1 - \frac{n}{q} ) - ( 1- \theta ) \frac{n}{r}\). Then there exists a constant \(c=c(n,p)\) such that for any ball \(B_\varrho (x_o)\subset \mathbb {R}^n\) with \(\varrho >0\) and any function \(u \in W^{1,q}(B_\varrho (x_o))\), we have
3 Energy bounds
In this section we provide a Caccioppoli inequality and a gluing lemma for solutions to the doubly non-linear system. Thereby, we include the whole range \(p>1\) and \(q>0\), since the proofs of these lemmas do not differ when p or q vary. We start with the Caccioppoli inequality.
Lemma 3.1
Let \(p>1\), \(q>0\) and u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on each cylinder \(Q_{\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) with \(0<\varrho \le 1\) and \(\lambda ,\theta >0\), and for all \(r\in [\varrho /2,\varrho )\) and all \(a\in \mathbb {R}^N\) the following energy estimate
holds true for a universal constant \(c=c(p,q,\nu ,L)\).
Proof
For \(v\in L^1(\Omega _T,\mathbb {R}^N)\), we define a mollification in time by
From the weak form (1.5) of the differential equation we deduce the mollified version
for any \(\varphi \in L^p(0,T;W^{1,p}_0(\Omega ,\mathbb {R}^N))\cap L^{q+1}(\Omega _T,\mathbb {R}^N)\). Throughout the rest of the proof we omit the reference to the center \(z_o=(x_o,t_o)\) in our notation. Let \(\eta \in C^1_0(B_\varrho ,[0,1])\) be a cut-off function with \(\eta \equiv 1\) in \(B_r\) and \(|D\eta |\le \frac{2}{\varrho -r}\), and define \(\zeta \in W^{1,\infty } (\Lambda ^{(\lambda ,\theta )}_\varrho (t_o),[0,1])\) by
Furthermore, for \(\varepsilon >0\) small enough and \(t_1 \in \Lambda ^{(\lambda ,\theta )}_r(t_o)\) we define the function \(\psi _\varepsilon \in W^{1,\infty } (\Lambda ^{(\theta )}_\varrho (t_o),[0,1])\) by
With these choices, (3.1) can be tested with
In the following, let 
. For the integral containing the time derivative we get
where we used the fact 
, which follows from the elementary identity
Since \(\llbracket u\rrbracket _h\rightarrow u\) in \(L^{q+1}_\mathrm{loc}(\Omega _T)\), we can pass to the limit \(h\downarrow 0\) in the integral on the right-hand side. We obtain
In the left-hand side of the preceding inequality we pass to the limit \(\varepsilon \downarrow 0\). For the term \(\mathrm {I}_{\varepsilon }\) we obtain for any \(t_1 \in \Lambda _{r}^{(\lambda ,\theta )}(t_o)\) that
whereas the term \(\mathrm {II}_\varepsilon \) can be estimated in the following way (observe that the boundary term is non-negative)
Next, we consider the diffusion term. After passing to the limit \(h\downarrow 0\), we use the ellipticity and growth assumption (1.4) for the vector-field \({\mathbf {A}}\), and later on Young’s inequality. In this way, we obtain
with a constant \(c=c(p,\nu ,L)\). Finally, we consider the right-hand side term, i.e. the term involving the inhomogeneity F. This term can be estimated with Young’s inequality. We conclude that
with a constant \(c=c(p,\nu )\). By standard properties of the mollification, we obtain
We combine these inequalities and pass to the limit \(\varepsilon \downarrow 0\). In this way we get that for almost every \(t_1 \in \Lambda _r^{(\theta )}(t_o)\) there holds
Here we pass to the supremum over \(t_1\in \Lambda _\varrho ^{(\lambda , \theta )}(t_o)\) in the first term on the left-hand side and then let \(t_1\uparrow t_o\) in the second one. Finally, we take mean values on both sides and recall the definition of \(\tau _\varrho ^{(\lambda ,\theta )}\). In view of Lemma 2.5, this yields the claimed energy estimate with a constant \(c=c(p,q,\nu ,L)\). \(\square \)
Now, we turn out attention to the gluing lemma.
Lemma 3.2
Let \(p>1\), \(q>0\) and u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1, where the vector-field \({\mathbf {A}}\) fulfills the growth and ellipticity assumptions (1.4). Then, on any cylinder \(Q_{\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) with \(0<\varrho \le 1\) and \(\lambda ,\theta >0\) there exists \({\hat{\varrho } \in } [\frac{\varrho }{2},\varrho ]\) such that for all \(t_1,t_2\in \Lambda _\varrho ^{(\lambda ,\theta )}(t_o)\) there holds
Proof
Let \(t_1,t_2\in \Lambda _\varrho ^{(\lambda ,\theta )}(t_o)\) with \(t_1<t_2\) and assume that \(r \in [\frac{\varrho }{2}, \varrho ]\). For \(\delta >0\) and \(0<\varepsilon \ll 1\), we define \(\xi _\varepsilon \in C^\infty _0(t_1-\varepsilon ,t_2+\varepsilon )\) by
and a radial function \(\Psi _\delta \in W^{1,\infty }_0(B_{r+\delta }(x_o))\) by \(\Psi _\delta (x){:}{=}\psi _\delta (|x-x_o|)\), where
For fixed \(i\in \{1,\dots ,N\}\) we choose \(\varphi _{\varepsilon ,\delta }=\xi _\varepsilon \Psi _\delta e_i\) as testing function in the weak formulation (1.5), where \(e_i\) denotes the i-th canonical basis vector in \(\mathbb {R}^N\). In the limit \(\varepsilon ,\delta \downarrow 0\) we obtain
We multiply the preceding inequality by \(e_i\) and sum over \(i=1,\dots ,N\). This yields
Here, we use the growth condition (1.4)\(_2\) and immediately get for any \(t_1,t_2\in \Lambda _\varrho ^{(\lambda ,\theta )}(t_o)\) and any \(r \in [\frac{\varrho }{2}, \varrho ]\) that there holds
Since
there exists a radius \({\hat{\varrho }} \in [\frac{\varrho }{2},\varrho )\) with
Therefore, we choose in the above inequality \(r={\hat{\varrho }}\) and then take means on both sides of the resulting inequality. This implies
for any \(t_1,t_2\in \Lambda _\varrho ^{(\theta )}(t_o)\) and with a constant \(c=c(L)\). \(\square \)
4 Parabolic Sobolev–Poincaré type inequalities
In this section we consider cylinders on which certain intrinsic or sub-intrinsic couplings with respect to u and its spatial gradient Du are satisfied. In order to cover all the needed cases of intrinsic and sub-intrinsic couplings, we now formulate the conditions that appear later in the course of the work. Later on, we will explicitly state which of the conditions must be satisfied. On a scaled cylinder \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) with radius \(0<\varrho \le 1\) and scaling parameters \(\lambda ,\theta >0\) we introduce couplings of the type
and
with \(p^\sharp {:}{=}\max \{p,q+1\}\) and some constants \(C_\lambda ,\, C_\theta \ge 1\). These inequalities are to be interpreted as follows. Assume that (4.1)\(_1\) resp. (4.2)\(_1\) is valid on \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) for a constant \(C_\lambda \) or \(C_\theta \). Then we call the scaled cylinder \(\lambda \)-sub-intrinsic (with constant \(C_\lambda \)) or \(\theta \)-sub-intrinsic (with constant \(C_\theta \)) . Similarly, if (4.1)\(_2\) resp. (4.2)\(_2\) holds, then we call \(Q_{\varrho }^{(\lambda ,\theta )}(z_o)\) \(\lambda \)-super-intrinsic (with constant \(C_\lambda \)) or \(\theta \)-sub-intrinsic (with constant \(C_\theta \)). Cylinders which are both \(\lambda \)-sub-intrinsic and \(\lambda \)-super-intrinsic are called \(\lambda \)-intrinsic. We use the same notation for \(\theta \)-sub-intrinsic and \(\theta \)-super-intrinsic cylinders. When we speak of a (sub-, super-) intrinsic coupling, we implicitly assume that a constant \(C_\lambda \) or \(C_\theta \) exists, so that the respective coupling holds. Note that (4.2)\(_1\) immediately implies
i.e. the subintrinsic coupling holds on \(Q_{\varrho }^{(\lambda ,\theta )}(z_o)\) with constant \(2^{n+p^\sharp +q+1}C_\theta \). A similar observation can be made for the case of (4.1)\(_1\), i.e. if (4.1)\(_1\) holds, then
The following corollary is an immediate consequence of Lemma 3.2.
