Self-improving property of nonlinear higher order parabolic systems near the boundary

Transkript

1 Nonlinear Differ Equ Appl 7 200), 2 54 c 2009 Birkhäuser Verlag Basel/Switzerland /0/ published online October 2, 2009 DOI 0007/s Nonlinear Differential Equations and Applications NoDEA Self-improving property of nonlinear higher order parabolic systems near the boundary Verena Bögelein and Mikko Parviainen Abstract We establish global regularity results for a wide class of non-linear higher order parabolic systems The model problem we have in mind is the parabolic p-laplacian system of order 2m, m, tu + ) m div m D m u p 2 D m u ) =0 with prescribed boundary and initial values We prove that if the boundary values are sufficiently regular, then D m u is globally integrable to a better power than the natural p The method also produces a global estimate Mathematics Subject Classification 2000) 35D0, 35G30, 35K50, 35K65 Keywords Global higher integrability, Degenerate parabolic systems, Higher order parabolic p-laplacian Introduction We study the global regularity properties of solutions to a wide class of nonlinear higher order parabolic systems In particular, the parabolic p-laplacian system of order 2m, m, t u + ) m div m D m u p 2 D m u ) =0 with prescribed initial and boundary values provides a basic example Under suitable conditions on the boundary data, the corresponding initial boundary value problem admits a solution u such that D m u is integrable to the power p Our aim is to show that D m u is actually globally integrable to a better power, that is, D m u L p+ε all the way up to the boundary provided that V Bögelein was supported by ESF Short Visit Grant and M Parviainen was supported by The Emil Aaltonen Foundation, The Fulbright Center, and The Magnus Ehrnrooth Foundation

2 22 V Bögelein and M Parviainen NoDEA the boundary values and the domain are sufficiently smooth We assume that the complement of the domain satisfies a uniform capacity density condition, which is essentially sharp for higher integrability results Moreover, the method produces an explicit estimate for the L p+ε -norm of D m u Higher integrability plays an important role in stability and partial regularity results for solutions and their gradients in both the elliptic and parabolic cases For elliptic regularity results with the standard and also non-standard growth conditions, see for example Acerbi-Mingione [ 3, 28] For recent parabolic applications of higher integrability in the framework of partial regularity and Calderón-Zygmund type estimates, see for example Acerbi-Mingione [4], Acerbi-Mingione-Seregin [5], Bögelein [0], Duzaar-Mingione [6], Duzaar- Mingione-Steffen [7] and Bögelein-Duzaar-Mingione [] The elliptic higher integrability techniques developed by Gehring [20], and Elcrat and Meyers [8] see also [2]) could not directly be carried over to the parabolic case Nevertheless, Giaquinta and Struwe proved a first parabolic analogue for systems with linear growth in [22] Higher integrability for more general parabolic systems with non-linear growth conditions remained open for some time: the first positive result for degenerate and singular second order parabolic p-growth systems was obtained by Kinnunen and Lewis [26] The proof employs the method of intrinsic scaling with respect to the gradient of the solution The idea to consider parabolic cylinders whose scaling depends on the solution itself goes back to DiBenedetto and Friedman [2 4] The local higher integrability result was recently extended to higher order parabolic systems in [9], and global higher integrability results for quasiminimizers and second order parabolic systems were obtained in [35 37] Our basic strategy follows the guidelines of the local result in [26] Indeed, we first derive a reverse Hölder inequality on intrinsic cylinders up to the boundary and then use a covering argument to extend the estimates to the whole space time cylinder The intuitive idea is to use cylinders whose space time scaling is roughly speaking comparable to the mean value of D m u 2 p on the same cylinder In a certain sense this space time scaling reflects the non-homogeneity of the parabolic system, which is not present in the elliptic case However, the boundary effects and lower order terms cause extra difficulties: the covering now consists of three kinds of intrinsic cylinders that may lie near the lateral or initial boundary, or inside the domain To estimate the lower order terms near the lateral boundary, we employ a boundary version of Poincaré s inequality iteratively This step exploits the uniform capacity density condition of the complement Near the initial boundary, we compare the solution with the mean value polynomial of the initial values To this end, the oscillation of weighted means of the solution and lower order derivatives between the different time slices needs to be controlled For the solution itself, we directly exploit the weak formulation of the parabolic system whereas for the derivatives, we utilize the suitable weighted means In the singular case, that is when p<2, the quadratic terms on the right-hand side of the Caccioppoli inequality usually cause technical difficulties Therefore, we employ an iteration method in order to absorb these terms

3 Vol 7 200) Self-improving property 23 at an early stage cf Lemmas 43 and 55) In this way, we later avoid additional terms in the scaling which simplifies the proof considerably Indeed, practically the same proof now runs in both the singular and degenerate cases This observation is useful even in the local second order higher integrability proof 2 Statement of the result We consider initial-boundary value problems of the type t u + ) m D α A α z,d m u)=0, in Ω T, α =m u = g, on p Ω T 2) Here, Ω is a bounded domain in n and Ω T =Ω 0,T) n+ stands for a parabolic cylinder The initial and lateral boundary values g of the solution are prescribed on the parabolic boundary p Ω T =Ω {0}) Ω 0,T)) of Ω T Moreover,u: Ω T N is a vector valued function and, as usual, we denote by t u = u t the derivative with respect to the time-variable t and by Du, respectively D k u = {D α u i } α =k i=,,n, the derivatives of order k) with respect to the space-variable x For convenience of notation, we identify D m u as a vector in l, l = N ) n+m m, and similarly for D k u Furthermore, we adopt the shorthand notation z =x, t) n+ For simplicity of notation, we write A = {A α } α =m, where A α :Ω T l N, and thus A :Ω T l l We assume that A is a Carathéodory function: z A z,w) is measurable for every w l, w A z,w) is continuous for almost every z Ω T, and satisfies the following p-growth conditions: A z,w),w ν w p, 22) A z,w) L w p + ), 23) for all z Ω T, w l and some constants 0 <ν and L< and 2n p>max{, n+2m } Above we have made several simplifications for expository reasons: we could add an inhomogeneity with controlled growth conditions into the right-hand side of 2) as well as additional functions to the growth bounds, cf [9] Nevertheless, the proofs would remain virtually the same The 2n restriction p>max{, n+2m } is necessary in the parabolic framework because of the Sobolev embedding W m, 2n n+2m L 2 as we have to deal with the L 2 -norm of u appearing in Caccioppoli s inequality There will naturally appear several exponents throughout the paper Set 2 = max{, 2n n+2m } and p = max{, pn n+2m },

