Higher integrability for doubly nonlinear parabolic systems

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>\frac{2n}{n+2}$$\end{document}p>2nn+2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0


INTRODUCTION
This paper studies regularity of the spatial gradient of weak solutions to doubly nonlinear parabolic equations (systems) of the type with 1 < p < ∞ in a space-time cylinder Ω T := Ω × (0, T ), where Ω ⊂ R n is a bounded domain, n ≥ 1, and T > 0. Equation (1.1) is a special case of the general doubly nonlinear parabolic equation with p > 1 and m > 0. This includes the parabolic p-Laplacian and the porous medium equation.Note that with the choice m = p − 1 we recover (1.1).Equation (1.2) has a different behavior when m < p − 1 and m ≥ p − 1.The first range is called the slow diffusion case, since disturbances propagate with a finite speed and free boundaries occur, while in the second range disturbances propagate with infinite speed and extinction in finite time may occur.This is called the fast diffusion case.In this sense, equation (1.1) represents the borderline case between the slow and fast diffusion ranges.One might expect that the regularity theory for the doubly nonlinear equation (1.1) is similar to the one for the heat equation.In fact, the equation is homogeneous, in the sense that solutions are invariant under multiplication by constants.In addition, a scale and location invariant parabolic Harnack's inequality holds true for non-negative weak solutions, see [23,15].However, in this case Harnack's inequality does not immediately imply Hölder continuity of solutions, which indicates that there is a difference compared to the heat equation.The main difficulty with (1.1) is that adding a constant to a solution destroys the property of being a solution.The general doubly nonlinear equation (1.2) is non-homogeneous and an intrinsic geometry is used in the regularity theory, i.e. the space-time scaling of cylinders depends either on the solution or the spatial gradient of the solution.The idea that the inhomogeneous behavior of a nonlinear parabolic equation can be compensated by an intrinsic geometry goes back to the pioneering work of DiBenedetto and Friedman, see for example the monograph [4].The regularity theory of weak solutions of (1.1) and (1.2) is reasonably developed, at least in the scalar case for non-negative solutions; see [23,10,15,5] for Harnack's inequality, [24,17,18] for Hölder regularity results, and finally [22] for Lipschitz regularity with respect to the spatial variable for solutions bounded from below by a positive constant.However, little is known about signed solutions, regularity of the gradient of a weak solution and systems.
The primary purpose of this paper is to establish a local higher integrability result for the spatial gradient of weak solutions to parabolic equations and systems of the type (1.1).We show that there exists a constant ε > 0, such that |Du| p(1+ε) ∈ L 1 loc (Ω T ), whenever u is a weak solution to the equation or the system.In particular, our result ensures that weak solutions of (1.1) belong to a slightly better Sobolev space than the natural energy space and therefore obey a self-improving property of integrability.Our result comes with a reverse Hölder type estimate, see Theorem 2.2.The higher integrability for the doubly nonlinear equation (1.1) has been an open problem for a long time.Here we give an answer to this question in the range max 2n n+2 , 1 < p < 2n (n−2)+ .This range may seem unexpected, but the lower bound also appears in the higher integrability for the parabolic p-Laplace system [16], while the upper bound is exactly the expected one for the porous medium system in the fast diffusion range.For n = 1 and n = 2 our result applies whenever 1 < p < ∞.It remains an open question whether the corresponding result holds true when n ≥ 3.
The key ingredient in the proof of our main result is a suitable intrinsic geometry.By now, variants of this idea have been successfully used in establishing the higher integrability for the parabolic p-Laplace system [16] and very recently for the porous medium equation [9] and system [2].Our idea is to consider space-time cylinders Q r,s (z o ) := B r (x o ) × (t o − s, t o + s), with z o = (x o , t o ), such that the quotient s r p satisfies This geometry involves the solution as well as its spatial gradient and therefore allows to balance the mismatch between |u| and |Du| in the equation.To our knowlegde this is the first time that such a geometry is used.On these cylinders we are able to prove Sobolev-Poincaré and reverse Hölder type inqualites.The construction of the cylinders is quite involved, since the cylinders on the right-hand side of (1.3) also depend on the parameter µ.In the course of the construction we modify the argument in [9]; see also [2].