Corollary 4.1
Let \(p>1\) and \(0<q\le 1\), and let u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1 and consider a cylinder \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) with \(0<\varrho \le 1\) and \(\lambda ,\theta >0\), satisfying the \(\theta \)-sub-intrinsic condition (4.2)\(_1\). Moreover, for \(r\in (0,\varrho )\) let \(a_r:[0,T]\rightarrow \mathbb {R}^N\) be given by 
, where 
denotes the slice-wise mean of 
on \(B_\varrho (x_o)\) as defined in (2.3). Then, there exists \({\hat{\varrho } \in } [\frac{\varrho }{2},\varrho ]\) such that for all \(t_1,t_2\in \Lambda _\varrho ^{(\lambda ,\theta )}(t_o)\) there holds
provided \(\max \{\frac{1}{p},\frac{p-1}{p}\}\le \vartheta \le 1\), where \(c=c(n,p,q,L,C_\theta )\).
Proof
For simplification we shall omit the reference point \(z_o\) throughout the proof in our notation. In turn, we apply Lemma 2.2 with \(\alpha =\frac{1}{q}\) and Lemma 3.2, to infer that there exists \({\hat{\varrho \in }} [\frac{\varrho }{2},\varrho ]\) such that for a.e. \(t,\tau \in \Lambda _\varrho ^{(\lambda , \theta )}\) there holds
for a constant c depending only on n, p, q and L. We take both sides to the power \(\vartheta p\), integrate with respect to t and \(\tau \) over \(\Lambda ^{\lambda ,\theta }_\varrho \), and then apply Hölder’s inequality, where we use \(0<q<1<p\). Then, using the very definition of \(\tau _\varrho ^{(\lambda ,\theta )}\) and the sub-intrinsic coupling (4.2)\(_1\) (in a form similar to (4.3) with \({\hat{\varrho }}\) instead of \(\varrho \)), we find that
with a constant \(c=c(n,p,q,L)\). This proves the claim. \(\square \)
The next lemma is a Poincaré type inequality for weak solutions. Corollary 4.1 will serve to overcome the lack of differentiability with respect to the time variable.
Lemma 4.2
Let \(p>1\) and \(0<q\le 1\), and let u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) with \(0<\varrho \le 1\) and \(\lambda ,\theta >0\) satisfying the \(\theta \)-sub-intrinsic coupling (4.2)\(_1\), the following Poincaré type inequality
holds true for any \(\max \{\frac{1}{p},\frac{p-1}{p}\}\le \vartheta \le 1\) and a universal constant \(c=c(n,p,q,L,C_\theta )\).
Proof
In the following we shall omit the reference point \(z_o\) in our notation. We also use the abbreviation
By \({\hat{\varrho } \in } [\frac{\varrho }{2},\varrho ]\) we denote the radius from Corollary 4.1. We now define the slice-wise means 
of 
as in (2.3) and \(a_r:[0,T]\rightarrow \mathbb {R}^N\) by 
for any \(t\in [0,T]\). By adding and subtracting \(a_{{\hat{\varrho }}}(t)\), we obtain
with the obvious meaning of the abbreviations \(\hbox {I}\) – \(\hbox {III}\). In the following, we treat the terms of the right side one after the other. We start with the term I. We apply Lemma 2.4 with the choices \(\alpha =\frac{1}{q}\), \(A=B_{{\hat{\varrho }}}\), \(B=B_\varrho \), 
and with 
in place of u. Since \(|B|/|A|\le 2^n\), we obtain
with \(c=c(n,p,q)\). In the last step, we used Poincaré’s inequality slice-wise. A similar argument applies to the third term \(\hbox {III}\), since \(\hbox {III}\le \hbox {I}\). The bound for II directly follows from Corollary 4.1. Indeed, we have
for some constant c depending only on n, p, q and L. The application of Corollary 4.1 is permitted, since we have assumed the \(\theta \)-sub-intrinsic coupling (4.2)\(_1\). At this point, we use the estimates for I, II, and III in (4.5) and arrive at
This yields the claim. \(\square \)
The following lemma can be interpreted as a Sobolev–Poincaré type inequality for weak solutions.
Lemma 4.3
Let \(p>1\) and \(0<q\le 1\), and let u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) satisfying the \(\lambda \)-intrinsic coupling (4.1) and the \(\theta \)-sub-intrinsic coupling (4.2)\(_1\) for some \(0<\varrho \le 1\) and some \(\lambda , \theta >0\), the following inequality
holds true for any given \(\epsilon \in (0,1]\) and universal constants \(c=c(n,p,q,L,C_\theta ,C_\lambda )\) and \(\beta =\beta (n,p,q)\). Here, the parameter \(\vartheta \) is defined by \( \vartheta {:}{=}\max \big \{\frac{n}{n+q+1},\frac{1}{p}, \frac{p-1}{p}\big \}. \)
Proof
In the following we shall omit again the reference point \(z_o\) in our notation and abbreviate \(a{:}{=}(u)_{\varrho }^{(\lambda ,\theta )}\). Moreover, we use the short-hand notations
With Lemma 2.4 and the \(\theta \)-sub-intrinsic coupling (4.2)\(_1\) in the form of (4.3) we conclude the bound
It suffices to consider the case \(\mathbf{E}>\mathbf{F}\), since otherwise, the asserted estimate is clearly satisfied. Under this assumption, estimate (4.7) implies
with a constant \(c=c(n,p,q,C_\theta )\). Moreover, Hölder’s inequality and the \(\lambda \)-sub-intrinsic coupling in the form (4.4) imply
where \(c=c(n,C_\lambda )\). Next, we apply Gagliardo-Nirenberg’s inequality as stated in Lemma 2.6, with \((p,q,r,\theta )\) replaced by \((p,\vartheta p,q+1,\vartheta )\), which is possible since \(\vartheta \ge \max \{\frac{n}{n+q+1},\frac{1}{p}\}\). Afterwards, we use Lemma 4.2. In this way we find
with a constant \(c=c(n,p,q,L,C_\theta )\). Next, we use Lemma 2.2 with \(\alpha =\frac{q+1}{2}\) for the estimate
We divide this inequality by \(\varrho ^{q+1}\) and take the mean value integral over \(B_\varrho \). Then we use Hölder’s inequality and estimate \(|a|=|(u)_\varrho ^{(\lambda ,\theta )}|\) by means of the \(\theta \)-sub-intrinsic coupling in the form (4.3). This leads to the bound
with a constant \(c=c(q,C_\theta )\). In the second to last line, we used the definition of \(\mathbf{S}\) and the definition of \(\tau _\varrho ^{(\lambda ,\theta )}\) according to (2.1), whereas in the last line we used inequality (4.8). Using this inequality to bound the right-hand side of (4.10), we arrive at the estimate
with a constant \(c=c(n,p,q,L,C_\theta )\). In the case \(p\ge 2\), we can use (4.9) to bound the negative powers of \(\lambda \) from above by powers of \(\mathbf{F}\), leading us to
for every \(\varepsilon \in (0,1]\) with a constant \(c=c(n,p,q,L,C_\theta ,C_\lambda )\), where the last estimate follows by two applications of Young’s inequality. Finally, we note that the exponent of \(\varepsilon \) is bounded from below by \(-\frac{1}{q}\), since \(\vartheta \ge \frac{p-1}{p}\). This yields the asserted inequality for exponents \(p\ge 2\).
In the remaining case \(p<2\), we note that (4.9) implies
with a constant \(c=c(n,p,q,C_\lambda )\). In this way, we deduce
for every \(\varepsilon \in (0,1]\), with a constant \(c=c(n,p,q,L,C_\theta ,C_\lambda )\). In the last line, we applied Young’s inequality for three factors, once with exponents \(\frac{q+1}{(1-\vartheta )p}\), \(\frac{q+1}{(2-p)(1+\vartheta q)}\), and \(\frac{q+1}{\vartheta (pq+p-2q)-(1-q)}\), and once with \(\frac{2}{(1-\vartheta )p}\), \(\frac{2}{(2-p)(1+\vartheta )}\), and \(\frac{1}{\vartheta (p-1)}\). Using \(\vartheta \ge \frac{1}{p}\) and \(p<2\), we can estimate the exponent of \(\varepsilon \) from (4.13) from below by \(-\frac{1}{q(p-1)}\). Since \(\lambda ^p\) can be estimated from above with the help of the \(\lambda \)-super-intrinsic coupling condition (4.1)\(_2\), estimate (4.13) implies the claim in the remaining case \(p<2\). \(\square \)
Moreover, we need a version of the Sobolev–Poincaré inequality to control the term in the energy estimate that stems from the time term of the parabolic equation.