4 24 V Bögelein and M Parviainen NoDEA and observe that when m =, we simply obtain the usual Sobolev exponents We will be able to combine the degenerate and singular cases by defining p = max{2,p}, p = max{2,p } and p = min{2, p p } Next we define the space V p β 0,T; Ω) for the initial and boundary values For β 0, we denote V p β {ϕ L 0,T;Ω)= p+β 0,T; W m,p+β Ω; N )) W, p +β 0,T; L p +β Ω; N )) } C[0,T); L 2 Ω; N )) : ϕ, 0) W m, p +β Ω; N ) The role of the continuity assumption is to fix the right representative Observe that even smooth boundary values lead to a nontrivial theory Next we specify the notion of a global solution Definition 2 Let p>2 A function u L p 0,T; W m,p Ω; N )) C[0,T); L 2 Ω; N )) is a global weak) solution to the initial-boundary value problem 2) if u ϕ t A z,d m u),d m ϕ dz = 0 24) Ω T for every test-function ϕ C0 Ω T ; N ) and, moreover, u g),t) W m,p 0 Ω; N ) for almost every t 0,T) 25) and h ux, t) gx, 0) 2 dx dt 0 as h 0 26) h 0 Ω for a given function g V p 0 0,T;Ω) Note that the space L p 0,T; W m,p Ω; N )) C[0,T); L 2 Ω; N )) seems natural in the light of the existence theorems see Lions [30] and Showalter [38] Chapter III, Proposition 2) We work on the parabolic cylinders of the form Q z0 ϱ, s) =B x0 ϱ) Λ t0 s) n+, where z 0 =x 0,t 0 ) n+, ϱ, s > 0andB x0 ϱ) denotes the open ball in n with center x 0 and radius ϱ and Λ t0 s) =t 0 s, t 0 + s) the interval of length 2s centered at t 0 In the case s = ϱ 2m, we write Q z0 ϱ) = Q z0 ϱ, ϱ 2m ) When no confusion arises, we shall omit the reference points Furthermore, we write αbϱ) =Bαϱ), αλs) =Λα 2m s), and αqϱ, s) =Qαϱ, α 2m s), for a ball, interval, and cylinder scaled by the factor α>0 Next we state our main theorem The global higher integrability is achieved under the assumption that the complement of the domain Ω satisfies a uniform capacity density condition This regularity condition guarantees that

5 Vol 7 200) Self-improving property 25 there is enough of complement near every boundary point The capacity density condition could be replaced for example by the stronger measure density condition For the precise formulation of the condition, see Definition 3 Theorem 22 Suppose that u is a global solution according to Definition 2 with boundary and initial data g V p β 0,T;Ω) for some β>0 and let n \ Ω be uniformly p-thick Then there exists ε = εn, N, m, p, L/ν) 0,β] such that u L p+ε 0,T; W m,p+ε Ω; N )) Moreover, for any parabolic cylinder Q 0 = B 0 Λ 0 = Q z0, 2 ) n+,we have the following boundary estimate ) ε+d)/d D m u p+ε c dz D m u p + G p )dz Q 0 Q 4 Q0 ΩT 0 Q 0 Ω T + c G p+ε dz + c Q 0 Q 0 Ω T ) p+ε)/ p +ε) cδ + D m g, 0) p +ε dx B 0 B 0 Ω where c = cn, N, m, p, L/ν) and δ =if 0 Λ 0 and δ =0otherwise Here, we have denoted G = D m g p + g p ) /p τ if B 0 \ Ω 27) and G =0otherwise and { 2 if p 2, d = n2 p) p if 2 <p<2 2m 3 Preliminaries 3 Variational p-capacity Let <p< and O be an open set The variational p-capacity of a compact set C O is defined to be cap p C, O) =inf f p dx, f O where the infimum is taken over all the functions f C0 O) such that f = in C To define the variational p-capacity of an open set U O, we take the supremum over the capacities of the compact sets belonging to U The variational p-capacity of an arbitrary set E O is defined by taking the infimum over the capacities of the open sets containing E For the capacity of a ball, we have cap p Bϱ),B2ϱ)) = cϱ n p 3) For further details, see Chapter 4 of Evans-Gariepy [9], Chapter 2 of Heinonen-Kilpeläinen-Martio [24], or Chapter 2 of Malý-Ziemer [3]

6 26 V Bögelein and M Parviainen NoDEA Next we introduce the uniform capacity density condition, which allows us to use a boundary version of a Sobolev Poincaré type inequality This condition is essentially sharp for our main result as shown by Kilpeläinen-Koskela [25] in the elliptic case and in [27] for the second order parabolic p-laplace equation Definition 3 AsetE n is uniformly p-thick if there exist constants µ, ϱ 0 > 0 such that cap p E B x ϱ),b x 2ϱ)) µ cap p B x ϱ),b x 2ϱ)), for all x E and for all 0 <ϱ<ϱ 0 If p>n, the condition is superfluous since then every set is uniformly p-thick The next lemma slightly extends the capacity estimate from the above definition cf [36], Lemma 38) Lemma 32 Let Ω be a bounded open set in n and suppose that n \ Ω is uniformly p-thick Choose y Ω such that B y 4ϱ/3) \ Ω Then there exists a constant µ = µµ, ϱ 0,n,p) > 0 such that cap p By 2ϱ) \ Ω,B y 4ϱ) ) µ cap p By 2ϱ),B y 4ϱ) ) A uniformly q-thick set is also uniformly ϑ-thick for all ϑ q Thisisa consequence of Hölder s and Young s inequalities Lemma 33 If a compact set E is uniformly q-thick, then E is uniformly ϑ- thick for all ϑ q The next theorem states that a uniformly p-thick set has a self-improving property This result was shown by Lewis in Theorem of [29] See also Ancona [7] and Mikkonen [33] Theorem 34 Let <p n IfasetE is uniformly p-thick, then there exists γ = γn, p, µ),p) for which E is uniformly γ-thick Next, we formulate a well-known version of the Sobolev-type inequality For the proof, see Chapter 0 of Maz ja s monograph [32] or Hedberg [23] and also [36] Later we combine this estimate with the boundary regularity condition and obtain a boundary version of Sobolev s inequality The lemma employs quasicontinuous representatives of Sobolev functions We call u W,p Ω) p-quasicontinuous if for each ε>0 there exists an open set U, U Ω B, such that cap p U, B 2 ) ε, and the restriction of u to the set Ω \ U is finite valued and continuous The p-quasicontinuous functions are closely related to the Sobolev space W,p Ω): for example, if u W,p Ω), then u has a p-quasicontinuous representative Adopting the usual notation for the mean value integral we have the following Bϱ) u q dx = Bϱ) Bϱ) u q dx

7 Vol 7 200) Self-improving property 27 Lemma 35 Let B = Bϱ) be a ball in n and suppose that u W,q B) is q-quasicontinuous Denote N Bϱ/2) u) ={x Bϱ/2) : ux) =0} Then there exists a constant c = cn, q) > 0 such that Bϱ) u q dx c cap q N Bϱ/2) u),bϱ)) Bϱ) Du q dx 32 Interpolation estimates When dealing with higher order problems, interpolation estimates play an essential role In several points, particularly when Poincaré s inequality cannot be applied they shall help us to treat the intermediate derivatives First, we provide an interpolation estimate for intermediate derivatives on the annulus, cf Adams [6], Theorem 44 or [8], Lemma B Note that the crucial point here is the right dependence on the width of the annulus Lemma 36 Let Br ) Br 2 ) be two concentric balls in n with 0 <r < r 2 and let u W m,p Br 2 )) with p Then for any 0 k m and 0 <ε there exists c = cn, m, p, /ε), such that Br 2)\Br ) ε D k u p dx r 2 r )m k)p D m u p dx + c Br 2)\Br ) Br 2)\Br ) u p dx r 2 r ) mp One of the difficulties in proving the main result is the fact that both powers 2 and p play a role in Caccioppoli s inequality We now state Gagliardo Nirenberg Sobolev s inequality see Nirenberg [34]) in a form, which helps us to combine the different powers Theorem 37 Let Bϱ) be a ball in n and u W m,q Bϱ)), m N and σ, q, r and θ 0, ) and 0 k m with k n σ θm n q ) θ) n r Then there exists c = cn, m, σ) such that θσ/q θ)σ/r D k σ u m dx c D j q r u dx u dx Bϱ) ϱ m k ϱ m j j=0 Bϱ) The following lemma will help us to absorb certain integrals into the lefthand side The proof employs a standard iteration argument, see for instance Giaquinta s monograph [2], Chapter V, Lemma 3 Lemma 38 Let 0 <ϑ<, A, B 0, α>0 and let f 0 be a bounded function satisfying ft) ϑfs)+as t) α + B for all 0 <r t<s ϱ Then there exists a constant c tech = c tech α, ϑ), such that fr) c tech Aϱ r) α + B ) Bϱ) ϱ m