In the stationary elliptic case the higher integrability was first observed by Elcrat & Meyers [19], see also the monographs [11,Chapter 11, Theorem 1.2] and [13,Section 6.5].The first higher integrability result, in the context of parabolic systems, can be found in [12,Theorem 2.1].The higher integrability for the gradient of solutions for general parabolic systems with p-growth has been established by Kinnunen & Lewis [16].This local interior result has been generalized in the meantime in various directions, e.g.global results, higher order parabolic systems (interior and at the boundary); see [20,1,3].For the porous medium equation, i.e. equation (1.2) with p = 2, the question of higher integrability turned out to be more challenging than for the parabolic p-Laplace equation, i.e. equation (1.2) with m = 1.The problem was solved only recently by Gianazza & Schwarzacher [9].They proved that non-negative weak solutions to the porous medium equation possess the higher integrability for the spatial gradient.Their proof, however, uses the method of expansion of positivity and therefore cannot be extended to signed solutions and porous medium type systems.A simpler and more flexible proof, which does not rely on the expansion of positivity, is given in [2], where higher integrability for porous medium type systems is achieved.As special case, signed solutions are included in this result.

NOTATION AND THE MAIN RESULT
2.1.Notation.Throughout the paper we use space-time cylinders of the form (2.1) with center z o = (x o , t o ) ∈ R n × R, radius ̺ > 0 and scaling parameter µ > 0, where Note that in both cases the cylinders (2.1) admit the scaling property (1.3) 1 .Moreover, they satisfy the inclusion In the case that µ = 1, we omit the scaling parameter in our notation and instead of Q xo;̺ (t) for µ = 1.Similarly, for a given measurable set E ⊂ Ω T of positive Lebesgue measure the mean value (u) E ∈ R N of u on E is defined by ̺ (zo) .Moreover, we often write u(t) := u(•, t) for notational convenience.For the power of a vector u ∈ R N , we use the short-hand notation u α := |u| α−1 u, for α > 0, which we interpret as u α = 0 in the case u = 0 and α ∈ (0, 1).Finally, we let p := max{p, 2}.

2.2.
Assumptions and the main result.We consider general systems of the type where the vector-field is a Carathéodory function satisfying the standard p-growth and coercivity conditions for a.e.z = (x, t) ∈ Ω T and any (u, ξ) ∈ R N × R N n , where 0 < ν ≤ L < ∞ are positive constants.In order to formulate our main result, we need to introduce the concept of weak solution.
Definition 2.1.Assume that the vector field is a weak solution to the doubly non-linear parabolic system (2.2) if and only if the identity The following theorem is our main result.
, where the right-hand side is interpreted as ∞ for the dimensions n = 1 and n = 2, and assume that σ > p.Then, there exists where ε 1 := min ε o , σ p − 1 .Moreover, for every ε ∈ (0, ε 1 ] and every cylinder where c = c(n, p, ν, L). ✷ Although Theorem 2.2 is proved for exponents p in the range (2.5), we indicate in each sub-step of the proof what are the exact restrictions on p that are needed in the particular step.In this way, the reader can easily retrace where restriction (2.5) occurs.
Lemma 3.3.For any α ≥ 1, there exists a constant c = c(α) such that, for all a, b ∈ R N , N ∈ N, we have The next lemma provides useful estimates for the boundary term Lemma 3.4.For any p ≥ 1 there exists a constant c = c(p) such that for any u, v ∈ R N , N ∈ N, we have Proof.The case 1 < p ≤ 2 follows from [2, Lemma 2.3 (i)] applied with m = 1 p−1 .Therefore it remains to consider the case p > 2. In the following we denote φ p (u) := p−1 < 2 we may apply Lemma 3.4 in the subquadratic case.In this way we obtain This finishes the proof of the lemma.
It is well known that mean values over subsets A ⊂ B are quasi-minimizers in the integral a → ´B |u − a| p dx.The following statement shows that mean values over subsets are still quasi-minimizing for u α with α ≥ 1 p .For p = 2 and A = B, the lemma has been proved in [6,Lemma 6.2]; see also [2,Lemma 2.6].Here, we state a general version for powers.As expected, the quasi-minimality constant depends on the ratio of the measures of the set and the subset.Lemma 3.5.Let p ≥ 1 and α ≥ 1 p .Then, there exists a constant c = c(α, p) such that whenever A ⊆ B ⊂ R k , k ∈ N, are two bounded domains of positive measure, then for any function u ∈ L αp (B, R N ) and any constant a ∈ R N , we have Proof.The key step in the proof is the estimate of the difference |(u) α A − a α |.In the case α ≥ 1, we use Lemmas 3.2 and 3.3 in order to obtain for a constant c = c(α, p) that Our next goal is to derive the same bound in the case 1 p ≤ α < 1.We begin by applying Lemma 3.2 to obtain We use this and the fact Now we join the two cases.In view of (3.4) and (3.5), the estimate (3.3) yields the bound We multiply this inequality by , apply Young's inequality with exponents 1  1−α , 1 α to the first term on the right-hand side, and Hölder's inequality with exponents αp, αp αp−1 to the second term.Note that both is possible in the case 1 p < α < 1, while in the case α = 1 p the application of Hölder's inequality is not necessary.This procedure results in the estimate .