Lemma 4.4
Let \(p>\frac{2n}{n+2}\) and \(0<q\le 1\), and let u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subseteq \Omega _T\) satisfying the \(\lambda \)-intrinsic coupling (4.1) and the \(\theta \)-sub-intrinsic coupling (4.2)\(_1\) for some \(0<\varrho \le 1\) and some \(\lambda ,\theta >0\), the following inequality
holds true for any given \(\varepsilon \in (0,1]\) and universal constants \(c=c(n,p,q,L,C_\theta ,C_\lambda )\) and \(\beta =\beta (n,p,q)>0\). Here, the parameter \(\vartheta \) is defined by \( \vartheta {:}{=}\frac{n}{n+2}. \)
Proof
We omit again the reference point \(z_o\) and write \(a{:}{=}(u)_\varrho ^{(\lambda ,\theta )}\) and continue to use the abbreviations introduced in (4.6). We begin with the case \(\frac{2n}{n+2}<p\le 2\). We split off a certain power of the spatial integrand and estimate it by the supremum over the time-slices. Then we use the estimate \(\frac{|a|}{\varrho }\le c\theta \), which follows by the sub-intrinsic coupling in the form (4.3) and apply Lemma 2.2. Finally, we apply the Sobolev inequality. This procedure leads to the estimate
with a constant \(c=c(n,p,q,L,C_\theta )\). We estimate the last integral by means of Lemma 4.2 with \(\vartheta =\frac{2n}{p(n+2)}\) and arrive at
In the last term we use the estimate \(\mathbf{F}\le c\lambda ^p\), which follows from the \(\lambda \)-sub-intrinsic coupling in the form (4.4). Then, we apply Young’s inequality with exponents \(\frac{n+2}{2}\), \(\frac{p(n+2)}{n(2-p)}\), and \(\frac{p(n+2)}{2n(p-1)}\) and obtain the bound
for every \(\varepsilon \in (0,1]\), with \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta )\). Since \(\lambda \) can be bounded from above by the \(\lambda \)-super-intrinsic coupling condition (4.1)\(_2\), this yields the claim in the case \(\frac{2n}{n+2}<p\le 2\).
For exponents \(p>2\), we use Lemma 2.2 and then use Young’s inequality to obtain
Now the claim follows by an application of Lemma 4.3. \(\square \)
Finally, we prove a third version of a Sobolev–Poincaré inequality that will be necessary in the case \(p<q+1\). We note that in the case \(p\ge q+1\), it could be replaced by Lemma 4.3. However, we state the lemma for all cases in order to reduce the case distinctions in the further proof to a minimum.
Lemma 4.5
Let \(p>\frac{n(q+1)}{n+q+1}\) and \(0<q\le 1\), and u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\theta )}(z_o)\subseteq \Omega _T\) satisfying the \(\lambda \)-intrinsic coupling (4.1) and the \(\theta \)-sub-intrinsic coupling (4.2)\(_1\) for some \(0<\varrho \le 1\) and some \(\lambda ,\theta >0\), the following inequality
holds true for any given \(\varepsilon \in (0,1]\) and universal constants \(c=c(n,p,q,L,C_\theta ,C_\lambda )\), \(\beta =\beta (n,p)>0\), and the parameter \( \vartheta {:}{=}\max \big \{\frac{n(q+1)}{p(n+q+1)},\frac{n}{n+q+1},\frac{1}{p}, \frac{p-1}{p}\big \}.\)
Proof
We again omit the reference point \(z_o\) and abbreviate \(a{:}{=}(u)_{\varrho }^{(\lambda ,\theta )}\) and
Moreover, we continue to use the abbreviations \(\mathbf{F}\) and \(\mathbf{S}\) introduced in (4.6). First, we consider the case \(p<q+1\), in which \(p^\sharp =q+1\). We start with the observation that the \(\theta \)-sub-intrinsic coupling in the form of (4.3) implies
with a constant \(c=c(n,p,q,C_\theta )\). Moreover, we can assume that
since otherwise, the assertion is obviously true. The combination of the two preceding inequalities yields
where \(c=c(n,p,q,C_\theta )\). For the proof of the Sobolev type inequality, we estimate a part of the spatial integral in \(\widetilde{\mathbf {E}}\) by its supremum over time and apply Lemma 2.2 to the other part. In a second step, we apply Sobolev’s inequality, with the result
We estimate the last term by Lemma 4.2 with \(\vartheta =\frac{n(q+1)}{p(n+q+1)}\) and then the \(\lambda \)-sub-intrinsic coupling in the form (4.4), which implies \(\mathbf{F}\le c\lambda ^p\). This leads to the bound
with \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta )\). We take the preceding estimate to the power \(\frac{2}{q+1}\) and multiply both sides with \(\lambda ^{p-2}\). In this way, we infer
Next, we use the lower bound (4.15) for \(\lambda ^{p-2}\theta ^2\), together with the fact \(q\le 1\). Then we apply Young’s inequality with exponents \(\frac{p(n+q+1)}{n(2-p)}\), \(\frac{n+q+1}{2}\) and \(\frac{p(n+q+1)}{2n(p-1)+p(q-1)}\) and obtain
for every \(\varepsilon \in (0,1]\), with constants \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta )\) and \(\beta =\beta (n,p,q)>0\). This yields the claim in the case \(p<q+1\), since \(\lambda ^p\) can be bounded from above by (4.1). In the remaining case \(p\ge q+1\), in which \(p^\sharp =p\), the asserted estimate is a consequence of Lemma 4.3. More precisely, Lemma 2.2 implies
In the case \(q+1\le p<2\), we use Lemma 4.2 and the \(\lambda \)-sub-intrinsic coupling in the form (4.4) to estimate
with \(c=c(n,p,q,L,C_\lambda )\). Combining the preceding estimates, we conclude
from which the assertion follows in the case \(q+1\le p<2\) by estimating the last integral with the help of Lemma 4.3. In the case \(p\ge 2\ge q+1\), we apply Young’s inequality to the right-hand side of (4.16) and deduce
for every \(\delta \in (0,1]\). Estimating the last term by Lemma 4.3, we deduce
for any \(\delta ,\kappa \in (0,1]\), with \(\vartheta {:}{=}\max \big \{\frac{n}{n+q+1},\frac{1}{p}, \frac{p-1}{p}\big \}\), \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta )\) and \(\alpha =\alpha (n,p,q)>0\). This implies the claim in the remaining case by choosing \(\delta =\frac{\varepsilon }{2}\) and \(\kappa =\frac{1}{2}\varepsilon \delta ^{\frac{p-2}{2}}\). \(\square \)
Lemma 4.6
Let \(p>\frac{n(q+1)}{n+q+1}\) and \(0<q\le 1\), and u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\theta )}(z_o)\subseteq \Omega _T\) that satisfies the \(\lambda \)-intrinsic coupling (4.1) and the \(\theta \)-intrinsic coupling (4.2) with constant \(C_\theta =1\), for some \(0<\varrho \le 1\) and some \(\lambda , \theta >0\), we have
where \(c=c(n,p,q,\nu ,L,C_\lambda )\).
