On the Calderón-Zygmund Theory for Nonlinear Parabolic Problems with Nonstandard Growth Condition

We prove Calderón-Zygmund estimates for a class of parabolic problems whose model is the non-homogeneous parabolic p(x, t)-Laplacian equation ∂tu − div ( |Du|p(x,t)−2Du ) = f − div ( |F|p(x,t)−2F ) . More precisely, we will show that the spatial gradient Du is as integrable as the inhomogeneities f and F, i.e. |F|p(x,t), | f | γ1 γ1−1 ∈ L loc ⇒ |F| p(x,t) ∈ L loc for any q > 1, where γ1 is the lower bound for p(x, t). Moreover, it is possible to use this approach to establish the CalderónZygmund theory for parabolic obstacle problems with p(x, t)-growth.


Introduction
In this manuscript, we consider nonlinear parabolic equations of the type where Ω ⊆ R n is a bounded domain of dimension n ≥ 2 and T > 0 is the height of the space-time cylinder Ω T .Moreover, ∂ t u denotes the partial derivative with respect to the time variable t, while Du denotes the one with respect to the space variable x.Furthermore, we write z := (x, t) for points in Ω T and ∂ P Ω T = ( Ω × {0}) ∪ (∂Ω × (0, T )) for the parabolic boundary of Ω T .The vector-field a(z, Du) satisfies certain nonstandard p(x, t)-growth and ellipticity conditions which we will note above.First of all, we want to mention the aim of this paper and the importance of nonstandard growth problems.The goal of this paper is to establish local Calderón-Zygmund estimates for Du of solutions to the parabolic problem (1), since these estimates imply that Du is as integrable as the inhomogeneities f and F, i.e.
|F| p(x,t) , x,t) ∈ L q loc for any q > 1, where γ 1 is the lower bound for p(x, t).Moreover, we want to mention that the approach we use here, could also be utilized to establish the Calderón-Zygmund theory for parabolic obstacle problems related to (1).Notice that in (Erhardt, 2014) the Calderón-Zygmund estimates for parabolic obstacle problem related to the model case with a(z, Du) = a(z)|Du| p(z)−2 Du, where µ ≤ a(z) ≤ L satisfies a certain VMO condition, has been already proved.But it is possible to combine the approach of this paper with the one in (Erhardt, 2014) to gain the Calderón-Zygmund estimates for parabolic obstacle problems with irregular obstacles related to the parabolic problem (1).

Physical Motivation
The motivation of considering parabolic partial differential equations is based on the fact that evolutionary equations and systems can be used to model physical processes, e.g.heat conduction or diffusion processes.The Navier-Stokes equations, the basic equations of fluid mechanics, is one important example.Also parabolic obstacle problems are motivated by numerous applications, e.g. in mathematical physics, mechanics, control theory or in mathematical biology.For more details and a good overview, we refer to the monograph (Scheven, 2011).Moreover, we want to highlight the importance on nonstandard growth in applications.The following system of a Navier-Stokes equation describes electro-rheological fluids where E(u) is the symmetric part of the gradient Du, π denotes the pressure and the variable growth exponent p(x) is a Hölder continuous function.Such fluids are of high technological interest because of their ability to change the mechanical properties under the influence of exterior electro-magnetic field, cf.(Ettwein & Růǔička, 2003), (Růǔička, 2000).For example, many electro-rheological fluids are suspensions consisting of solid particles and a carrier oil.These suspensions change their material properties dramatically if they are exposed to an electric field, see (Růǔička, 2004).Most of the known results concern the stationary models, see for example (Acerbi & Mingione, 2001& 2002a,b).Moreover, the non-stationary case, i.e. the model depending on variable exponents p(x, t) has been studied in (Acerbi, Mingione & Seregin, 2004).Here, we are going to study similar parabolic problems with p(x, t)-growth and we will establish a further regularity result.Other applications are the models for flows in porous media, see (Antontsev & Shmarev, 2005).