The second-last term can be re-absorbed into the left-hand side, which leads us to This is the estimate (3.2) now also for the case 1 p ≤ α < 1.In any case, we can apply either (3.2) or (3.6) to conclude which proves the claim.
Finally, we state Gagliardo-Nirenberg's inequality in the form we will use in the sequel.
Lemma 3.6.Let 1 ≤ p, q, r < ∞ and θ ∈ (0, 1) such that Then there exists a constant c = c(n, p) such that for any ball B ̺ (x o ) ⊂ R n with ̺ > 0 and any function u ∈ W 1,q (B ̺ (x o )), we have

ENERGY BOUNDS
In this section we exploit the doubly nonlinear system (2.2) in order to deduce an energy estimate and a gluing lemma.These are the only points in the proof where the fact that u is a solution of (2.2) is used.Proof.For v ∈ L 1 (Ω T , R N ), we define the following mollification in time From the weak form (2.4) of the differential equation we deduce the mollified version Furthermore, for ε > 0 small enough and t 1 ∈ Λ s (t o ) we define the function In (4.1) we choose the testing function In the following we abbreviate w p−1 := u p−1 h and omit in the notation the reference to the center z o = (x o , t o ).For the integral in (4.1) containing the time derivative we compute ¨ΩT where we used the identity ∂ t w p−1 = − 1 h (w p−1 − u p−1 ) and recall the definition of the boundary term b in (3.1).Since u p−1 h → u p−1 in L p p−1 (Ω T ) we can pass to the limit h ↓ 0 in the integral on the right-hand side.We therefore get We now pass to the limit ε ↓ 0. For the term I ε we obtain for any t 1 ∈ Λ s that Taking into account that the boundary term b[u, a] is non-negative, the term II ε can be estimated independently from ε, since Next, we consider the diffusion term.After passing to the limit h ↓ 0, we use the ellipticity and growth assumption (2.3) for the vector-field A, and subsequently Young's inequality.
In this way, we obtain Next, we consider the right-hand side term involving the inhomogeneity F .With the help of Young's inequality we find that Finally, for the last integral in (4.1), the convergence of the mollifications and the fact ϕ(0) = 0 imply Combining the preceding results and passing to the limit ε ↓ 0 we obtain for almost every for a constant c = c(p, ν, L).Here we pass to the supremum over t 1 ∈ Λ s in the first term on the left-hand side.In the second one we let t 1 ↑ t o + s.Finally we take mean values on both sides and apply Lemma 3.4 twice.This leads to the claimed energy estimate.
Next, we deduce a gluing lemma for the doubly nonlinear system.
Lemma 4.2.Let p > 1 and u be a weak solution to (2.2) in Ω T in the sense of Definition 2.1.Then, on any cylinder and a radial function where For fixed i ∈ {1, . . ., N } we choose ϕ ε,δ = ξ ε ψ δ e i as testing function in the weak formulation (2.4), where e i denotes the i-th canonical basis vector in R N .In the limit ε, δ ↓ 0 we obtain ˆBr(xo) Multiplying the preceding inequality by e i and summing over i = 1, . . ., N yields ˆBr(xo) Due to growth condition (2.3) 2 we get for any In the above inequality, we choose r = r and then take means on both sides of the resulting estimate.This implies ) and with a constant c = c(L).

PARABOLIC SOBOLEV-POINCAR É TYPE INEQUALITIES
One of the difficulties in the parabolic setting is that weak solutions are not necessarily differentiable with respect to time.As a consequence, the Sobolev-Poincaré inequality on R n+1 is not applicable.Since such an inequality is indispensable in the proof of the higher integrability we will derive some type of Poincaré and Sobolev-Poincaré inequality which is valid for weak solutions.The idea is to use the Gluing Lemma 4.2 in order to manage the lack of differentiability with respect to time.