8 28 V Bögelein and M Parviainen NoDEA 33 Mean value polynomials In order to prove a higher integrability result for the mth derivative of u, we shall approximate the function up to order m For this aim, we shall employ mean value polynomials of order m Let B x0 r) be a ball in n and f W m, B x0 r); N ) Then its mean value polynomial P r f) : n N of degree m is defined uniquely by the condition δp r f) ) x0;r =δf) x0;r, 32) where δu =u,du,,d m u) denotes the vector of lower order derivatives and f) x0;r = f dz B x0 r) denotes the mean-value of f on B x0 r) Therefore, 32) can be rewritten as D k P r f) ) x0;r =D k f) x0;r for k =0,,m The mean value polynomial can be expressed in terms of the mean values of f as P r f) x) = b β α! Dα+β f) x0;r x x 0 ) α, where α m α+β m, if β =0 b β = b β γ y x 0 ) γ dy, if β γ! 0<γ β B x0 r) For more details, see for instance Duzaar Gastel Grotowski [5] Due to the defining property of P r f), we can replace in the above representation D α+β f) x0;r by D α+β P r f) ) x0;r Moreover, we can show that b β cn, m) r β for all multi-indices β with 0 β m This observation leads us to the estimate m P x) cn, m) k D k P ) x0;r for all x B x0 ), 33) k=0 valid for any polynomial P : n N of order m and balls B x0 r), B x0 ) in n with 0 <r See [8], Lemma A, for a more detailed proof From this estimate, we can deduce a bound for the difference of the mean value polynomials on two different balls The proof applies the definition of the mean value polynomials together with Poincaré s inequality Lemma 39 Let B x0 r), B x0 ) be two balls in n with /2 r<and suppose that f W m, B x0 ); N ) Denote by P r f),p f) : n N the mean value polynomials of f of degree m Then there exists c = cn, N, m) such that x) P f) x) c m D m f dx for all x B x0 ) P r f) B x0 )

9 Vol 7 200) Self-improving property 29 Proof To estimate the difference of the polynomials, we use 33) with P r f) P f) f) ) instead of P and exploit the defining property of the polynomial P r to find m P r f) P f) )x) c k D k P r f) P f) )dy k=0 B x0 r) m = c k D k f P f) )dy k=0 B x0 r) Next we enlarge the domain of integration in the integrals on the right-hand side and recall that B x0 ) / B x0 r) 2 n since /2 r Finally, applying Poincaré s inequality m k times to D k f P f) ) which is allowed since D k f P f) )) x 0,r = 0 leads us to m P r f) P f) )x) c k D k f P f) ) dx c m D m f dx k=0 This is the desired estimate B x0 ) B x0 ) 34 Steklov-means Since weak solutions do not a priori possess any differentiability properties with respect to the time variable t, it is standard to use a mollification in time Therefore, given a function f L Ω T ), we define its Steklov-mean by t+h fx, s)ds, t 0,T h), f h x, t) =[f] h x, t) = h t 0, t T h, T ), for 0 <h<t and x, t) Ω T Using Steklov-means, we get for ae t 0,T) an equivalent system: t u h,t) ϕ + [A,D m u)] h,t),d m ϕ dx =0, 34) Ω for all ϕ L 2 Ω; N ) W m,p 0 Ω; N ) 4 Estimates near the lateral boundary In this chapter, we derive estimates on parabolic cylinders lying near the lateral boundary Ω 0,T) For notational convenience, in this chapter we will combine the boundary terms and the constant coming from the growth bounds as follows G = D m g p + g p ) /p τ + Since we now are in the lateral boundary situation we have G = G +, where G is from 27) As usual, the first step when proving higher integrability is

10 30 V Bögelein and M Parviainen NoDEA to derive suitable Caccioppoli s inequality Although we state it for arbitrary cylinders in n+, it will be needed later only for cylinders intersecting the lateral boundary Lemma 4 Let u be a global solution according to Definition 2 Then there exists c Cac = c Cac n, m, p, L/ν) such that for all parabolic cylinders Q z0 r, s), Q z0, S) n+ with 0 </2 r<, s = λ 2 p r 2m, S = λ 2 p 2m, λ>0 there holds sup t Λ t0 s) 0,T ) c Cac B x0 r) Ω Q z0,s) Ω T λ p 2 u g),t) 2 dx + u g r) m 2 D m u p dz Q z0 r,s) Ω T p + u g r) m + G p dz Proof We choose r r <r 2 and η C0 B x0 r 2 )), ζ C ) tobe two cut-off functions with { η inbx0 r ), 0 η, D k c η η r 2 r ) for all 0 k m; k ζ 0on,t 0 S),ζ on t 0 s, ), 0 ζ, 0 ζ 2 S s 4) Choosing the test-function ϕ h = ηζ 2 u h g h ) in the Steklov-formulation 34) and integrating with respect to τ over 0,t), we get for t 0,T) τ u h ϕ h + [A,D m u)] h,d m ϕ h dz =0, 42) Ω t where we abbreviated Ω t =Ω 0,t) For the first term on the left-hand side, we find that τ u h ϕ h dz = τ u h g h ) ϕ h + τ g h ϕ h dz Ω t Ω t 2 u g),t) 2 ηζt) 2 dx u g 2 ηζζ dz + τ g u g)ηζ 2 dz, Ω Ω t Ω t as h 0 Here we have also taken into account that the initial boundary term vanishes at τ = 0 because of the initial condition The last integral on the right-hand side is now further estimated with the help of Young s inequality with exponents 2, 2) if p<2, respectively p, p/p )) when p 2 Note also that r 2 r, respectively λ 2 p r 2 r ) 2m S We get τ g u g)ηζ 2 dz g p p 2 u g 2 u g p τ +λ + dz Ω t Q z0,s) Ω T r 2 r ) 2m r 2 r ) mp Passing to the limit h 0 also in the second term on the right side of 42), we find [A,D m u)] h,d m ϕ h dz Ω t A,D m u),d m u ηζ 2 A,D m u),d m g ηζ 2 + A,D m u), lot ζ 2 dz, Ω t