Proof
Again, we omit the center \(z_o\) in the notation and write \(a_\sigma =(u)_{\sigma }^{(\lambda ,\theta )}\) for \(\sigma \in \{\frac{\varrho }{2},\varrho \}\). We use the \(\theta \)-super-intrinsic coupling (4.2)\(_2\) with constant \(C_\theta =1\), Minkowski’s inequality and finally, Lemma 2.4. This yields the bound
where \(c=c(p,q)\). For the estimate of the second last integral, we distinguish between the cases \(p\ge q+1\) and \(p<q+1\). In the first case, in which \(p^\sharp =p\), we use in turn Lemma 2.3, Lemma 4.2 with \(\vartheta =1\), and then the \(\lambda \)-sub-intrinsic coupling (4.1)\(_1\) to deduce
with \(c=c(n,p,q,L,C_\lambda )\). In the case \(p<q+1\), we use Lemma 4.5 instead. In view of the \(\lambda \)-sub-intrinsic coupling (4.1)\(_1\) we infer
for every \(\varepsilon \in (0,1]\), with \(c=c(n,p,q,L,C_\lambda )\) and \(\beta =\beta (n,p,q)>0\). We estimate the supremum on the right-hand side by the energy estimate from Lemma 3.1 and then by Lemma 2.4 with \(a=0\) and obtain
with \(c=c(p,q,\nu ,L,C_\lambda )\), where we applied the coupling conditions (4.1)\(_1\) and (4.2)\(_2\) in the last step. Joining the two preceding estimates and applying Young’s inequality, we infer
which holds in the case \(p<q+1\). Using (4.18), respectively (4.19) in (4.17), we deduce that in both cases the estimate
holds for every \(\varepsilon \in (0,1]\), with \(c=c(n,p,q,\nu ,L,C_\lambda )\) and \(\beta =\beta (n,p,q)\). At this stage, we choose the parameter \(\varepsilon \in (0,1]\) in dependence on \(n,p,q,\nu ,L,\) and \(C_\lambda \) so small that
This allows us to absorb the first term from the right-hand side into the left-hand side. After dividing by \(1-c\varepsilon ^{\frac{q+1}{2}}\), we arrive at the bound
with \(c=c(n,p,q,\nu ,L,C_\lambda )\). This completes the proof of the lemma. \(\square \)
5 Reverse Hölder inequality
Proposition 5.1
Let \(p>\frac{2n}{n+2}\) and \(0<q\le 1\), and u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\lambda , \theta )}(z_o)\Subset \Omega _T\) that satisfies the \(\lambda \)-intrinsic coupling (4.1) and the \(\theta \)-intrinsic coupling (4.2) for some \(0<\varrho \le 1\), \(\lambda \ge 1\) and \(\theta >0\), the following reverse Hölder type inequality holds true
for \(\vartheta {:}{=}\max \big \{\frac{n(q+1)}{p(n+q+1)},\frac{n}{n+q+1},\frac{1}{p}, \frac{p-1}{p}\big \}\) and some universal constant c that depends on \(n,p,q,\nu , L,C_\lambda ,\) and \(C_\theta \).
Proof
Once again, we omit the reference to the center \(z_o\) in the notation. We consider radii r, s with \(\varrho \le r<s\le 2\varrho \) and write \(a_\sigma {:}{=}(u)_\sigma ^{(\lambda ,\theta )}\) with \(\sigma \in \{r,s\}\). From the energy estimate in Lemma 3.1 we obtain
with the obvious abbreviation of I–III. The dependencies of the constant are given by \(c=c(p,q,\nu ,L)\). With the \(\theta \)-super-intrinsic coupling (4.2)\(_2\) we first estimate
We introduce the abbreviation
For the estimate of the first term in (5.2), we use in turn (5.3) and the assumption \(0<q\le 1\), then Hölder’s inequality and finally, Lemma 2.4 to replace \(a_r\) by \(a_s\). This procedure results in the estimate
with \(c=c(n,p,q,\nu ,L,C_\theta )\). In the term \(\mathrm {II}\), we also apply Lemma 2.4 and obtain
Collecting the estimates and using Lemmas 4.3, 4.4, and 4.5 to estimate the individual terms, we arrive at
for any \(\varepsilon \in (0,1]\), with \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta )\). Here, we choose \(\varepsilon =\frac{1}{2c{\mathcal {R}}_{r,s}^{p^\sharp }}\) and obtain
To re-absorb the term \(\frac{1}{2}(\dots )\) from the right-hand side into the left-hand side, we apply the Iteration Lemma 2.1. This leads to the claimed reverse Hölder type inequality (5.1) and finishes the proof of Proposition 5.1. \(\square \)
Proposition 5.2
Let \(p>\frac{2n}{n+2}\) and \(0<q\le 1\), and u be a weak solution to (1.3) in \(\Omega _T\) in the sense of Definition 1.1. Then, on any cylinder \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\Subset \Omega _T\) satisfying the \(\lambda \)-intrinsic coupling (4.1) and a coupling condition of the form
for some \(0<\varrho \le 1\), \(\lambda \ge 1\) and \(\theta >0\) and some constants \(C_\theta , K_\theta \ge 1\), the following reverse Hölder type inequality holds true
with \(\vartheta {:}{=}\max \big \{\frac{n(q+1)}{p(n+q+1)},\frac{n}{n+q+1},\frac{1}{p}, \frac{p-1}{p}\big \}\), and where the constant c depends on n, p, q, \(\nu ,L,C_\lambda ,C_\theta ,\) and \(K_\theta \).
Proof
The largest part of the proof is identical to that of Proposition 5.1. We again start with the energy estimate (5.2) and estimate the term \(\mathrm {II}\) therein by (5.5). The only difference is the estimation of \(\mathrm {I}\), which is now based on (5.6). More precisely, we use (5.6) in combination with (4.4) to deduce
For the estimate of \(\mathrm {I}\), we use in turn the preceding estimate, then Hölder’s inequality, Lemma 2.4 and finally, Young’s inequality with exponents \(\frac{2}{1-q}\) and \(\frac{2}{q+1}\) to infer
with a constant \(c=c(n,p,q,\nu ,L,C_\lambda ,K_\theta )\). Using this estimate in (5.2) and recalling (5.5), we arrive at
The first term on the right-hand side can be estimated with the help of the \(\lambda \)-super-intrinsic coupling (4.1)\(_2\) in the form
and the integral of \(|Du|^p\) can be absorbed into the left-hand side. Next, we estimate the first integral on the right-hand side of (5.8) by Lemma 4.5 and the second one by Lemma 4.3 and obtain the bound
for any \(\varepsilon \in (0,1]\), with \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta ,K_\theta )\). In the preceding estimate, we choose the parameter \(\varepsilon \) in the form \(\varepsilon =\frac{1}{2c{\mathcal {R}}_{r,s}^{2\vee p}}\) and arrive at
where \(c=c(n,p,q,\nu ,L,C_\lambda ,C_\theta ,K_\theta )\). Now, we apply the Iteration Lemma 2.1 to re-absorb the first two terms on the right-hand side into the left. This yields the claimed reverse Hölder inequality (5.7) and finishes the proof of the Proposition. \(\square \)
6 Proof of the higher integrability
We consider a fixed cylinder \(Q_{4R} \equiv Q_{4R}^{(1,1)}\) with \(R\in (0,1]\) such that \(Q_{8R}\subset \Omega _T\) and some
with the scaling deficit d introduced in (1.8), i.e.
Moreover, we fix \(\lambda \ge \lambda _o\) and define \(R_o{:}{=}\min \{1,\lambda ^{\frac{p-q-1}{q+1}}\}R\). Then, for any \(z_o\in Q_{2R}\), any \(\theta \ge \lambda \) and every \(\varrho \le R_o\), we have \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o)\subset Q_{4R}\).
6.1 Construction of a non-uniform system of cylinders
The following construction of a non-uniform system of cylinders is similar to the one in [11]. Let \(z_o\in Q_{2R}\). For a radius \(\varrho \in (0,R_o]\) we now define
where we again use the notation \(p^\sharp =\max \{p,q+1\}\). If the meaning is clear from the context we write \({\widetilde{\theta }}_\varrho \) instead of \({\widetilde{\theta }}_{z_o;\varrho }^{(\lambda )}\). Of course, this assumes that \( z_o \) and \(\lambda \) are fixed in the respective context. We note that the set of those \(\theta \ge \lambda \) for which the condition above is satisfied is obviously not empty. In fact, in the limit \(\theta \uparrow \infty \), by definition of \(Q_{\varrho }^{(\lambda ,\theta )}(z_o)\) the integral converges to zero in the case \(0<q<1\) and stays bounded for \(q=1\), while the right-hand side blows up with speed \(\theta ^{p^\sharp +q-1}\).