Throughout this section we consider scaled cylinders Q ̺ (z o ) ⊆ Ω T as defined in (2.1) on which certain intrinsic, respectively sub-intrinsic couplings with respect to u and its spatial gradient Du hold true.For ̺, µ > 0 we assume that (5.1) holds true for a constant K ≥ 1. Recall that p = max{2, p}.Such cylinders are termed µ-sub-intrinsic.Furthermore, we assume that either Finally, a cylinder which is µ-suband µ-super-intrinsic is called µ-intrinsic.In the following we distinguish the cases whether the growth exponent p is sub-or superquadratic.In order to emphasize the stability of the proof when p → 2, we include the quadratic case p = 2 in both subsections.

The case max{
As a first preliminary result, we compare the first and the second term on the right-hand side of the energy inequality in Lemma 4.1.It turns out that for p ∈ (1, 2] on µ-sub-intrinsic cylinders the second term can easily be bounded in terms of the first one.Lemma 5.1.Let 1 < p ≤ 2 and u be a weak solution to (2.2) in Ω T in the sense of Definition 2.1.Then, on any cylinder where c = c(p, K).
Proof.For simplicity in notation, we omit the reference point z o .Due to Lemma 3.2 applied with α = 2 p and Hölder's inequality, we obtain .
For the last integral, hypothesis (5.1) yields Inserting this above proves the claim with a constant c depending only on p and K.
The next lemma should be interpreted as a parabolic Poincaré inequality for solutions on µ-sub-intrinsic cylinders.The fact that weak solutions do not necessarily possess a weak time derivative is compensated by the Gluing Lemma 4.2.However, the gluing lemma provides an estimate for time differences of slice-wise means of u p−1 rather than u.Therefore, mean values of u p−1 and u have to be estimated very carefully against each other.Lemma 5.2.Let 1 < p ≤ 2 and u be a weak solution to (2.2) in Ω T in the sense of Definition 2.1.Then, on any cylinder , holds true for any q ∈ [1, p] and a constant c = c(n, p, L, K).
Proof.In the proof we renounce again to consider the center z o in the notation.With ̺ ∈ [ ̺ 2 , ̺) we denote the radius from Lemma 4.2.We start by estimating the left-hand side with the help of the quasi-minimality of the mean value as follows where we abbreviated Next, we treat the terms I and II of the right-hand side.For the term I we first recall that Therefore, the application of Lemma 3.5 with α = 1 p−1 ≥ 1 q and subsequently Poincaré's inequality leads to for a constant c = c(n, p).Note that q ∈ [1, p] and the constant in Poincaré's inequality depends continuously on q.Now we will treat II.An application of Lemma 3.2 with α = 1 p−1 ≥ 1 and subsequently Hölder's inequality yields , for a constant c(p).We continue estimating the right-hand side with the help of the Gluing Lemma 4.2, the µ-sub-intrinsic coupling (5.1) and Hölder's inequality.In this way we find , for a constant c(p, L, K).Joining the preceding estimates for I and II finally proves the claim.
Our next aim is to derive a Sobolev-Poincaré type inequality.It has to be understood in the following way.Lemma 5.1 allows to bound the second term on the right-hand side of the energy inequality in terms of the first one.Therefore, in our Sobolev-Poincaré type inequality we will derive an upper bound for this term.In this bound we would like to have the integral of |Du| q for some q < p on the right-hand side.However, due to the nonhomogeneous behavior of the underlying differential equation some extra terms show up.Fortunately they have exactly the form of the left-hand side of the energy estimate so that they can be re-absorbed later on.Note that the estimate of the term II 2 in the proof of Lemma 5.3 is the only point in the paper where the condition p > 2n n+2 is needed.
Then, we obtain In the following, it remains to consider the second term on the right-hand side.For the estimate of µ we use hypothesis (5.2).If (5.2) 1 is satisfied, we first apply Lemma 5.2 with q = p to obtain Together with the µ-super-intrinsic coupling (5.2) 1 this yields with a constant c depending on n, p, L and K. On the other hand, if (5.2) 2 is satisfied, then (5.3) holds true with c = K 1/p .Consequently, we have inequality (5.3) in any case and therefore obtain where and For the estimate of II 1 we apply Lemma 3.3 with α = 2 p , Sobolev's inequality and Lemma 5.2.In this way we find , where c = c(n, p, L, K).Now we turn our attention to the second term.With the help of Lemma 3.2 applied with α = 2 p and Sobolev's inequality, we find that for a constant c = c(n, p).The term involving |u − a| q is now treated as above with Lemma 5.2, so that holds true with a constant c depending only on n, p, L, and K. Inserting the preceding estimates above and applying Young's inequality, we derive the desired inequality.