11 Vol 7 200) Self-improving property 3 where we abbreviated the lower order terms by m ) m lot = D m k η D k u g) k k=0 From the ellipticity 22) ofa, we infer for the first term that A,D m u),d m u ηζ 2 dz ν D m u p ηζ 2 dz, Ω t Ω t while for the second one, we obtain by the growth bound 23) ofa and Young s inequality for ε>0 that A,D m u),d m g ηζ 2 dz ε D Ω t Ω m u p +)ηζ 2 dz + c ε D m g p dz, t Ω t where c ε = c ε p, L, /ε) Similarly, for the third term, we get A,D m u), lot ζ 2 dz ε D m u p +)ζ 2 dz+c ε lot Ω t Ω t spt η Ω p ζ 2 dz, t where c ε = c ε p, L, /ε) To estimate the integral involving the terms of lower order, we first note that D k η = 0 on B x0 r ) for k Due to our boundary condition 25) we can extend u g by zero outside Ω T to an L p W m,p function, ie for the extended function we know u g L p 0,T; W m,p B x0 r 2 ); N ) This allows us to replace the domain of integration by B x0 r 2 )\B x0 r ) 0,t) and then apply the Interpolation-Lemma 36 slicewise on the annulus B x0 r 2 )\B x0 r ) This yields for 0 < ε that m t lot p ζ 2 D k u g) p dz c Ω t k=0 0 B x0 r 2)\B x0 r ) r 2 r ) pm k) ζ2 dz t t ε D m u g) p ζ 2 u g p dz + c ε 0 B x0 r 2)\B x0 r ) 0 B x0 r 2) r 2 r ) mp ζ2 dz, where c ε = c ε n, m, p, / ε) Combining the previous observations with 42), recalling that η = on B x0 r ) and choosing ε with respect to p, L and ε we infer for ae t 0,T) that t 2 u g),t) 2 ζ 2 t)dx + ν D m u p ζ 2 dx dτ B x0 r ) Ω 0 B x0 r ) Ω t 3ε D m u p ζ 2 dz + c 0 B x0 r 2) Ω λ,s) ΩT Qz0 p 2 u g 2 u g p + r 2 r ) 2m r 2 r ) mp + G p dz, where c = cn, m, p, L, /ε) Now, we choose ε = ν/6 and multiply with /ν Applying Lemma 38, we get rid of the term involving D m u on the right-hand side Then, we take the supremum over t Λ t0 s) 0,T) in the first term on the left-hand side of the resulting inequality and choose t = min{t 0 + S, T } in

12 32 V Bögelein and M Parviainen NoDEA the second term Finally we recall that ζ onλ t0 s) to conclude desired Caccioppoli s inequality Next, we derive a Poincaré type inequality for solutions on parabolic cylinders intersecting the lateral boundary Ω 0,T) The capacity density condition and the boundary condition allow us to apply Poincaré s inequality slicewise to u g Therefore, in contrast to the local situation, we do not need to compare mean value polynomials between different time slices Lemma 42 Let u be a global solution according to Definition 2 and suppose that n \ Ω is uniformly p-thick Furthermore, let Q z0 ϱ, s) n+ be a parabolic cylinder such that B x0 ϱ/3) \ Ω Then there exist γ = γn, p, µ),p) such that for all 0 k m and γ ϑ p, we have D k u g) ϑ dz cϱ ϑm k) D m u ϑ + G ϑ dz, Q z0 ϱ,s) Ω T Q z0 ϱ,s) Ω T where c = cn, m, N, µ, ϱ 0,ϑ) and G was defined in 27 Proof Let γ = γn, p, µ),p) be the constant from Theorem 34 Then we know that n \ Ω is uniformly γ-thick, and therefore also uniformly ϑ-thick by Lemma 33 Then we extend u g by zero outside of Ω T, use the same notation for the extension We fix k j m andt Λs) 0,T)and denote N j Bϱ/2) = {x Bϱ/2) : Dj u g)x, t) =0} From Lemma 35, we get here, we consider for the moment the ϑ-quasicontinuous representative of u) D j u g),t) ϑ dx = D j u g),t) ϑ dx Bϱ) Ω Bϱ) cϱ n cap ϑ N j Bϱ/2),Bϱ)) D j+ u g),t) ϑ dx, Bϱ) with c = cn, N, ϑ) Since n \ Ω is uniformly ϑ-thick, Lemma 32 and 3) imply ) cap ϑ N j Bϱ/2),Bϱ) ) µ cap ϑ Bϱ/2),Bϱ) = cϱ n ϑ Note that µ = µn, µ, ϱ 0,ϑ) Combining this capacity estimate with the previous one, we conclude D j u g),t) ϑ dx cϱ ϑ D j+ u g),t) ϑ dx, Bϱ) Ω Bϱ) Ω where c = cn, N, µ, ϱ 0,ϑ) Integrating with respect to t over Λs) 0,T)and iterating the resulting estimate for j = k,,m, we deduce the following Poincaré s inequality D k u g) ϑ dz cϱ ϑm k) D m u g) ϑ dz Qϱ,s) Ω T Qϱ,s) Ω T The assertion now follows by Young s inequality and the definition of G

13 Vol 7 200) Self-improving property 33 Also in the singular case, ie when p<2, we will have to estimate the L 2 -norm of u, since it appears on the right-hand side of Caccioppoli s inequality in Lemma 4 Therefore, we prove a suitable L 2 -estimate in the following lemma This lemma simplifies the proof in the singular case considerably since we absorb the additional terms into the left-hand side Indeed, due to this lemma, we can apply the same scaling as in the degenerate case Lemma 43 Let κ, 2 <p<2, andu be a global solution according to Definition 2 Furthermore, let Q = Q z0 ϱ, s) n+ with 0 <ϱ, s = λ 2 p ϱ 2m, andλ>0 be a parabolic cylinder such that B x0 2ϱ/3) \ Ω If D m u p + 2Q G p) dz κλ p, 43) 2Q Ω T then there exists c = cn, N, m, p, L/ν, µ, ϱ 0,κ) such that u g 2 dz cϱ 2m λ p Proof We first extend u g by zero outside of Ω T Next, we choose α < α 2 2 and denote α i Q = Q z0 α i ϱ, αi 2m s) fori =, 2 Applying Gagliardo Nirenberg Sobolev s inequality, ie Theorem 37 with σ, q, θ, r, k) replaced by 2,p,p/2, 2, 0) slicewise to u g),t), we obtain u g 2 dz = u g 2 dz α Q Ω T α Q 2 p)/2 m cϱ mp D k p u g) ϱ m k dx u g 2 dx dt cϱ mp α Λ k=0 m k=0 α Q α B D k u g) ϱ m k p α B dz sup t α Λ α B u g),t) 2 dx 2 p)/2 44) To estimate the integrals in the sum on the right-hand side, we recall that α Q 2Q and u g =0on2Q \ Ω T Therefore, we can replace the domain of integration by 2Q Ω T which allows us to apply Poincaré s inequality from Lemma 42 on 2Q Ω T Finally, using hypothesis 43) and the fact that 2Q =2 n+2m Q we infer for 0 k m that α Q D k u g) ϱ m k p dz c D m u p + G p dz cλ p Q, 2Q Ω T where c = cn, m, N, µ, ϱ 0,κ) We now come to the estimate for the sup-term in 44) Here, we first apply Caccioppoli s inequality, ie Lemma 4 Then we use Young s inequality, note that p<2, to estimate the second term on the right-hand side and the assumption 43) to estimate the term involving