Observe that the condition in the infimum can be re-written in the form
Therefore, we either have
or otherwise
Note that \({\widetilde{\theta }}_{R_o}\ge \lambda \ge \lambda _o\ge 1\). If \( \lambda <{\widetilde{\theta }}_{R_o}\), then by (6.2) we have
Therefore, in any case we end up with the bound
Next, we establish that the mapping \((0,R_o]\ni \varrho \mapsto {\widetilde{\theta }}_\varrho \) is continuous. To this end, consider \(\varrho \in (0,R_o]\) and \(\varepsilon >0\), and define \(\theta _+{:}{=}{\widetilde{\theta }}_\varrho +\varepsilon \). Then, there exists \(\delta =\delta (\varepsilon ,\varrho )>0\) such that
for any \(r\in (0,R_o]\) with \(|r-\varrho |<\delta \). In fact, the preceding strict inequality holds by the very definition of \({\widetilde{\theta }}_\varrho \) with \(r=\varrho \), since the integral on the left-hand side decreases with the replacement of \({\widetilde{\theta }}_\varrho \) by \(\theta _+\) (note that \(\Lambda _{\varrho }^{(\lambda , \theta _+)}(t_o) \subset \Lambda _{\varrho }^{(\lambda , {\widetilde{\theta }}_\varrho )}(t_o) \), so that the integral decreases), while the right-hand side strictly increases. The claim now follows, since both the integral on the right and the left-hand side are continuous functions with respect to the radius r. In other words, we have shown that \({\widetilde{\theta }}_r\le \theta _+={\widetilde{\theta }}_\varrho +\varepsilon \) for radii r as above. It remains to prove \({\widetilde{\theta }}_r\ge \theta _-{:}{=}{\widetilde{\theta }}_\varrho -\varepsilon \) for r close to \(\varrho \). If \(\theta _-\le \lambda \) holds, then \({\widetilde{\theta }}_r\ge \lambda \ge \theta _-\) for any r. In the other case, after diminishing \(\delta =\delta (\varepsilon ,\varrho )>0\) if necessary, we get
for all \(r\in (0,R_o]\) with \(|r-\varrho |<\delta \). For \(r=\varrho \), this is a direct consequence of the definition of \({\widetilde{\theta }}_\varrho \) (here, the right-hand side strictly decreases in the transition from \({\widetilde{\theta }}_\varrho \) to \(\theta _-\), while the left-hand side increases, since \(\Lambda _{\varrho }^{(\lambda , {\widetilde{\theta }}_\varrho )}(t_o) \subset \Lambda _{\varrho }^{(\lambda , \theta _-)}(t_o)\)), and for r with \(|r-\varrho |<\delta \) small enough the claim follows from the continuity of both sides as a function of r. The preceding inequality implies that \({\widetilde{\theta }}_r\ge \theta _-{:}{=}{\widetilde{\theta }}_\varrho -\varepsilon \). This completes the proof of the continuity of \((0,R_o]\ni \varrho \mapsto {\widetilde{\theta }}_\varrho \).
Unfortunately, the mapping \({\widetilde{\theta }}_\varrho \) might not be monotone in \(\varrho \). For this reason we modify \({\widetilde{\theta }}_\varrho \), so that the modification—denoted by \(\theta _\varrho \)—becomes monotone. More precisely, we define
We only use the abbreviation of \(\theta _{z_o;\varrho }^{(\lambda )}\) by \(\theta _\varrho \) in a context in which \(z_o\) and \(\lambda \) are fixed, so that no confusion is possible. By construction the mapping \((0,R_o]\ni \varrho \mapsto \theta _\varrho \) is continuous and monotonically decreasing. Moreover, we claim that for radii \(\varrho \le s\), the cylinders \(Q_{s}^{(\lambda ,\theta _\varrho )}(z_o)\) are \(\theta \)-sub-intrinsic with constant \(C_\theta =1\), i.e.
In fact, the definition of \(\theta _s\) and the monotonicity of \(\theta _\varrho \) imply \({\widetilde{\theta }}_s\le \theta _{s}\le \theta _{\varrho }\), so that \(Q_{s}^{(\lambda ,\theta _{\varrho })}(z_o)\subset Q_{s}^{(\lambda ,{\widetilde{\theta }}_{s})}(z_o)\). Therefore, we have
This proves the claim (6.4). We now define
In particular, for any \(\sigma \in [\varrho ,{\widetilde{\varrho }}]\) we have \(\theta _\sigma ={\widetilde{\theta }}_{\tilde{\varrho }}\). Next, we claim that
If \(\theta _\varrho =\lambda \), then also \(\theta _s=\lambda \), so that (6.6) trivially follows. Therefore, it remains to consider the case \(\theta _\varrho >\lambda \). Then, for any \(s\in [{\widetilde{\varrho }},R_o]\) the monotonicity of \(s\mapsto \theta _s\), (6.2) and (6.4) imply
which yields (6.6). If \(s\in (\varrho ,{\widetilde{\varrho }}]\), then \(\theta _\varrho =\theta _{s}\), and the claim (6.6) follows again. Thus, we have established (6.6) in any case.
Next, we apply (6.6) with \(s=R_o\). Since \(\theta _{R_o}={\widetilde{\theta }}_{R_o}\), estimate (6.3) for \({\widetilde{\theta }}_{R_o}\) yields
In the following, we consider the system of concentric cylinders \(Q_{\varrho }^{(\lambda , \theta _{z_o;\varrho }^{(\lambda )})}(z_o)\) with radii \(\varrho \in (0,R_o]\) and \(z_o\in Q_{2R}\). The cylinders are nested in each other, in the sense that
The inclusion holds true due to the monotonicity of the mapping \(\varrho \mapsto \theta _{z_o;\varrho }^{(\lambda )}\). The disadvantage of this system of nested cylinders is that in general the cylinders are not \(\theta \)-intrinsic, but merely \(\theta \)-sub-intrinsic with \(C_\theta =1\).
6.2 Covering property
Here, we will prove a Vitali type covering lemma for the cylinders constructed in the last section. More precisely, we will show the following.
Lemma 6.1
Let \(\lambda \ge \lambda _o\). There exists a constant \({\hat{c}}={\hat{c}}(n,p,q)\ge 20\) such that the following holds true: Let \({\mathcal {F}}\) be any collection of cylinders \(Q_{4r}^{(\lambda , \theta _{z;r}^{(\lambda )})}(z)\), where \(Q_{r}^{(\lambda ,\theta _{z;r}^{(\lambda )})}(z)\) is a cylinder of the form as constructed in Sect. 6.1 with radius \(r\in (0,\tfrac{R_o}{{\hat{c}}})\). Then, there exists a countable subfamily \({\mathcal {G}}\) of disjoint cylinders in \({\mathcal {F}}\) such that
where \({\widehat{Q}}\) denotes the \(\frac{1}{4}{\hat{c}}\)-times enlarged cylinder Q, i.e. if \(Q=Q_{4r}^{(\lambda ,\theta _{z;r}^{(\lambda )})}(z)\), then \(\widehat{Q}=Q_{{\hat{c}} r}^{(\lambda ,\theta _{z;r}^{(\lambda )})}(z)\).
Proof
Let \({\hat{c}}\ge 20\) be a parameter at our disposal. Later on we fix \({\hat{c}}\) in a universal way in dependence on n, p, and q. For \(j\in {\mathbb {N}}\) we define
The subfamilies \({\mathcal {G}}_j\subset {\mathcal {F}}_j\) are constructed according to the following recursive scheme: Let \({\mathcal {G}}_1\) be any maximal disjoint collection of cylinders in \({\mathcal {F}}_1\). Note that \({\mathcal {G}}_1\) is finite, since by (6.7) and the definition of \({\mathcal {F}}_1\) the \({\mathcal {L}}^{n+1}\)-measure of each cylinder \(Q\in {\mathcal {G}}_1\) is uniformly bounded from below. Now, assume that \({\mathcal {G}}_1, {\mathcal {G}}_2, \dots , \mathcal G_{k-1}\) have already been inductively selected up to some \(k\in {\mathbb {N}}_{\ge 2}\). Then, from \({\mathcal {F}}_k\) we remove those cylinders Q which intersect one of the previously chosen cylinders \(Q^*\in {\mathcal {G}}_1\cup \ldots \cup {\mathcal {G}}_{k-1}\). Amongst the remaining cylinders we then choose a maximal disjoint sub-collection \({\mathcal {G}}_k\). In other words, we choose \(\mathcal G_k\) as a maximal disjoint collection of cylinders in
Note again that \({\mathcal {G}}_k\) is finite. Finally, we define
By construction, \({\mathcal {G}}\) is a countable collection of disjoint cylinders and \({\mathcal {G}}\subset {\mathcal {F}}\).