5.2.The case p ≥ 2 p ≥ 2 p ≥ 2. Now, we turn our attention to the superquadratic case p ≥ 2. We emphasize that all results of this section hold true for the full range p ≥ 2. The restriction p < 2n (n−2)+ will be necessary later on in the covering argument.Contrary to the subquadratic case in Lemma 5.1, we find in the superquadratic case a straight-forward bound on µ-sub-intrinsic cylinders for the first term on the right-hand side of the energy inequality in Lemma 4.1 in terms of the second one.
Lemma 5.4.Let p ≥ 2 and u be a weak solution to (2.2) in Ω T in the sense of Definition 2.1.Then, on any cylinder Proof.As before, we omit the reference point z o in our notation.Applying Lemma 3.2 with α = p 2 , Hölder's inequality, and finally hypothesis (5.1), we obtain , which proves the claim.
The next lemma is the analogue of Lemma 5.2 for the superquadratic case and should be interpreted as a parabolic Poincaré type inequality.
Proof.Throughout the proof we omit the reference to the center z o in our notation.Similar to the proof of Lemma 5.2, we find that where we abbreviated ) denoting the radius from the Gluing Lemma 4.2.Recall the abbreviation u (µ) for the slice-wise mean.For the estimate of I we in turn apply Lemma 3.5 with α = 1 p−1 ≥ 1 q and ̺ ∈ [ ̺ 2 , ̺] and subsequently Poincaré's inequality.This leads to for a constant c = c(n, p).Now we will treat II.We start with an application of Lemma 3.3 with α = p−1 ≥ 1 and subsequently the Gluing Lemma 4.2 with R = µ 2−p p ̺ and S = ̺ p .This gives If hypothesis (5.2) 2 is satisfied, the lemma is proven.Therefore, it remains to consider the case where (5.2) 1 is in force.Here, we apply Lemma 3.5 to deduce that Having arrived at this point, we take the last inequality to the power q(p − 2) and obtain with the abbreviations and For the estimate of II 2 , we proceed as follows.We first insert the definition of II, then use Lemma 3.2 with α := p − 1 ≥ 1 and finally apply the Gluing Lemma 4.2.This leads to With Hölder's inequality and hypothesis (5.2) 1 , we finally obtain Inserting the preceding estimates above, we have shown that For the estimate of II 1 , we use (5.4) and hypothesis (5.2) 1 to obtain . (5.6) In the case q = p, we use Young's inequality with exponents p−1 p−2 and p − 1 and obtain Inserting this into (5.5) and reabsorbing the first term of the right-hand side into the left yields the desired Poincaré type inequality in the case q = p.At this point it remains to consider the case q ∈ [p − 1, p).Here, we use in (5.6) the Poincaré type inequality for q = p to conclude Together with (5.5) this finishes the proof in the remaining case q ∈ [p − 1, p).
As final result of this section we derive a Sobolev-Poincaré type inequality, which should be seen as the analogue of Lemma 5.3 for the superquadratic case.Lemma 5.6.Let p ≥ 2 and u be a weak solution to (2.2) in Ω T in the sense of Definition 2.1.Then, on any cylinder Q (µ) ̺ (z o ) ⊆ Ω T with ̺, µ > 0 satisfying (5.1) and (5.2) and for any ε ∈ (0, 1], we have with q = max{ np n+2 , p−1} and c = c(n, p, L, K).Proof.As before, we omit the reference point z o in our notation.Moreover, we abbreviate a := (u) (µ) ̺ .Applying Gagliardo-Nirenberg's inequality in Lemma 3.6 with (p, q, r, θ) replaced by (p, q, 2, q p ) and Lemma 5.5, we find that We now exploit assumption (5.2) in order to obtain an upper bound for µ.If (5.2) 2 is satisfied we have µ ≤ K 1/p .On the other hand, if (5.2) 1 is in force we apply Lemma 5.5 to infer that which in combination with the µ-super-intrinsic coupling (5.2) 1 yields .
This shows that holds true in any case.Inserting this upper bound for µ into (5.7)yields with the obvious abbreviations and .
For the first term, we use Hölder's inequality and Lemma 3.3 to infer that with a constant c = c(n, p, L, K).Now we turn our attention to the second term.With the help of Lemma 3.2 applied with α = p 2 and Hölder's inequality, we find that , for a constant c = c(n, p).We add the resulting inequalities for I 1 and I 2 and apply Young's inequality.This yields the desired result.

REVERSE H ÖLDER INEQUALITY
Our aim in this section is to derive a reverse Hölder type inequality for weak solutions of (2.2).It will be a consequence of the energy estimate in Lemma 4.1 and the Sobolev-Poincaré type inequality in Lemma 5.3, respectively Lemma 5.6.