14 34 V Bögelein and M Parviainen NoDEA G Finally, we recall that Q =2ϱ 2m λ 2 p B Proceeding this way, we obtain for ae t α Λ that u g),t) 2 dx = α n B u g),t) 2 dx α B c Cac B α B Ω α2q ΩT λ p 2 u g 2 α 2 ϱ α ϱ) 2m + u g p α 2 ϱ α ϱ) mp + G p dz c λ B α2q ΩT p 2 u g 2 α 2 ϱ α ϱ) 2m + λp dz = c u g Q α2q ΩT 2 α 2 α ) 2m dz + cϱ2m λ 2, where c = cn, m, p, L/ν, κ) Joining the previous estimates with 44), applying Young s inequality and recalling that s = λ 2 p ϱ 2m, we arrive at u g 2 dz 2 u g 2 c dz + α Q Ω T α 2Q Ω T α 2 α ) 2m2 p)/p ϱ2m Q λ p, where c = cn, N, m, p, L/ν, µ, ϱ 0,κ) Applying Lemma 38, we deduce the desired estimate Now, we have the prerequisites to prove a reverse Hölder type inequality for parabolic cylinders lying near the lateral boundary Lemma 44 Let κ, andu be a global solution according to Definition 2 and suppose that n \Ω is uniformly p-thick Furthermore, let Q = Q z0 ϱ, s) n+ with 0 <ϱ and s = λ 2 p ϱ 2m, λ be a parabolic cylinder such that B x0 4ϱ/3) \ Ω Suppose that λ p κ D m u p + Q G p )dz, D m u p + Q Ω T 8Q G p )dz κλ p 8Q Ω T 45) and let γ = γn, p, µ) be the constant from Lemma 42 Then, for any q with max{γ, p } q<pthere exists c = cn, N, m, p, L/ν, µ, ϱ 0,κ) such that ) p/q c D m u p dz D m u q dz + c 4Q 4Q Ω T 4Q Gp dz 4Q Ω T Proof First, we extend u g by zero outside of Ω T and use the same notation for the extension From Caccioppoli s inequality, ie Lemma 4, weget D m u p dz c 2 p Cac λ p 2 u g Q 2Q Ω T ϱ m + u g ϱ m + G p dz 46) In the following, we will infer bounds for the first two terms on the right-hand side Therefore, we abbreviate note that u g =0on2Q \ Ω T ) σ I σ = λ p σ u g ϱ m dz 2Q

15 Vol 7 200) Self-improving property 35 for σ =2orσ = p First, observe that u g),t) W m,p 2B; N ) when extended by zero on 2B \ Ω Now we fix q [max{γ, p },p), apply Gagliardo Nirenberg Sobolev s inequality, ie Theorem 37 in the case r = 2and θ = q/σ slicewise to u g),t) and take the supremum over t 2Λ in the second integral to infer where I σ cn, m, p) λ p σ m k=0 2Q J =sup t 2Λ 2B D k u g) ϱ m k u g),t) ϱ m 2 q dx dz J σ q)/2, 47) Let us first observe that we can replace the domain of integration in the above integrals by 2Q Ω, respectively 2B Ω, since u g = 0 on the set 2Q \ Ω T We first consider the sum of integrals on the right-hand side of 47) Here, we apply Poincaré s inequality from Lemma 42 to find for 0 k m that 2Q D k u g) ϱ m k q dz c 4Q 4Q Ω T D k u g) ϱ m k c D m u q + G q dz, 4Q 4Q Ω T where c = cµ, ϱ 0,n,m,N,κ) Next, we derive an estimate for J Using Q = 2λ 2 p ϱ 2m B and applying Caccioppoli s inequality, Lemma 4, weget J c 2 Cac u g p Q 4Q Ω T ϱ m + λ 2 p u g ϱ m + λ 2 p Gp dz Our aim is to bound the right-hand side in terms of λ 2 For the first term we either apply Lemma 43 when p<2, which is applicable due to the second inequality in 45), or when p 2, we in turn apply Poincaré s inequality from Lemma 42, Hölder s inequality and the second inequality in 45) To estimate the second term on the right-hand side, we use Poincaré s inequality from Lemma 42 and the second inequality in 45) in any case Finally, the term involving G p is estimated in terms of λ p also due to our assumption 45) Observe that here we utilize the fact that the scaling also takes the boundary terms into account Proceeding this way we find that J cλ 2 Combining this and the second last estimate with 47) and applying Young s inequality, we obtain for ε>0 that I σ cλ p σ 4Q ελ p cε + 4Q q dz D m u q + G q )dz λ σ q 4Q Ω T ) p/q D m u q dz + c ε G p dz, 4Q Ω T 4Q 4Q Ω T

16 36 V Bögelein and M Parviainen NoDEA where c ε = c ε n, N, m, p, L/ν, µ, ϱ 0,κ,/ε) Inserting this estimate for σ =2 and σ = p in 46), we arrive at D m u p dz ) p/q 2c Cac ελ p cε + D m u q dz + c ε 4Q 4Q Ω T 4Q Gp dz, 4Q Ω T where c ε = c ε n, N, m, p, L/ν, µ, ϱ 0,κ,/ε) Now we use the first inequality in 45) to bound λ p in terms of the integral on the left-hand side of the preceding inequality Choosing ε small enough, we can absorb this term on the left Proceeding this way, we deduce the desired reverse Hölder inequality 5 Estimates near the initial boundary In this chapter, we are concerned with parabolic cylinders lying near the initial boundary Ω {0} We shall use the following abbreviation for the initial values g 0 x) =gx, 0) for x Ω Due to our assumptions the initial values are well defined Moreover, given a ball B x0 r) in n, we denote by P r g0) : n N the mean value polynomial of g 0 of degree m onb x0 r) defined by δp r g0) ) x0;r =δg 0 ) x0;r, asintroduced in Sect 33 As usual, we first prove suitable Caccioppoli s inequality for parabolic cylinders lying near the initial boundary Lemma 5 Suppose that u is a global solution according to Definition 2 Then there exists c Cac = c Cac n, m, p, L/ν) such that for all parabolic cylinders Q z0 r, s), Q z0, S) n+ with 0 </2 r<, s = λ 2 p r 2m, S = λ 2 p 2m, λ satisfying B x0 ) Ω and 0 Λ t0 S) there holds sup u P g0) ),t) 2 dx + D t Λ t0 s) 0,T ) B x0 r) Q m u p dz z0 r,s) Ω T c Cac λ p 2 u P g0) 2 u P g0) p r) m + r) m + dz Q z0,s) Ω T + c Cac n+2m B x0 ) D m g 0 2 dx Proof Since the proof is very much similar to the one of Lemma 4, we only point out the differences We again start with the Steklov-formulation 42) of the parabolic system But now we take ϕ h = ηζ 2 u h P g0) ) as test-function with cut-off functions η, ζ as in 4) The main difference compared to the proof of Lemma 4 is related to the first term on the left-hand side of 42) Therefore, we shall only accomplish how to treat this term Taking into 2/2