At this point it remains to prove that for each \(Q\in {\mathcal {F}}\) there exists a cylinder \(Q^*\in {\mathcal {G}}\) such that \(Q\cap Q^*\not =\emptyset \) and \(Q\subset \widehat{Q}^*\). To this aim we choose some arbitrary \(Q=Q_{4r}^{(\lambda ,\theta _{z;r}^{(\lambda )})}(z)\in {\mathcal {F}}\). Then, there exists an index \(j\in {\mathbb {N}}\) such that \(Q\in {\mathcal {F}}_j\). The maximality of \({\mathcal {G}}_j\) ensures the existence of some cylinder \(Q_*=Q_{4r_*}^{(\lambda ,\theta _{z_*;r_*}^{(\lambda )})}(z_*)\in \bigcup _{i=1}^{j} {\mathcal {G}}_i\) with \(Q\cap Q_*\not =\emptyset \). Observe that \(r\le \tfrac{R_o}{2^{j-1}{\hat{c}}}\) and \(r_*>\tfrac{R_o}{2^j{\hat{c}}}\), so that \(r<2r_*\). In the following we shall establish an estimate relating the size of \(\theta _{z_*;r_*}^{(\lambda )}\) to the one of \(\theta _{z;r}^{(\lambda )}\). The precise estimate is as follows:
Let \(\eta {:}{=}16\). Denote by \({\widetilde{r}}_*\in [r_*,R_o]\) the radius associated to the cylinder \(Q_{r_*}^{(\lambda ,\theta _{z_*;r_*}^{(\lambda )})}(z_*)\); see (6.5) for the construction. Recall that either \(Q_{\widetilde{r}_*}^{(\lambda ,\theta _{z_*;r_*}^{(\lambda )})}(z_*)\) is intrinsic in the sense of (6.2)\(_2\) or \(\widetilde{r}_*=R_o\) and \(\theta _{z_*;r_*}^{(\lambda )}=\lambda \). In the latter case we have
by definition of \(\theta _{z;r}^{(\lambda )}\), so that the claim (6.9) is clearly satisfied. Therefore, it only remains to consider the case that \(Q_{\widetilde{r}_*}^{(\lambda ,\theta _{z_*;r_*}^{(\lambda )})}(z_*)\) is intrinsic, more precisely
Now, we distinguish between the cases \({\widetilde{r}}_*\le \frac{R_o}{\eta }\) and \({\widetilde{r}}_*> \frac{R_o}{\eta }\). In the latter case we exploit (6.10), the definition of \(\lambda _o\) and the facts \(\theta _{z;r}^{(\lambda )}\ge \lambda \ge \lambda _o\) in order to obtain
where the last estimate follows by distinguishing between the cases \(p\ge q+1\) and \(p<q+1\). The preceding inequality shows that
proving (6.9) in the case \(\widetilde{r}_*>\frac{R}{\eta }\). It remains to consider the case \(\widetilde{r}_*\le \frac{R_o}{\eta }\). Since \({\tilde{r}}_*\ge r_*\) and \(|x-x_*|<4r+4r_*\le 12r_*\), we know \(B_{4{\tilde{r}}_*}(x_*)\subset B_{16{\tilde{r}}_*}(x)= B_{\eta \tilde{r}_*}(x)\). Moreover,
At this point, we can assume \(\theta _{z;r}^{(\lambda )}\le \theta _{z_*;r_*}^{(\lambda )}\). Otherwise (6.9) trivially holds. The monotonicity of \(\varrho \mapsto \theta _{z;\varrho }^{(\lambda )}\) and the fact that \(r\le 2r_*\le 2 {\widetilde{r}}_*\le \eta {\widetilde{r}}_*\) implies that
Therefore, we can conclude that
and this immediately implies the inclusion
Altogether, we have shown
Using (6.10), the inequalities (6.12), and (6.4) with \(\varrho =s=\eta {\tilde{r}}_*\), we obtain
We conclude
so that (6.9) is established in all cases. From (6.9) we get
if we choose the constant \({\hat{c}}={\hat{c}}(n,p,q)\ge 20\) suitably. Moreover, we note that
Therefore, we have shown that
This proves the claim (6.8) and completes the proof. \(\square \)
6.3 Stopping time argument
We now let
Obviously, this condition implies the previously demanded requirement (6.1). For \(\lambda >\lambda _o\) and \(r\in (0,2R]\), we define the superlevel set of |Du| by
In this context, of course, the concept of Lebesgue points of |Du| must be adapted to the system of cylinders constructed in Sect. 6.1. For radii \(R\le R_1<R_2\le 2R\), we consider the concentric parabolic cylinders \(Q_R\subseteq Q_{R_1} \subset Q_{R_2}\subseteq Q_{2R}\). Note that the inclusion \(Q_{2\varrho }^{(\lambda ,\theta )}(z_o) \subset Q_{4R}\) holds true, whenever \(z_o\in Q_{2R}\), \(\theta \ge \lambda \) and \(\varrho \in (0,R_o]\). We fix \(z_o\in {\varvec{E}}(R_1,\lambda )\) and abbreviate \(\theta _s\equiv \theta _{z_o;s}^{(\lambda )}\) for \(s\in (0,R_o]\) throughout this section. By definition of \(\varvec{E}(R_1,\lambda )\), we have that
In the following, we consider values of \(\lambda \) satisfying
where \({\hat{c}}={\hat{c}}(n,p,q)\) denotes the constant from the Vitali-type covering Lemma 6.1. For s with
we have, by definition of \(\lambda _o\), that
In the individual intermediate steps we have used in turn (6.7), (6.16) and (6.15), and distinguished between the cases \(p\ge q+1\) and \(p<q+1\). On the other hand, on behalf of (6.14) we find \(0< s < \tfrac{R_2-R_1}{\varvec{\mathfrak {m}}}\) sufficiently small such that the integral in (6.14) with \(Q_{s}^{(\lambda ,\theta _s)}(z_o)\) as domain of integration, possesses a value larger than \(\lambda ^{p}\). Consequently, by the absolute continuity of the integral there exists a maximal radius \(0<\varrho _{z_o} < \tfrac{R_2-R_1}{\varvec{\mathfrak {m}}}\) such that
The choice of \(\varrho _{z_o}\) as maximal radius now guarantees
More generally, for any \(\varrho _{z_o}<s\le R_o\), the monotonicity of \(\varrho \mapsto \theta _\varrho \), estimate (6.6) and (6.18) imply
Moreover, \(Q_{{\hat{c}}\varrho _{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\subset Q_{R_2}\). In fact, this inclusion follows from the fact \(\varrho _{z_o}<\frac{R_2-R_1}{\varvec{\mathfrak {m}}}\) since this implies
6.4 A Reverse–Hölder inequality
As before, we consider \(z_o\in {\varvec{E}}(R_1,\lambda )\) with \(\lambda \) as in (6.15). Since \(\lambda \) and \(z_o\) are fixed, it is possible to use the abbreviation \(\theta _{\varrho _{z_o}} {:}{=} \theta _{z_o;\varrho _{z_o}}^{(\lambda )}\) without causing confusion. We keep in mind that by construction \(0<\varrho _{z_o}< \frac{R_2-R_1}{\varvec{\mathfrak {m}}}\). We choose \({\widetilde{\varrho }}_{z_o}\in [\varrho _{z_o},R]\) according to (6.5). This choice allows us to pass from the possibly sub-intrinsic cylinder \(Q_{\varrho _{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\) to the intrinsic cylinder \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\). By construction we have \(\theta _s=\theta _{\varrho _{z_o}}\) for any \(s\in [\varrho _{z_o}, {\widetilde{\varrho }}_{z_o}]\) and in particular, \(\theta _{{\widetilde{\varrho }}_{z_o}}=\theta _{\varrho _{z_o}}\). Our aim now is to prove the following reverse Hölder inequality
with \(\vartheta {:}{=}\max \big \{\frac{n(q+1)}{p(n+q+1)},\frac{n}{n+q+1},\frac{1}{p}, \frac{p-1}{p}\big \}<1\) and \(c=c(n,p,q,\nu , L)\). We distinguish between the two cases in which \({\widetilde{\varrho }}_{z_o}\le 2\varrho _{z_o}\) or \({\widetilde{\varrho }}_{z_o}> 2\varrho _{z_o}\).