In contrast to Section 5 we now consider two concentric cylinders We suppose that a µ-sub-intrinsic coupling of the type (6.1) is satisfied for some K ≥ 1.Furthermore, we assume that either (6.2) Proof.Once again, we omit the reference to the center z o in the notation.We consider radii r, s with ̺ ≤ r < s ≤ 2̺ and let (6.3) R r,s := s s − r .
Note that hypothesis (6.1) and ( 6.2) imply that the coupling conditions (5.1) and ( 5.2) are satisfied on with the constant 2 2n+3p K in place of K. From now on we distinguish between the cases max{ 2n n+2 , 1} < p ≤ 2 and p ≥ 2. The case max{ 2n n+2 , 1} < p ≤ 2.Here the energy estimate from Lemma 4.1 reads as with the obvious meaning of I-III.The constant c depends only on p, ν, and L. We estimate II with the help of Lemma 3.5, Lemma 5.1 and Young's inequality with the result that µ p−2 s p dxdt holds true for any δ ∈ (0, 1].Taking into account that (s − r) p ≤ s p − r p , we obtain due to Lemma 3.5 that µ p−2 s p dxdt.
We add both inequalities and apply Lemma 5.3 on In this way we obtain where q = max{ 2n n+2 , 1}.We insert this inequality into (6.4).Then, we choose and apply the Iteration Lemma 3.1 to re-absorb the term 1 2 [. . .] from the right-hand side into the left.This leads to the claimed reverse Hölder type inequality, i.e. to and finishes the proof of Proposition 6.1 in the case max{ 2n n+2 , 1} < p ≤ 2. The case p ≥ 2. In this case, the energy estimate from Lemma 4.1 yields with the obvious meaning of I-III.Now, we estimate the term I by using the fact that (s − r) p ≤ s p − r p , Lemma 3.5, Lemma 5.4 and Young's inequality.In this way we obtain for any δ ∈ (0, 1].Moreover, from Lemma 3.5 we know that We combine the preceding estimates and apply Lemma 5.6 with ε = δ p 2 in order to obtain where q = max{ np n+2 , p−1}.As before, we insert this inequality into (6.6),choose δ ∈ (0, 1] of the form (6.5) and apply the Iteration Lemma 3.1.This allows to re-absorb the term 1 2 [. . .] into the left-hand side and yields the desired reverse Hölder type inequality in the remaining case p ≥ 2. This finishes the proof of the proposition.

HIGHER INTEGRABILITY: PROOF OF THEOREM 2.2
In this section we finally prove the higher integrability result of Theorem 2.2.We consider a fixed cylinder .
7.1.Construction of a non-uniform system of cylinders.The main difficulty now is to construct a covering of the λ-superlevel set of |Du| by cylinders on which the reverse Hölder type inequality from Proposition 6.1 is applicable.This means that the scaled cylinders have to satisfy hypothesis (6.1) and (6.2).The following construction of a nonuniform system of cylinders is inspired by the one in [21,9].Let z o ∈ Q 2R .For a radius ̺ ∈ (0, R] we now define In particular, the restriction p < 2n n−2 for n > 2 ensures that p − β > 0 in any case.However, we note that in dimensions n > 2, the exponent of µ tends to zero in the limit p ↑ 2n n−2 .This is the only point where the restriction p < 2n n−2 enters the proof.If z o and λ are fixed and if the meaning is clear from the context we write µ ̺ instead of µ (λ) zo;̺ .Observe that the set of those µ ≥ 1 for which the condition in the infimum is satisfied is not empty.In fact, in the limit µ ↑ ∞ the integral on the left-hand side converges to zero (note that the measure of Q (µ) ̺ (z o ) shrinks to 0), while the right-hand side blows up with speed µ p−β (recall that p − β > 0).We point out that the condition in the infimum is equivalent to Therefore, we either have Using this observation for ̺ = R, we have that either µ R = 1, or µ R > 1 and Therefore, in any case we have the bound Our next aim is to ensure that the mapping (0, R] ∋ ̺ → µ ̺ is continuous.To this end, we consider ̺ ∈ (0, R] and ε > 0, and define µ + := µ ̺ +ε.Then, there exists δ = δ(ε, ̺) > 0 such that for any r ∈ (0, R] with |r − ̺| < δ.In fact, due to the definition of µ ̺ the preceding strict inequality holds for r = ̺, since µ + > µ ̺ and Q The claim now follows, since the left-hand side depends continuously on the radius r.Recalling the very definition of µ r , the last inequality implies µ r ≤ µ Due to the continuity of the left-hand side with respect to r, this implies the claim for r with |r − ̺| < δ small enough.The preceding inequality implies that µ r ≥ µ − := µ ̺ − ε.This completes the proof of the continuity of (0, R] ∋ ̺ → µ ̺ .