Historical Background -a Short Overview
The history of the Calderón-Zygmund theory for nonlinear problems starts in the elliptic case.The result for the stationary p-Laplacian equation was established in (Iwaniec, 1983), while the result for the p-Laplacian system was proved in (DiBenedetto & Manfredi, 1993).The fact, that for elliptic equations with VMO coefficients, the Calderón-Zygmund estimates for the gradient on L q -level for q > p are valid, was shown in (Kinnunen & Zhou, 1999).Notice that, VMO-type conditions are very weak assumptions on the regularity of the coefficients.General elliptic equations, also involving nonstandard growth conditions, have been treated in (Acerbi & Mingione, 2005) who built on previous ideas of Caffarelli and Peral in (Cafferelli & Peral, 1998) valid for homogeneous equations with highly oscillating coefficients.For the case of higher order systems with nonstandard p(x)-growth conditions we refer to (Habermann, 2008).In (Eleuteri & Habermann, 2010) some Calderón-Zygmund results for equations and systems with nonstandard growth conditions, mainly for obstacle problems with p(x)-growth, were established.
The result for the parabolic evolutionary p-Laplacian system has finally been achieved by Acerbi and Mingione in (Acerbi & Mingione, 2007) who introduced the necessary new tools for developing a local Calderón-Zygmund theory for the time dependent, parabolic case, see also (Misawa, 2005) for the special case F ∈ BMO.Later on, extensions to general parabolic systems have been obtained in (Duzaar, Mingione & Steffen, 2011), see also (Scheven, 2010).Moreover, a Calderón-Zygmund theory for obstacle problems was first established in (Bögelein, Duzaar & Mingione, 2011) and then, Scheven extends this result to obstacle problems with VMO-coefficients in (Scheven, 2011(Scheven, & 2014)).Furthermore, the Calderón-Zygmund theory for elliptic and parabolic measure data equations are proved in (Mingione, 2007a,b) and (Baroni & Habermann, 2012), respectively.The global gradient estimates for degenerate and singular parabolic systems are again established in (Bögelein, 2013).
In the context of nonstandard p(x, t)-growth we want to mention that the Calderón-Zygmund theory for parabolic p(x, t)-Laplacian systems was shown in (Baroni & Bögelein, 2013), see also the monograph (Baroni, 2013), while the Calderón-Zygmund theory for parabolic obstacle problems related to the parabolic p(x, t)-Laplacian was established in (Erhardt, 2014), see also the monograph (Erhardt, 2013).As we already mentioned above the approach we use here, could also applied to parabolic obstacle problems with p(x, t)-growth.Therefore, we are able to prove the result in (Erhardt, 2014) for parabolic obstacle problem with irregular obstacles related to (1).

The Function Spaces
The spaces L p (Ω), W 1,p (Ω) and W 1,p 0 (Ω) stand for the usual Lebesgue and Sobolev spaces.Parabolic Lebesgue-Orlicz spaces.We start by the definition of the nonstandard p(z)-Lebesgue space.The space L p(z) (Ω T , R k ) is defined as the set of those measurable functions v : Parabolic Sobolev-Orlicz spaces.Next, we introduce nonstandard parabolic Sobolev spaces.By W p(•) g (Ω T ) we denote the Banach space ).Finally, we shall assume that Definition 1.We identify a function u ∈ L 1 (Ω T ) as a weak solution of the parabolic equation ( 1), if and only if Notice that the existence of a weak solution of (1) is guaranteed by the result in (Erhardt, 2013b), see also (Erhardt, 2013a).
Our next aim is to introduce the dual space of W p(•) 0 (Ω T ).Therefore, we denote by W p(•) (Ω T ) ′ the dual of W p (•)  0 (Ω T ).Here and in the following, we write ⟨⟨•, •⟩⟩ Ω T for the pairing between W p(•) (Ω T ) ′ and W p(•) 0 (Ω T ), see also (Erhardt, 2013a) } we mean that there exists The previous equality makes sense due to the inclusions As a consequence of this embedding, functions w ∈ W(Ω T ) that vanish on the lateral boundary also satisfy C 0 ([0, T ]; L 2 (Ω)).More precisely, we refer the following lemma which is established in (Erhardt, 2013a,b).