17 Vol 7 200) Self-improving property 37 account that t P g0) = 0 and the initial condition 26), we find in the limit h 0 ) τ u h ϕ h dz = τ u h P g0) ϕ h dz Ω t Ω t ),t) 2 ηζt) 2 g 0 P g0) 2 ηζ0) 2 dx 2 Ω u P g0) u P g0) Ω t 2 ηζζ dz To estimate the second integral on the right-hand side we iterate Sobolev Poincaré s inequality [recall that spt η B) andη, ζ, that D k g 0 P g0) )] =0for0 k m by the definition of P g0) and that D m P g0) =0) g 0 P g0) 2 ηζ0) 2 dx c B) 2 Ω B) Dg 0 P g0) cn, m) B) 2m B) ) 2n/n+2) dx D m g 0 2 dx 2/2 n+2)/n The remaining terms in 42) are estimated similarly as in the proof of Lemma 4 with P g0) instead of g note again that D m P g0) = 0) This leads us to the desired Caccioppoli s inequality In our notion of a global weak solution we did not impose any differentiability assumption with respect to time Therefore, we cannot apply usual Poincaré s inequality Nevertheless, we can exploit the parabolic system to gain the needed regularity with respect to time Indeed, in the next lemma, we will show that the weighted means D k u) η t) ofd k u,t) defined below possess an absolutely continuous representative This is first deduced for the weighted means of u by using the system The result then extends to the weighted means of derivatives D k u with integration by parts We say η C0 B x0 ϱ)) is a nonnegative weight-function on B x0 ϱ) n,if η 0, η dx = and D l η c η /ϱ l for 0 l 2m 5) B x0 ϱ) Note that the smallest possible value of c η depends on n and mletq z0 ϱ, s) n+ be a parabolic cylinder and v L Q z0 ϱ, s); k ), k N Then we define

18 38 V Bögelein and M Parviainen NoDEA the weighted mean of v,t)onb x0 ϱ) for ae t Λ t0 s) by v) η t) = v,t) η dx 52) B x0 ϱ) Lemma 52 Suppose that u is a global solution according to Definition 2 and let Q z0 ϱ, s) n+ be a parabolic cylinder with 0 <ϱ, s>0 and B x0 ϱ) Ω Letη C0 B x0 ϱ)) be a nonnegative weight-function satisfying 5) Then there exists c = cn,l,c η ) such that for the weighted means of D k u, 0 k m, andaet,t 2 Λ t0 s) 0,T), there holds D k u) η t 2 ) D k u) η t ) c ϱ m+k t2 t B x0 ϱ) D m u p + ) dx dt Proof Let i {,,N} and η be as above and choose ϕ: n+ N with ϕ i = η, ϕ j 0forj i as a test-function in the Steklov-formulation 34) For ae t,t 2 Λ t0 s) 0,T), we get [u i ] h ) η t 2 ) [u i ] h ) η t )= = t2 t t2 t [u i ] h ) η dt t B x0 ϱ) [A i,d m u)] h,d m η dx dt, where A i :Ω T l l/n denotes a component of A To infer the assertion for the case k = 0, we use the growth conditions 23) fora and the fact that D m η c/ϱ m After passing to the limit h 0 and summing over i =,,N we obtain u) η t 2 ) u) η t ) cl ϱ m t2 t B x0 ϱ) D m u p + ) dx dt For the general case, we consider a multi-index α of order k and obtain with integration by parts D α u) η t) = D α u,t)ηdx = ) k u,t)d α η dx = ) k u) D α ηt) B x0 ϱ) B x0 ϱ) Therefore the asserted estimate follows from the case k = 0 by exchanging η with D α η and summing over α = k Since we have achieved some regularity with respect to time of our solution u, we are now in a position to prove a Poincaré type inequality Lemma 53 Suppose that u is a global solution according to Definition 2 and let Q = Q z0 ϱ, s) n+ be a parabolic cylinder with 0 <ϱ and

19 Vol 7 200) Self-improving property 39 s = λ 2 p ϱ 2m, λ> and B = B x0 ϱ) Ω and 0 Λ t0 s) Then for all 0 k m and ϑ p there exists c = cn, N, m, L, ϑ) such that Q Q Ω T D k u P g 0) dz c ϑ ϱ ) ϱ m k + [ D m u ϑ dz λ2 p Q D m u p +) dz + Q Ω T B ϑ D m g 0 dx Proof Let η C 0 B) be a nonnegative weight-function satisfying 5) For k j m and ae t Λ 0,T) and 0 <h<t 0 + s, we decompose B D j u,t) P g 0) ϱ ) ϑ dx 4 ϑ Dj u,t) P g 0) B ϱ ) h + D j u) η t) D j u) η τ)dτ 0 h + D j u) η τ) D j g 0 ) η dτ 0 + D j g 0 ) η D j P g 0) ϱ ) D j u,t) P g 0) dx ) η ϑ ϑ ϑ ϑ ϱ ) η =4 ϑ It)+IIt)+III + IV ) 53) To estimate It), we apply Poincaré s inequality slicewise to D j u P ϱ g0) ),t) and find for ae t Λ 0,T) It) cn, ϑ) ϱ ϑ D j+ u,t) P g0) ϑ dx B ϱ ) To estimate IIt), we use Lemma 52 note that Q =2ϱ 2m λ 2 p B ) It implies for ae t Λ 0,T) that IIt) ess sup t,t 2 Λ 0,T ) D j u) η t ) D j u) η t 2 ) ϑ λ cϱ ϑm j) 2 p ) ϑ D m u p +)dz Note that the previous bound is independent of h To estimate III,wefirst consider a multi-index α of order j With integration by parts, we obtain

20 40 V Bögelein and M Parviainen NoDEA h h D α u) η τ) D α g 0 ) η dτ = D α u,τ) g 0 )η dx dτ 0 0 B h = u,τ) g 0 )D α η dx dτ 0 c ϱ j B h 0 B u,τ) g 0 dx dτ 0 as h 0 by our initial condition 26) Observe that the weight function helped us to complete the previous step Summing over all indices α of order j, we therefore conclude that III 0ash 0 Finally, to estimate IV, we recall that η cn, m) and apply Poincaré s inequality m j times to D j g 0 P ϱ g0) ) note that D l g 0 P ϱ g0) )) B =0forl = j,,m andd m P ϱ g0) =0)to infer ϑ ϑ IV D j g 0 P g0) dx cϱ ϑm j) D m g 0 dx B ϱ ) Combining the previous estimates for It) IV with 53), passing to the limit h 0 and integrating with respect to t over Λ 0,T), we get for k j m D j u P g0) ϱ ) ϑ dz cϱϑ D j+ u P g0) ϱ ) ϑ dz ϑ + cϱ ϑm j) λ2 p D m u p +)dz + D m g 0 dx, where c = cn, N, m, L, ϑ) Iterating this estimate for j = k,,m, we find D k u P ϱ g0) ) ϑ dz Q Q Ω T cϱϑ Q cϱ2ϑ Q Q Ω T Q Ω T D k+ u P g0) ϱ ) D k+2 u P g0) ϱ ) B B ϑ dz + cϱ ϑm k) ) ϑ ϑ dz + cϱ ϑm k) ) ϑ cϱϑm k) D m u ϑ dz + cϱ ϑm k) ) ϑ, with the obvious meaning of ) ϑ This proves the asserted Poincaré type inequality The integral D m u p dz) ϑ on the right-hand side of the previous Poincaré type inequality has the wrong exponent Therefore, we shall exploit