Unfortunately, the mapping ̺ → µ ̺ might not be monotone.For this reason we modify µ ̺ in such a way that the modification -denoted by µ ̺ -becomes monotone.Therefore, we define As before, we abbreviate µ (λ) zo;̺ by µ ̺ if z o and λ are fixed, so that no confusion is possible.By construction the mapping (0, R] ∋ ̺ → µ ̺ is continuous and monotonically decreasing.Moreover, the cylinders Q (µ̺) s (z o ) are µ-sub-intrinsic (with constant K = 1) whenever ̺ ≤ s.More precisely, we have1 In fact, the definition of µ s and the monotonicity of µ ̺ imply µ s ≤ µ s ≤ µ ̺ , so that In the last step we used the fact p − β > 0. We now define (7.5) Note that µ s = µ ̺ for any s ∈ [̺, ̺] and in particular µ ̺ = µ ̺ .Next, we claim that If µ ̺ = 1, then also µ s = 1, so that (7.6) trivially holds.Therefore, it remains to consider the case µ ̺ > 1.If s ∈ (̺, ̺], then µ ̺ = µ s , and (7.6) obviously holds true.Otherwise, if s ∈ ( ̺, R], then (7.2), the monotonicity of s → µ s and (7.4) imply This proves the claim (7.6).We now apply (7.6) with s = R. Since µ R = µ R , the bound (7.3) for µ R yields In the following, we consider the system of concentric cylinders Q (µ (λ)  zo;̺ ) ̺ (z o ) with radii ̺ ∈ (0, R] and z o ∈ Q 2R .The cylinders are nested, in the sense that The inclusions hold true due to the monotonicity of the mapping ̺ → µ (λ) zo;̺ and the fact that µ zo;̺ is that the associated cylinders are in general only µ-sub-intrinsic with K = 1, but not µ-intrinsic.

7.2.
Covering property.The system of cylinders Q (µ (λ)  zo;r ) r (z o ) constructed above satisfies a Vitali type covering property.This will be proven in the following lemma.Lemma 7.1.There exists a constant ĉ = ĉ(n, p) ≥ 20 such that whenever λ ≥ λ o and F is any collection of cylinders Q (µ (λ)  z;r ) 4r (z), where Q (µ (λ)  z;r ) r (z) is a cylinder of the form as constructed in Section 7.1 with radius r ∈ (0, R ĉ ], then there exists a countable subfamily G of disjoint cylinders in F such that Proof.Throughout the proof we abbreviate µ z;r := µ (λ) z;r .We let ĉ ≥ 20 be a parameter that will be chosen later.For j ∈ N we define and select G j ⊂ F j by the following procedure: We choose G 1 to be any maximal disjoint collection of cylinders in F 1 .Note that G 1 contains only finitely many cylinders, since by the definition of F 1 and (7.7) the L n+1 -measure of each cylinder Q ∈ G 1 is uniformly bounded from below.Now, assume that for some k ∈ N ≥2 the collections G 1 , G 2 , . . ., G k−1 have already been inductively selected.Then, we choose a maximal disjoint sub-collection of cylinders from F k which do not intersect any of the cylinders Q * from one of the collections G j , j ∈ {1, . . ., k − 1}.More precisely, we choose a maximal disjoint collection of cylinders in Note again that G k is finite.Finally, we let By construction, G ⊂ F is a countable subfamily of disjoint cylinders in F .At this point it remains to prove that for each Q ∈ F there exists a cylinder To this aim we consider some arbitrary cylinder Q = Q (µz;r) 4r (z) ∈ F .Then, there exists an index j ∈ N such that Q ∈ F j .The maximality of G j ensures that there exists a cylinder Then, we have r < 2r * , since r ≤ R 2 j−1 ĉ and r * > R 2 j ĉ .The main difficulty now is to establish a bound for µ z * ;r * in terms of µ z;r .We claim that the following estimate holds true: and proves (7.9) in this case.Therefore it remains to consider radii r * ≤ R η .Note that we can assume µ z;r ≤ µ z * ;r * .Otherwise (7.9) trivially holds.Therefore, the monotonicity of ̺ → µ z;̺ and the fact that r ≤ 2r * ≤ 2 r * ≤ η r * imply (7.11) µ z;η r * ≤ µ z;r ≤ µ z * ;r * .