Intrinsic Geometry
Before we are able to state the result, we have to mention a very important concept in the parabolic regularity theory.Therefore, we introduce symmetric parabolic cylinders with center in z 0 = (x 0 , t 0 ) ∈ Ω T of the form Q ϱ (z 0 ) := B ϱ (x 0 ) × (t 0 − ϱ 2 , t 0 + ϱ 2 ), where (t 0 − ϱ 2 , t 0 + ϱ 2 ) ⊂ (0, T ) and B ϱ (x 0 ) ⊂ Ω denotes a ball with radius ϱ > 0 and center x 0 .To obtain the relevant (scaling invariant) local estimates we will use, in order to re-balance the non-homogeneity of parabolic problems, certain scaled cylinders, i.e. so-called intrinsic cylinders of the form where λ > 0 and p 0 := p(z 0 ).The reason for such scaled cylinder is based on the fact (explained by the easiest problem), that a multiple c • u of a solution to ∂ t u − div(|Du| p−2 Du) = 0 is no longer a solution, except c ∈ {0, 1}, p = 2 or u ≡ 0. Such kind of intrinsic cylinders were introduced in the case p =const. in the pioneering work of DiBenedetto and Friedman in (DiBenedetto & Friedman, 1985a,b).The way we use the idea of intrinsic cylinders goes back to Bögelein and Duzaar in (Bögelein & Duzaar, 2012).The delicate aspect in this technique relies in the fact that the cylinders will be constructed in such a way, that the scaling parameter λ > 0 and the average of |Du| p(•) over Q (λ)  ϱ (z 0 ) are coupled in the following way:

Strategy of the Proof and Preliminary Results
In this section, we establish the result and describe the strategy of the proof.

The statement and the strategy of the proof
First of all, we state the main result of this paper.
Theorem 3. Assume that p : Ω T → [γ 1 , γ 2 ] satisfies ( 6)-( 8) and the assumptions (2)-( 5) and ( 9) on the vector-field a : Ω T × R n → R n are valid.Moreover, the inhomogeneities (10) are given.Additionally, we suppose that the data has the higher integrability properties Then, every weak solution u ∈ C 0 ([0, T ]; L 2 (Ω)) ∩ W p(•) (Ω T ) of the parabolic equation (1) satisfies Moreover, for κ, K ≥ 1 there exists a radius r 0 > 0 depending on n, γ 1 , γ 2 , q, µ, L, σ, κ, K, ω(•) and a constant c = c(n, γ 1 , γ 2 , q, µ, L, σ, κ, K, ω(•)), such that the following holds: If is satisfied, then for every parabolic cylinder Q 2r ≡ Q 2r (z 0 ) Ω T with radius r ∈ (0, r 0 ], there holds where p 0 = p(z 0 ) and d(p 0 ) is defined in Now, we briefly describe the plan of the paper and the strategy of the proof.As we mentioned in the introduction the desired result of this manuscript is to prove the gradient estimate (13).This estimate implies that Du is as integrable as the inhomogeneities F and f , cf.Theorem 3.For the proof of the gradient estimate we will use some comparison arguments to derive the needed a priori estimate.In Section 2.3, we will compare the solution of (1) with the solution of a certain homogeneous parabolic Cauchy-Dirichlet problem with nonstandard growth, then we will compare the solution of the Cauchy-Dirichlet problem with the solution of a parabolic problem with frozen exponent, i.e. with exponent p 0 = p(z 0 ).Moreover, we will need several technical tools to gain the desired gradient estimate, e.g. the higher integrability result of Bögelein and Duzaar, cf.Theorem 6, a localization argument of Baroni and Bögelein, cf.Lemma 7, and the Lipschitz bound, which is a consequence of the C 1,α -regularity of DiBenedetto and Friedman, cf.Theorem 9. Later on, we will transfer these a priori estimates via comparison argument to our nonstandard growth problem.All these preliminary results and many more are stated in Section 2.2.Finally, in Section 3 we will prove the main result.Here, we start with a so-called stopping time argument.This argument is very important in the regularity of parabolic problems, since there we will choose the correct cylinders, cf.(Kinnunen & Lewis, 2000) and (Acerbi & Mingione, 2005) for instance.Then, we will apply the comparison estimates of Section 2.3 on certain intrinsic cylinders.The next step is to derive estimates on the level sets, followed by the final estimate, which is in principle the desired gradient estimate.The final step of the proof of Theorem 3 is to adjust the exponent.In this step we will conclude the validity of (13).