21 Vol 7 200) Self-improving property 4 the intrinsic scaling of the cylinders, which depends via hypothesis 54) on the solution itself This will help us to compensate the degeneracy Corollary 54 Let κ and u be a global solution according to Definition 2 Furthermore, let Q = Q z0 ϱ, s) n+ be a parabolic cylinder with 0 <ϱ, λ>0 and s = λ 2 p ϱ 2m and B = B x0 ϱ) Ω and 0 Λ t0 s) Suppose that D m u p +)dz κλ p 54) Then there exists c = cn, N, m, L, ϑ, κ) such that for all 0 k m and ϑ p, we have D k u P g0) ϑ ϑ ϱ ) ϱ m k dz c λ + D m g 0 dx Proof We first apply Poincaré s inequality from Lemma 53 note that s/ϱ 2m = λ 2 p ) Then we use Hölder s inequality and hypothesis 54) to infer D k u P g0) ϑ ϱ ) ϱ m k dz [ λ c D m u ϑ 2 p dz + D m u p +) dz ϑ + D m g 0 dx B ϑ c λ ϑ + λ 2 p λ p + D m g 0 dx = c λ + D m g 0 dx B where c = cn, N, m, L, ϑ, κ) This proves the desired estimate B ϑ, The next lemma is an analogue of Lemma 43 for parabolic cylinders near the initial boundary Later, in the proof of the reverse Hölder inequality it will help us to bound the L 2 -norm of u in the case p<2 Lemma 55 Let κ, 2 <p<2 and let u be a global solution according to Definition 2 Furthermore, let Q = Q z0 ϱ, s) n+ be a parabolic cylinder with 0 <ϱ, λ>0 and s = λ 2 p ϱ 2m and 2B = B x0 2ϱ) Ω and 0 Λ t0 s) If D m u p +)dz κλ p, 55) 2Q 2Q Ω T B

22 42 V Bögelein and M Parviainen NoDEA then there exists c = cn, N, m, p, L/ν, κ) such that u P ϱ g0) 2 dz cϱ 2m λ 2 + D m g 0 2 dx 2B 2/2 Proof Let α < α 2 2 Applying Gagliardo Nirenberg Sobolev s inequality, ie Theorem 37 with σ, q, θ, r, k) replaced by 2,p,p/2, 2, 0) slicewise to u P α g0) ϱ ),t), we obtain α Q Ω T u P g0) 2 dz cϱ mp α ϱ m k=0 α Q Ω T D k u P g0) p α ϱ ) ϱ m k dz J 2 p)/2, 56) where J = sup t α Λ 0,T ) α B u,t) P g0) α ϱ 2 dx We first estimate the integrals in the sum on the right-hand side of 56) For this aim we apply Corollary 54 on α Q [note that the hypothesis 54) follows from 55) since α Q 2Q and 2Q / α Q 2 n+2m ] yielding that Q α Q Ω T D k u P g0) p α ϱ ) ϱ m k dz c λ + α B p D m g 0 dx 57) We now come to the estimate of J Lemma 39 provides the following estimate for difference of the mean value polynomials on α Q and α 2 Q: P g0) α ϱ x) P g0) α 2ϱ x) cϱ m α B D m g 0 dx cϱ m 2B D m g 0 2 dx /2, for all x α 2 B and thus we can use the Caccioppoli inequality, Lemma 5 to obtain the following estimate for J: J c ) g0) α n B λ p 2 u P α 2ϱ 2 α 2Q Ω T α 2 ϱ α ϱ) 2m + u P α g0) 2ϱ p + dz α 2 ϱ α ϱ) mp 2/2 + cϱ 2m D m g 0 2 dx 2B c Q α 2Q Ω T g u P 0) α 2ϱ 2 dz + cϱ2m α 2 α ) 2m λ 2 + D m g 0 2 dx 2B 2/2

23 Vol 7 200) Self-improving property 43 Here, in the last line, we have used Young s inequality and the fact that s = λ 2 p ϱ 2m and λ similarly as in the proof of Lemma 43) Joining the previous estimate and 57) with 56) and applying Young s inequality, we find α Q Ω T u P α g0) ϱ 2 dz 2 α 2Q Ω T λ 2 Q + Applying Lemma 38 we obtain the desired estimate u P α g0) 2ϱ 2 c dz+ ϱ2m α 2 α ) 2m2 p)/p D m g 0 2 dx 2/2 Now we are in a position to prove a reverse Hölder inequality on cylinders intersecting the initial boundary We take G from Sect 4 into account in the scaling because then we can later use the same scaling in all the cases Lemma 56 Let κ and let u be a global solution according to Definition 2 Furthermore, let Q = Q z0 ϱ, s) n+ be a parabolic cylinder with 0 <ϱ, λ>0 and s = λ 2 p ϱ 2m and 8B Ω, where8b = B x0 8ϱ) and 0 2Λ t0 s) Suppose that λ p κ Q Q Ω T D m u p + G p )dz, 2B D m u p + 8Q G p )dz κλ p, 8Q Ω T 58) for some G L p 8Q) with G Then there exists a constant c = cn, N, m, p, L/ν, κ), such that for any q with max{p, p } q<pthere holds ) p/q c D m u p dz D m u q dz 4Q 4Q Ω T + c 4Q Gp dz + cλ p p D m g p 0 dx p/ p 4Q Ω T Proof From Caccioppoli s inequality, ie Lemma 5, weget D m u p dz c Cac λ p 2 u P g0) 2 2ϱ u P g0) p 2ϱ Q 2Q Ω T ϱ m + ϱ m + dz + c Cac λ p 2 D m g 0 2 dx 2/2 59) We still have to bound the first two terms on the right-hand side Therefore, we abbreviate I σ = λp σ u P g0) σ 2ϱ Q ϱ m dz 2Q Ω T 2B 4B

24 44 V Bögelein and M Parviainen NoDEA for σ =2orσ = p Now,wefixq [max{p, p },p) In order to estimate I σ we apply Gagliardo Nirenberg Sobolev s inequality, ie Theorem 37 with r =2andθ = q/σ slicewise to u P g0) 2ϱ ),t) yielding that m I σ cn, m, p) λp σ D k u P g0) 2ϱ ) q 2Q k=0 2Q Ω T ϱ m k dz J σ q)/2, 50) where u,t) P g0) 2 2ϱ J sup ϱ m dx t 2Λ 0,T ) 2B To estimate the integrals in the sum on the right-hand side, we enlarge the domain of integration from 2Q to 4Q and apply Poincaré s inequality from Lemma 53 to infer for 0 k m that recall that G ) D k u P g0) 2ϱ ) q [ 2Q 2Q Ω T ϱ m k dz c D m u q dz 4Q 4Q Ω T q + λ2 p D m u + 4Q G) p dz + D m g 0 dx 4Q Ω T 4B To further estimate the second term on the right-hand side, we use Hölder s inequality and hypothesis 58) ie the first inequality in 58) when p<2, respectively the second inequality in 58) when p>2) to find λ 2 p D m u + 4Q G) p dz = λ 2 p ) /p ) ) /p ) 4Q Ω T λ 2 p 4Q cλ 2 p λ pp 2) p = c ) p 2)/p D m u + G) p dz D m u + 4Q Ω T 4Q G) q dz 4Q Ω T ) /q D m u + 4Q G) q dz 4Q Ω T ) /q D m u + 4Q G) q dz 4Q Ω T ) /q Inserting this above to bound the second term on the right-hand side, we deduce D k u P g0) 2ϱ ) q 2Q 2Q Ω T ϱ m k dz q ) /q c D m u q + 4Q G q )dz + D m g 0 dx 5) 4Q Ω T The estimate of J now is similar to the one from Lemma 55 We first apply Caccioppoli s inequality in Lemma 5 with Q z0 r, s) andq z0, S) replaced by 2Q and 4Q, note that Q =2λ 2 p ϱ 2m B and then estimate the terms 4B