Next, we claim that (7.12) For the proof of (7.12) a distinction must be made between the cases p ≤ 2 and p ≥ 2. We first consider exponents max{  This establishes the claim (7.12) also for the remaining case 2 ≤ p < 2n (n−2)+ .Now we can finish the proof of (7.9).Due to (7.10), (7.12), (7.4) applied with ̺ = s = ηr * , and (7.11), we obtain This finishes the proof of (7.9).It remains to show the inclusion  (7.13).In any case we have thus established the claim (7.8).This completes the proof of the Vitali type covering property.Here, we mean Lebesgue points of |Du| with respect to the system of cylinders constructed in Section 7.1.
On the other hand, due to (7.15) we find a sufficiently small radius 0 < s < R2−R1 ĉ such that the integral in (7.15) possesses a value larger than λ p .By the continuity of ̺ → µ ̺ and the absolute continuity of the integral, there exists a maximal radius 0 < ̺ zo < R2−R1 Moreover, due to the monotonicity of ̺ → µ ̺ and (7.6) we have 7.4.A Reverse Hölder Inequality.As before, we consider z o ∈ E(R 1 , λ) with λ as in (7.16).Since λ and z o are fixed, we once again use the abbreviation µ ̺z o := µ (λ) zo;̺z o .We keep in mind that by construction 0 < ̺ zo < R2−R1 ĉ .According to (7.5) we construct ̺ zo ∈ [̺ zo , R] and recall that, at least in the case ̺ zo < R, the cylinder Q   n, p, L).After re-absorbing 1  2 µ ̺z o λ into the left-hand side, we find that µ ̺z o ≤ c(n, p, L).This ensures that (6.2) 2 is satisfied with K = c(n, p, L).Therefore, we are allowed to apply Proposition 6.1 on the cylinder Q  |Du| q dxdt p q −1 .
In view of Hölder's inequality and (7.19) we find that We insert this inequality above.Then, we choose η = ( 1 2c ) 1 p and re-absorb 1 2 λ p into the left-hand side.Multiplying the result by Q (z i ) cover the super-level set E(R 1 , λ) and are still contained in Q R2 .More precisely, we have Since the cylinders Q (µz i ;̺z i ) 4̺z i (z i ) are pairwise disjoint we obtain with (7.holds true with a constant c = c(n, p, ν, L).This is the reverse Hölder inequality on super-level sets we are looking for.7.6.Proof of the gradient estimate.At this point the quantitative higher integrability estimate follows in a standard way from the reverse Hölder inequality on super-level sets by multiplying (7.23) by λ εp−1 and then integrating with respect to λ.For the sake of completeness we nevertheless provide the details.The just described procedure would lead on the left to an integral of |Du| p(1+ε) on Q R1 , while on the right the same integral appears with factor 1  2 and Q R2 as domain of integration.If both integrals are finite the one on the right could be re-absorbed in view of Lemma 3.1.However, it is not clear in advance that these integrals are finite.For this reason we use a truncation argument in order to avoid powers of |Du| that are larger than p.The rigorous argument is as follows: For k > λ 1 we define the truncation of |Du| by |Du| k := min{|Du|, k}, and for r ∈ (0, 2R] the corresponding super-level set by Note that |Du| k ≤ |Du| a.e., as well as E k (r, λ) = ∅ for k ≤ λ and E k (r, λ) = E(r, λ) for k > λ.Therefore, (7.23) implies ¨Ek (R1,λ) |Du| p−q k |Du| q dxdt ≤ c ¨Ek (R2,λ) λ p−q |Du| q dxdt + c ¨F (R2,λ) |F | p dxdt.
The idea now is to exchange the order of integration in each of the integrals by an application of Fubini's theorem.For the integral on the left-hand side Fubini's theorem shows Note that c = c(n, p, ν, L).A straightforward covering argument now yields the claimed quantitative estimate.This completes the proof of Theorem 2.2.

1 p
|u| p .With the abbreviations a = u p−1 and b

Lemma 4 . 1 .
Let p > 1 and u be a weak solution to (2.2) in Ω T in the sense of Definition 2.1.Then, on any cylinder Q R,S (z o ) := B R (x o ) × Λ S (t o ) ⊆ Ω T with R, S > 0, and for all r ∈ [R/2, R), s ∈ [S/2 p , S) and a ∈ R N , we have sup p (R − r) p + |F | p dxdt, where c = c(p, ν, L).