Technical Tools and Preliminary Results
In this section, we cite several important technical tools, which we need for the proof of the desired Calderón-Zygmund estimates.Moreover, we will prove some tools which are important to derive our main result.
An iteration lemma.In order to "re-absorb" certain terms, we will use the following iteration lemma, which is a standard tool and can be found in (Giaquinta, 1983).The iteration results reads as follows.
Lemma 4. Let 0 < ϑ < 1, A, C ≥ 0 and β > 0.Then, there exists a constant c = c(β, ϑ), such that there holds: For any non-negative bounded function satisfying A version of the Vitali's covering Theorem.Moreover, we will need a version of the Vitali's covering Theorem for non-uniformly intrinsic parabolic cylinders, which is stated in (Bögelein & Duzaar, 2011).The result is the following: Lemma 5. Assume that M ≥ 1, λ ≥ 1 and p : Ω T → (γ 1 , γ 2 ) satisfies the conditions ( 6)-( 8).Then, there exists a constant χ = χ(n, L 1 , γ 1 ) ≥ 5, such that the following is true: Let F = {Q i } i∈I be a family of axially parallel parabolic cylinders of the from with uniformly bounded radii, uniformly in the sense that where ϱ 0 := Iwaniec's inequality.Here, we state a useful estimate which is a consequence of Iwaniec's inequality for Orlicz spaces, see (Iwaniec & Verde, 1999).Let ϑ > 0, Q ⊂ R n+1 and g ∈ L ς (Q) for some ς > 1.Then, there holds Thereby, the constant c(ϑ, ς) blows up as ς ↓ 1.Moreover, c(ϑ, ς) depends continuously on ϑ and therefore, it can be replaced by a constant c(γ . A higher integrability estimate.The higher integrability result will play a key role in the proof of the Calderón-Zygmund estimates.In the case p(•) =const.the needed higher integrability estimate goes back to Kinnunen and Lewis (Kinnunen & Lewis, 2000).In the following, we cite higher integrability estimate for degenerate parabolic equations with nonstandard growth from (Erhardt, 2013c), which is a little modification of the result in (Bögelein & Duzaar, 2011), because they considered parabolic problem of the form instead of (1).Notice that this modification is a simple exercise.The statement reads as follows: Theorem 6.Let σ > 0 and p : 6)-( 8).Then, there exists ) is a weak solution of the parabolic equation ( 1), where ( 2)-( 3) are in force and f Moreover, for any K ≥ 1 there exists a radius ϱ 1 ≡ ϱ 1 (n, γ 1 , γ 2 , L 1 , K, ω(•)) > 0, such that there holds: If ( 12) and ε ∈ (0, ε 0 ], then for any parabolic cylinder , where d is defined in ( 14).
To apply this result, we need a further technical tool, more precisely a non uniform intrinsic geometry argument.This we will mention in the following lemma, where we provide a parabolic localization technique.This lemma goes back to (Baroni & Bögelein, 2013).Obviously the difficulty stems from the necessity to couple the technique of intrinsic geometry with the localization needed to treat the variable exponent growth conditions.
Lemma 7. Let κ, K, H ≥ 1 and p : Ω T → [γ 1 , γ 2 ] satisfy ( 6) and ( 8).Then, there exists a radius Then, we have where p 0 := p(z 0 ), p 1 := inf p(•) and p 2 := sup p(•) and Our next aim is to adapt the local estimate from Theorem 6 to intrinsic cylinders of the form p 0 ϱ 2 ).We will determine the parameter λ ≥ 1 in dependence on the solution itself and we will use this intrinsic coupling to compensate the anisotropic scaling behaviour of the parabolic inequality.On cylinders of this type, the higher integrability estimate from the above theorem takes the following from.