The case \(\varvec{{\widetilde{\varrho }}_{z_o}\le 2\varrho _{z_o}}\). Here we wish to apply Proposition 5.1 on the cylinder \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\). In order to check that this is possible, we first note that \(\theta _{\varrho _{z_o}}>\lambda \), since otherwise we would have \({\widetilde{\varrho }}_{z_o}=R_o>2\varrho _{z_o}\). Therefore, we can apply (6.2). Keeping in mind that \(\theta _{\varrho _{z_o}}=\theta _{\tilde{\varrho }_{z_o}}=\tilde{\theta }_{\tilde{\varrho }_{z_o}}\) and using additionally (6.4) with \(s=2{\widetilde{\varrho }}_{z_o}\), we deduce
This shows that \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\) is \(\theta \)-intrinsic. More precisely, the coupling condition (4.2) is satisfied with constant \(C_\theta =1\). On the other hand, estimate (6.19) with \(s=2{\widetilde{\varrho }}_{z_o}\le 4\varrho _{z_o}\) and the identity (6.17) imply
with a constant \(c=c(n,p,q)\). This ensures that the \(\lambda \)-intrinsic coupling condition (4.1) is satisfied for the cylinder \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\), with a constant \(C_\lambda \) depending on n, p, and q. Altogether, we have shown that the assumptions of Proposition 5.1 are satisfied. Applying this proposition and using the assumption \({\widetilde{\varrho }}_{z_o}\le 2\varrho _{z_o}\), we infer the reverse Hölder inequality
where \(c=c(n,p,q,\nu ,L)\). This establishes (6.20) in the first case.
The case \(\varvec{{\widetilde{\varrho }}_{z_o}> 2\varrho _{z_o}}\). Here, we will apply Proposition 5.2 on the cylinder \(Q_{\varrho _{z_o}}^{(\theta _{\varrho _{z_o}})}(z_o)\). This is only permitted, however, if the hypotheses (4.1) and (5.6) are satisfied on this cylinder. First, we notice that (4.1) (with \(C_\lambda =1\)) follows from (6.17) and (6.18). Moreover, the condition (5.6)\(_1\) (with \(C_\theta \equiv 1\)) is an immediate consequence of (6.4). For the verification of (5.6)\(_2\), we consider two cases. In the case \({\widetilde{\varrho }}_{z_o}\in (\frac{R_o}{2},R_o]\), we use estimate (6.7) with \(\varrho ={\widetilde{\varrho }}_{z_o}\) and then (6.17) to deduce
with \(c=c(n,p,q)\), which yields (5.6)\(_2\) in the first case. In the second case \({\widetilde{\varrho }}_{z_o}\in (2\varrho _{z_o},\frac{R_o}{2}]\), we observe that by (6.2) with \(\varrho ={\widetilde{\varrho }}_{z_o}\) and by (6.4) with \(s=2{\widetilde{\varrho }}_{z_o}\), the cylinder \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\) satisfies the \(\theta \)-intrinsic coupling condition (4.2) with constant \(C_\theta =1\). Because of (6.17), applied with \({\widetilde{\varrho }}_{z_o}\) in place of \(\varrho _{z_o}\), the cylinder \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\) is \(\lambda \)-intrinsic as well. Because of \(2{\widetilde{\varrho }}_{z_o}\le R_o\), the cylinder with doubled radius is still contained in \(Q_{4R}\subset \Omega _T\). Therefore, the assumptions of Lemma 4.6 are satisfied on \(Q_{{\widetilde{\varrho }}_{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\), and we infer the estimate
with \(c=c(n,p,q,\nu ,L)\), where the last estimate holds since \({\widetilde{\varrho }}_{z_o}>2\varrho _{z_o}\) ensures that (6.4) is applicable with \(s={\widetilde{\varrho }}_{z_o}/2\). The last term can be absorbed into the left-hand side, and we conclude that the bound
is satisfied in all cases, with a constant \(c=c(n,p,q,\nu ,L)\). The last identity is due to (6.17). This proves that (5.6)\(_2\) is satisfied, and we have verified all assumptions of Proposition 5.2 for the cylinder \(Q_{\varrho _{z_o}}^{(\theta _{\varrho _{z_o}})}(z_o)\). Therefore, we obtain the reverse Hölder inequality
Overall, we have shown that in any case the reverse Hölder inequality (6.20) holds.
6.5 Estimate on super-level sets
Up to this point, we have shown: If \(\lambda \) satisfies (6.15), then for every \(z_o\in E(R_1,\lambda )\) there exists a cylinder \(Q_{\varrho _{z_o}}^{(\lambda ,\theta _{z_o;\varrho _{z_o}}^{(\lambda )})}(z_o)\) such that the \({\hat{c}}\) times enlarged cylinder \(Q_{{\hat{c}}\varrho _{z_o}}^{(\lambda ,\theta _{z_o;\varrho _{z_o}}^{(\lambda )})}(z_o)\) is still contained in \( Q_{R_2}\), and such that (6.17), (6.18) and (6.20) hold on this specific cylinder. As before, we write \(\theta _{\varrho _{z_o}}\equiv \theta _{z_o;\varrho _{z_o}}^{(\lambda )}\). Next we define the superlevel set of the inhomogeneity |F| by
Using (6.17) and (6.20) we obtain, for \(\eta \in (0,1]\) to be specified later, that
for a constant \(c=c(n,p,q,\nu ,L)\). In the preceding term we used the abbreviation
For the estimation of \(\mathrm I\) we use Hölder’s inequality and (6.19). This leads to
with \(c=c(n,p,q)\). We insert this above and choose \(\eta =(\frac{1}{2c})^{\frac{1}{p}}\). This allows to re-absorb the resulting term \(\frac{1}{2}\lambda ^{p}\) into the left-hand side. Then, we multiply the result by \(\big |Q_{4\varrho _{z_o}}^{(\lambda ,\theta _{\varrho _{z_o}})}(z_o)\big |\). This leads to the inequality
again with \(c=c(n,p,q,\nu ,L)\). Now, (6.19) for the choice \(s={\hat{c}}\varrho _{z_o}\) allows us to estimate \(\lambda ^{p}\) from below by
Inserting this above and keeping in mind that \({\hat{c}}\) depends only on n, p, and q, we deduce
To summarize: so far we have shown that for any value \(\lambda >B\lambda _o\) the associated super-level set \(\varvec{E}(R_1,\lambda )\) is covered by a family
of parabolic cylinders with center \(z_o\in \varvec{E}(R_1,\lambda )\) which are contained in \(Q_{R_2}\), and such that (6.21) holds true on each of these cylinders. At this point, we use the Vitali type covering property from Sect. 6.2 and obtain a countable subfamily
consisting of pairwise disjoint cylinders, such that the \(\frac{1}{4}{\hat{c}} \)-times enlarged cylinders \(Q_{{\hat{c}}\varrho _{z_i}}^{(\lambda ,\theta _{z_i;\varrho _{z_i}}^{(\lambda )})}(z_i)\) cover the super-level set \({\varvec{E}}(R_1,\lambda )\) and are still contained in \(Q_{R_2}\), i.e.
Since the \(Q_{4\varrho _{z_i}}^{(\lambda ,\theta _{z_i;\varrho _{z_i}}^{(\lambda )})}(z_i)\) are pairwise disjoint we obtain with (6.21) that
for a constant \(c=c(n, p, q, \nu , L)\). On \(\varvec{E}(R_1,\eta \lambda )\setminus {\varvec{E}}(R_1,\lambda )\) we have \(|Du|\le \lambda \) and therefore
We add the two preceding inequalities and obtain
In this inequality we replace \(\eta \lambda \) by \(\lambda \) and recall that \(\eta =\eta (n,p,q,\nu ,L)<1\). With this replacement we obtain for any \(\lambda \ge \eta B\lambda _o{=}{:}\lambda _1\) that
holds true with a constant \(c=c(n,p,q,\nu ,L)\). With this we have derived the desired reverse Hölder inequality on super-level sets.
6.6 Proof of the gradient estimate
In principle, the quantitative higher integrability estimate would follow from the reverse Hölder inequality on super-level sets by multiplying (6.22) by \(\lambda ^{\varepsilon p-1}\) and then integrating with respect to \(\lambda \). This procedure would lead to an integral of \(|Du|^{p(1+\varepsilon )}\) over \(Q_{R_1}\) on the left-hand side, while on the right-hand side the same integral appears with \(Q_{R_2}\) as domain of integration. However, it is not clear in advance that these integrals are finite for \(R\le R_1<R_2\le 2R\). For this reason one must avoid powers of |Du| that are larger than p. This can be achieved by a truncation argument. The rigorous argument is as follows: For \(k> \lambda _1\) the truncation of |Du| is defined by
and for \(r\in (0,2R]\) the corresponding super-level set by
With these definitions, we infer the following version of (6.22) for the truncated gradient.