Notice that there is a similar result is given by (Baroni & Bögelein, 2013).The main difference is, that they consider the parabolic system ∂ t v − div(a(z)|Dv| p(•)−2 Dv) = 0, where a : Ω T → R is a measurable function with µ ≤ a(z) ≤ L for any z ∈ Ω T , while we consider again the equation ( 1).Furthermore, in the standard growth case, there is also as similar result given in (Scheven, 2014).
Proof.First, we assume that z 0 = 0.Then, we rescale the problem from intrinsic cylinders 2ϱ to the standard parabolic cylinders Q ϱ , Q 2ϱ .Therefore, we have to transform in time and then, we could apply the Theorem 6.Here, we have to discuss two cases.On the one hand the case p 0 := p(0) ≥ 2 and on the other hand p 0 := p(0) < 2. We start with the case p 0 := p(0) ≥ 2 and define for (x, t) ∈ Q 2ϱ the rescaled exponent p(x, t) = p(x, λ 2−p 0 p 0 t).Furthermore, we consider the rescaled function ṽ(x, t) := λ Moreover, we observe the rescaled vector-field ã(•) and the rescaled inhomogeneities F and f as follows: for all (x, t) ∈ Q 2ϱ and w ∈ R n .Then, ṽ is a weak solution of the parabolic equation i.e. for the rescaled parabolic problem of the parabolic equation (1).The next step is to ensure that the rescaled exponent p and the rescaled vector-field ã(•) satisfy the conditions (6) and also the structure condition, i.e. the growth and monotonicity property (2) and (3).Since, p 0 ≥ 2, λ ≥ 1 and therefore λ 2−p 0 2p 0 ≤ 1, we have where we used (6).Thus, we have shown for any choice of (x 1 , t 1 ), (x 2 , t 2 ) ∈ Q 2ϱ .Next, we check the assumptions ( 2) and (3) for the vector-field ã.Therefore, we apply (2) to the vector-field ã(•).This yields , where we finally used (20).Thus, we have for all (x, t) ∈ Q 2ϱ and w ∈ R n .Finally, we can conclude that where we used (20) for the last step.This yields the desired assumption for all (x, t) ∈ Q 2ϱ and w, w 0 ∈ R n with s replaced by s := sλ − 1 p 0 ∈ (0, 1).Now, we are in the situation to apply Theorem 6 with ( µ ĉ , ĉL) instead of (µ, L).Therefore, we conclude that Dṽ ∈ L p(•)(1+ε 0 ) loc (Q 2ϱ , R n ) and moreover the following quantitative estimate holds: Notice that p 0 = p(0) = p(0).Next, we transform from v to ṽ, use the previous estimate and then, we scale back from ṽ to v.This yields, using (20), ( 12) and ( 21), that with a constant c = c(n, γ 1 , γ 2 , µ, L, K, c * , ĉ).This finishes the proof of the lemma in the case p 0 ≥ 2.
Finally, we have to prove the case 2n n+2 < p 0 < 2. Here, we define p, ṽ, F, f and ã similarly as above, i.e. p(x, t) := p ) , and ã(x, t, w) := λ shows that ṽ is a weak solution of the equation where ã, ṽ, p, F and f are the time quantities defined just above.Notice, that ã satisfies the growth and monotonicity condition ( 24) and ( 25).Similar to (23), we can also conclude that | p(z 1 ) − p(z 2 )| ≤ ω (d P (z 1 , z 2 )) is valid, for every z 1 , z 2 ∈ Q 2 ρ, since p 0 < 2 and λ ≥ 1. Applying again Theorem 6 and repeating the computations from above we obtain the assertion of the lemma also in the case p 0 < 2.
A priori estimates.For the proof of the Calderón-Zygmund estimates for the spatial gradient we will need a gradient estimate.To this aim, we refer in the next theorem Lipschitz bounds for solutions to parabolic equations with standard growth, that will be employed for suitable comparison problems.These Lipschitz bounds will be very important for the proof of the gradient estimate via comparison and they are due to the fundamental contributions of DiBenedetto and Friedman (DiBenedetto & Friedman, 1985a,b) and can be retrieved from (DiBenedetto, 1993, Chapter 8).Later on, we will transfer these a priori estimates via comparison argument to our nonstandard growth problem and mainly, to our parabolic obstacle problem.Therefore, we denote Note that the scaling of cylinders C (λ) ϱ (z 0 ) does not depends on the center z 0 .Later on, we will apply the subsequent Theorem with the choice p = p 0 ≡ p(z 0 ).Thus, the cylinder C (λ)  ϱ (z 0 ) becomes the intrinsic cylinder Q (λ) ϱ (z 0 ).The precise statement of this result reads as follows and was established in (Scheven, 2014, Theorem 5.3).
Theorem 9. Suppose that the vector-field b :

and a(•) replaced by b(•) and furthermore, that b(•) is differentiable with respect to the spatial variable with
is a weak solution of the parabolic equation ∂ t w − div b(z, Dw) = 0 on Ω T and that C (λ)  2ϱ (z 0 ) ⊂ Ω T is an intrinsic cylinder with Then, there holds sup where the constant c Lip depends only on p, n, µ, L, γ and c * .
Step 3: Proof of the second comparison estimate.Next, we want to conclude a comparison estimate between Dv and Dw.To compare the solution of ( 34) and ( 35), we have to consider the difference of their weak formulation (see Definition 1 with f, F = 0 and u replaced by v respectively w), i.e. ∫ Next, we have to define for h > 0 and τ By ( 42) we know that Dv ∈ L p 0 (Q (λ) ϱ (z 0 )) with ϱ ≤ ϱ 3 .Thus, we have Dv − Dw ∈ L p 0 (Q (λ) 2ϱ (z 0 )) and v = w on ∂ P Q (λ)  ϱ (z 0 ), we are (formally) allowed to choose φ = (v − w)χ h in the preceding identity.Then, it follows ∫ This implies ∫ Moreover, we gain that ∫ Using the monotonicity property of the vector-field a(z 0 , •), i.e. (3) with p 0 = p(z 0 ), we have Next, we apply the continuity condition ( 9) and Hölder's inequality with exponents p 0 and p ′ 0 to the right-hand side of the previous estimate.This yields with a constant c = c(γ 1 , γ 2 , p 0 , L), where we used that we have for all Lemma 7 yields ω(4ϱ) ≤ ω(Γϱ) for γ 1 ≥ 2, since Γ ≥ 4, see (18).Moreover in the case γ 1 < 2 we have ω(4λ ) .
Next, we note that the monotonicity of the logarithm implies log(e + ab) ≤ log(e + a) + log(e + b) for all a, b ≥ 0. Since the logarithm is monotone increasing and by the last inequality, we can conclude that ) and also that ) is in force.Combining the last three estimate, we derive at ) dz with the obvious labeling of I − V I. Here, we want to estimate III and V by the inequality ( 16).Therefore, we have to choose on the one hand g 1 := (1 + |Dv|) p ′ 0 (p 2 −1) and and on the other hand g 2 := (1 + |Dv|) p 0 and ς 2 := (1 + ε 1 ) = c(n, γ 1 , γ 1 , µ, L, σ) > 1 with respect to ( 40) and ( 42).Moreover, we choose Q = Q (λ) ϱ (z 0 ).Thus, we get and . Now, we want to use ( 42) to bound III and IV.The term IV, we can immediately bound by ( 42), i.e The term III, we can bound by (42) as follows.First, we use the Hölder's inequality.This yields , where we used (42), λ ≥ 1 and (39).Next, we estimate I by ( 43).Thus, we have , where we again used