This inequality is obviously true for values \(k\le \lambda \), since then \({\varvec{E}}_k(r,\lambda )=\emptyset \), while for values \(k>\lambda \) the claim immediately follows from (6.22), since in this case \({\varvec{E}}_k(r,\lambda )={\varvec{E}}(r,\lambda )\), and \(|Du|_k\le |Du|\) a.e. Now, we multiply (6.23) by \(\lambda ^{\varepsilon p-1}\), with a parameter \(\epsilon \in (0,1]\) that will be chosen later in a universal way. We integrate the result with respect to \(\lambda \) over the interval \((\lambda _1,\infty )\) and obtain
Here, we exchange the order of integration in the individual integrals by an application of Fubini’s theorem. For the integral on the left-hand side, Fubini’s theorem implies
while for the first integral on the right-hand side we find that
Finally, for the second integral on the right-hand side we obtain
We insert the preceding estimates above and multiply by \(\varepsilon p\). This leads to
The previous inequality is now combined with the corresponding inequality on the complement \(Q_{R_1}\setminus \varvec{E}_k(R_1,\lambda _1)\), i.e. with the inequality
Taking into account that \(|Du|_k\le |Du|\), the result is the estimate
where \(c_*=c_*(n,p,q,\nu ,L)\ge 1\). Now, we choose
Note that \(\varepsilon _o\) depends only on \(n,p,q,\nu \), and L. Furthermore, \(\lambda _1^\varepsilon \equiv (\eta B\lambda _o)^\varepsilon \le B \lambda _o^\varepsilon \), since \(B\ge 1\), \(\eta <1\), and \(\varepsilon \le 1\). Moreover, we recall the definition of B in (6.15). Therefore, the last inequality implies that for each pair of radii \(R_1\), \(R_2\) with \(R\le R_1<R_2\le 2R\) the following estimate holds:
Now, we apply the Iteration Lemma 2.1 to the last inequality and arrive at the estimate
In the left-hand side, we now pass to the limit \(k\rightarrow \infty \) with the help of Fatou’s lemma. Subsequently we take means on both sides. This gives
Recalling the definition of \(\lambda _o\) from (6.13) the preceding inequality turns into
Note that \(c=c(n,p,q,\nu ,L)\). For the estimate of the right-hand side, we use the energy estimate from Lemma 3.1 with \(a=0\) and \(\lambda =\theta =1\) in the form
Joining the two preceding estimates, we arrive at
with \(c=c(n,p,q,\nu ,L)\), provided \(Q_{8R}\subset \Omega _T\). This implies the asserted quantitative estimate (1.7) by a straightforward covering argument. If we choose \(\varepsilon \le \varepsilon _1{:}{=}\min \{\varepsilon _o,\frac{\sigma }{p}-1\}\), then the right-hand side is finite, and we deduce \(Du\in L_{\mathrm {loc}}^{(1+\varepsilon _1)p}(\Omega _T)\). This completes the proof of Theorem 1.2.
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Alonso, R., Santillana, M., Dawson, C.: On the diffusive wave approximation of the shallow water equations. Euro. J. Appl. Math. 19(5), 575–606 (2008)
Bamberger, A., Sorine, M., Yvon, J.P.: Analyse et contrôle d’un réseau de transport de gaz. (French) Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977), II, pp. 347–359, Lecture Notes in Phys., 91, Springer, Berlin-New York, 1979
Bögelein, V.: Higher integrability for weak solutions of higher order degenerate parabolic systems. Ann. Acad. Sci. Fenn. Math. 33(2), 387–412 (2008)
Bögelein, V., Duzaar, F.: Higher integrability for parabolic systems with non-standard growth and degenerate diffusions. Publ. Mat. 55(1), 201–250 (2011)
Bögelein, V., Duzaar, F., Kinnunen, J., Scheven, C.: Higher integrability for doubly nonlinear parabolic systems. J. Math. Pures Appl. (9) 143 (2020), 31–72
Bögelein, V., Duzaar, F., Korte, R., Scheven, C.: The higher integrability of weak solutions of porous medium systems. Adv. Nonlinear Anal. 8(1), 1004–1034 (2019)
Bögelein, V., Duzaar, F., Scheven, C.: Higher integrability for the singular porous medium system. J. Reine Angew. Math. 767, 203–230 (2020)
Bögelein, V., Parviainen, M.: Self-improving property of nonlinear higher order parabolic systems near the boundary. NoDEA Nonlinear Differ. Equ. Appl. 17(1), 21–54 (2010)
DiBenedetto, E., Friedman, A.: Hölder estimates for non-linear degenerate parabolic systems. J. Reine Angew. Math. (1985), 1–22
DiBenedetto, E.: Degenerate parabolic equations. Springer-Verlag, Universitytext xv, 387, New York, NY, 1993
Gianazza, U., Schwarzacher, S.: Self-improving property of degenerate parabolic equations of porous medium-type. Am. J. Math. 141(2), 399–446 (2019)
Gianazza, U., Schwarzacher, S.: Self-improving property of the fast diffusion equation. J. Funct. Anal. 277 (2019), no. 12, 108291, 57 pp
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)
Giaquinta, M., Struwe, M.: On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179(4), 437–451 (1982)
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Company, Tuck Link, Singapore (2003)
Kalashnikov, A.S.: Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. (Russian) Uspekhi Mat. Nauk 42 (1987), no. 2(254), 135–176, 287
Kinnunen, J., Lewis, J.L.: Higher integrability for parabolic systems of \(p\)-Laplacian type. Duke Math. J. 102(2), 253–271 (2000)
Leugering, G., Mophou, G.: (2018) Instantaneous Optimal Control of Friction Dominated Flow in a Gas-Network. In: Schulz V., Seck D. (eds) Shape Optimization, Homogenization and Optimal Control. International Series of Numerical Mathematics, vol 169. Birkhäuser, Cham
Mahaffy, M.W.: A three-dimensional numerical model of ice sheets: Tests on the Barnes ice cap, northwest territories. J. Geophys. Res 81(6), 1059–1066 (1976)
Meyers, N.G., Elcrat, A.: Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Math. J. 42, 121–136 (1975)
Parviainen, M.: Global gradient estimates for degenerate parabolic equations in nonsmooth domains. Ann. Mat. Pura Appl. (4), 188 (2009), no. 2, 333–358
Saari, O., Schwarzacher, S.: A reverse Hölder inequality for the gradient of solutions to Trudinger’s equation. NoDEA Nonlinear Differential Equations Appl. 29 (2022), no. 3, Paper No. 24, 23 pp
Schwarzacher, S.: Hölder-Zygmund estimates for degenerate parabolic systems. J. Differ. Eqs. 256(7), 2423–2448 (2014)
Singer, T., Vestberg, M.: Local boundedness of weak solutions to the diffusive wave approximation of the shallow water equations. J. Differ. Eqs. 266(6), 3014–3033 (2019)
Singer, T., Vestberg, M.: Local Hölder continuity of weak solutions to a diffusive shallow medium equation. Nonlinear Anal. 185, 306–335 (2019)
Sturm, S.: Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data. Manuscripta Math. 157(3–4), 295–322 (2018)
Urbano, J.M.: The method of intrinsic scaling. A systematic approach to regularity for degenerate and singular PDEs. Lecture Notes in Mathematics, 1930. Springer-Verlag, Berlin, 2008
Vázquez, J.L.: Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. Oxford Lecture Series in Mathematics and its Applications, 33. Oxford University Press, Oxford, 2006
Acknowledgements
V. Bögelein has been supported by the FWF-Project P31956-N32 “Doubly nonlinear evolution equations”. The second and third authors would like to thank the Department of Mathematics at Paris Lodron University Salzburg for their hospitality during the completion of the manuscript.
Funding
Open access funding provided by Austrian Science Fund (FWF).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest nor competing interests.
Additional information
This article is part of the section "Applications of PDEs" edited by Hyeonbae Kang.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bögelein, V., Duzaar, F. & Scheven, C. Higher integrability for doubly nonlinear parabolic systems. Partial Differ. Equ. Appl. 3, 74 (2022). https://doi.org/10.1007/s42985-022-00204-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s42985-022-00204-0