λ ≥ 1.The expression II can be also bounded by (42).This yields Furthermore, we consider the following ) p 0 2 with a constant c = c(n, γ 1 , γ 1 , µ, L), where we utilized (43) and Lemma 7.This and (43) we apply to IV to conclude that Note we can always assume c ) p 0 2 ≥ e by possibly reducing the value of ϱ 3 .This allows to deduce that where we also used the fact that log(cx) ≤ c log(x) for c ≥ 1. Combining the last two estimate, we get with a constant c = c(n, γ 1 , γ 1 , µ, L).By the same arguments and (42), we can also conclude that with a constant c = c(n, γ 1 , γ 1 , µ, L, c * , ĉ, σ).In the case p 0 ≥ 2, we get by the Young's inequality the following comparison estimate , while in the case p 0 < 2 we have to estimate the left-hand side from below as in the proof of Lemma (10).This yields for any κ ∈ (0, 1] and with a constant c κ = c(κ, n, γ 1 , γ 1 , µ, L, c * , ĉ, σ).Using Hölder's inequality with exponents 1 1+ε 1 , ε 1 1+ε 1 and ( 42), then we derive the third comparison estimate for any number κ ∈ (0, 1) and with constants Finally, we apply the fact |Dw| p 0 ≤ 2 p 0 −1 (|Dv| p 0 + |Dv − Dw| p 0 ) and again Hölder's inequality, ( 42) and ( 46) .This yields the energy estimate for Dw, i.e.
Step 4: Proof of the Lipschitz bound.Next, we want to derive a Lipschitz bound to Dw.Therefore, we choose Here, we have to mention that we need the dependency of δ on the term c κ H later for the comparison estimate between Du and Dw, which we derive in the next and final step.The next arguments are also true without c κ H . Therefore, this implies the desired energy estimate Hence, we can apply Theorem 9 which implies the following Lipschitz bound sup with a constant c Lip = c(n, µ, L, p 0 , c * ).Since, the dependence upon p 0 is continuous it can be replaced by a lager constant depending on γ 1 and γ 2 instead of p 0 , i.e. c Lip = c(n, γ 1 , γ 2 , µ, L, c * ).

Proof of the Main Result
Proof.The proof is divided into several steps.
Step 1: Choice of the intrinsic cylinders.Here, we will use the stopping time argument.Therefore, let K ≥ 1 and suppose that (12) is valid.Then, we observe a standard parabolic cylinder Q r ≡ Q r (z 0 ), such that Q 2r Ω T .First, we define and d(•) is defined in ( 14).Moreover, we consider the concentric parabolic cylinders Q r ⊆ Q r 1 ⊂ Q r 2 ⊆ Q 2r for fixed radii r ≤ r 1 < r 2 ≤ 2r, all the cylinders sharing the same center z 0 .Next, we shall consider λ, which satisfies λ > Bλ 0 (52) with where χ = χ(n, L 1 , γ 1 ) ≥ 5 denotes the corresponding constant from Lemma 5.In addition, we observe radii s, which are conform to where p 0 = p(z 0 ).Notice that the maximal radius R 0 is chosen, such that for all points z 0 ∈ Q r 1 and radii s ≤ R 0 the inclusion Q (λ) s (z 0 ) ⊂ Q r 2 is fulfilled.Next, we want to prove that for any z 0 ∈ Q r 1 , there holds From this fact and the definition of λ 0 we can conclude that holds, where we used (51) for the last estimate.Next, we have to treat the two cases 2 ≤ p 0 ≤ γ 2 and γ 1 ≤ p 0 < 2 separated.In the case p 0 ≥ 2, we have d(p 0 ) = p 0 2 and min Hence, this yields by ( 56), ( 54), ( 52) and ( 53) the following In the case that γ 1 ≤ p 0 < 2, there is d(p 0 ) = 2p 0 p 0 (n+2)−n respectively 1 d(p 0 ) = n+2 2 − n p 0 and min 2p 0 , we can conclude, in the same way as in the preceding case, that where we finally used (53).This addict the same estimate as in the case 2 ≤ p 0 ≤ γ 2 .The availability can be shown easily.For the purpose, we consider the following calculation .