25 Vol 7 200) Self-improving property 45 appearing on the right-hand side as follows: in the case p 2, we now use Corollary 54 to bound the first integral on the right-hand side, while in the case p<2 we use Lemma 55 Moreover, from Corollary 54, we also infer a bound for the second integral on the right-hand side Note that the second inequality in 58) ensures that the hypothesis of Corollary 54 and Lemma 55 are satisfied Proceeding this way we arrive at sup t 2Λ 0,T ) 2B u,t) P g0) 4ϱ ϱ m 2 dx c λ 2 + D m g 0 2 dx 4B 2/2 Since by Lemma 39, we can bound the difference of the mean value polynomials on 2B and 4B as g0) x) P x) cϱ m D m g 0 dx for all x 4B, P g0) 2ϱ 4ϱ 4B this leads us to J c λ 2 + D m g 0 2 dx 4B 2/2 Inserting the previous estimate and 5) in50) for σ = 2 and σ = p and applying Young s inequality, we obtain for ε>0 that ) p/q I σ ελ p + c ε D m u q + 4Q G q )dz + 4Q Ω T 4B D m g 0 2 dx p/2, where c ε = c ε n, N, m, p, L/ν, κ, /ε) Inserting this estimate for σ =2and σ = p in 59) and using Young s inequality for the term involving g 0 note that ) p/2 +λ p 2 ) 2/2 ελ p +c ε λ p max{2,p} ) max{2,p}/2 ), we arrive at ) p/q D m u p dz 3c Cac ελ p cε + D m u q + 4Q G q )dz 4Q Ω T Q + c ε λ p p 4B D m g 0 2 dx p/2 Now we use the first inequality in 58) to bound λ p in terms of the integral on the left-hand side of the preceding inequality Choosing ε small enough we can absorb this term on the left Finally, applying Hölder s inequality to the term involving G and to the initial term if necessary ie if p>2) we deduce the desired reverse Hölder inequality

26 46 V Bögelein and M Parviainen NoDEA 6 Proof of the higher integrability In this chapter, we prove the global higher integrability result from Theorem 22 To deduce the desired estimate on the set Ω T, we will cover Ω T by intrinsic cylinders Therefore, we have to take into account three different configurations, that is, cylinders lying in the interior of Ω T and those lying near the lateral or initial boundaries For the latter two, we have proved reverse Hölder type inequalities in Lemmas 44 and 56 For the interior cylinders there holds an analogue of Lemma 56 without the initial boundary term, of course cf [9], Lemma 3) Proof of Theorem 22 We fix a cylinder Q 0 = Q z0, 2 ) n+ which might intersect the complement of Ω T As usual, we denote 4 Q 0 = Q z0 /4, /4) 2 ) and also B 0 = B z0 ) and Λ 0 =Λ t0 2 ) At the end, a choice Ω T 4 Q 0 leads to the global higher integrability result To begin with, we cover Q 0 by Whitney-type cylinders Q i = Q zi r i,ri 2m ), i =, 2,, where r i is comparable to the parabolic distance of Q i to the boundary Q 0 of Q 0 see, for example, page 5 of [39]) The parabolic distance of two sets E,F n+ is defined to be dist p E,F)=inf { } x x + t t /2m) :x, t) E,x, t) F In addition, the cylinders Q i are of bounded overlap, meaning that every z belongs at most to a fixed finite number of cylinders depending only on n and m, and 5Q i Q 0 Later we shall divide Q 0 into a good and a bad set, ie into certain level sets according to a level λ>0 In order to apply the reverse Hölder inequality from Lemma 44, respectively Lemma 56, we aim to find cylinders having the scaling factor λ 2 p and satisfying 45), respectively 58) around each point lying in the bad set For this we first set λ 0 = Q 0 Q 0 Ω T D m u p + G p) dz) /d, where d was defined in the statement of Theorem 22 and G is defined by { G p if B 0 Ω = G p + ifb 0 \ Ω We now choose λ such that λ>max{λ 0, } = λ 0 For z Q 0 Ω T, we define hz) = c Q 0 min{ Q i /d : z Q /d i } D m uz),

27 Vol 7 200) Self-improving property 47 where c will be fixed later Further, we consider z Q 0 Ω T such that h z) >λ and fix a Whitney-cylinder Q i = Q zi r i,ri 2m ) such that z Q i Ω T We define ) /d Q0 α = α z) =, Q i and θ = θ z) =λα z)) 2 p Now we will use the stopping time argument to find an intrinsic cylinder around z of the type Q z r, θr 2 ) on which the assumptions 45), respectively 58) of Lemma 44, respectively 56 are satisfied To begin with, we show that the first inequality in 45), respectively the first inequality in 58), is valid for suitably small cylinders due to Lebesgue s differentiation theorem Indeed, for almost every z Q i Ω T such that h z) >λ,wehave lim r 0 Qr,θr 2m ) Q zr,θr 2m ) Ω T D m u p + G p) dz>c p α p λ p 6) Note that Q z r,θr 2m ) Ω T = Q z r,θr 2m )forr > 0 small enough Our next aim is to show that the second inequality in 45), respectively the second inequality in 58), is valid due to the definition of λ 0 For this we have to distinguish the cases p 2andp<2since the scaling factor θ is smaller that one in the former case, respectively larger than one in the latter case We start considering the case p 2,wherewehaved =2andθ For an intrinsic cylinder Q z r, θr 2m ), with radius r such that r i /64 r r i, note that Q z r, θr 2m ) 2Q i Q 0 ), we obtain due to the definition of λ 0,α and d Qr, θr 2m ) c Q 0 Q i θ Q 0 Q zr,θr 2m ) Ω T Q 0 Ω T D m u p + G p) dz D m u p + G p) dz c p α p λ p, 62) where c is chosen to be large enough This fixes the constant c inthe definition of h in dependence of n, m and p Now we want to use a similar reasoning for the case 2 <p<2 Here, the basic difference compared to the case p 2 is that the scaling factor θ for the parabolic cylinders is now larger than Nevertheless, we still have to ensure that the considered intrinsic cylinders are contained in Q 0 To accomplish this, we consider radii r with θ /2m r i /64 r θ /2m r i Thus,Q z r, θr 2m ) Q 0, and due to the definition of λ 0,α and d note that d = p n2 p)/2m) in the present case), we obtain Qr, θr 2m ) c p α p λ p Q zr,θr 2m ) Ω T D m u p + G p) dz c Q 0 Q i θn/2m) λ d 63)

28 48 V Bögelein and M Parviainen NoDEA According to 6) and 62) when p 2, respectively 63) when p<2 and due to the fact that the integrals above depend continuously on the radius of the cylinder, there exists one largest radius ϱ = ϱ z) with ϱ 0,r i /64] when p 2, respectively ϱ 0,θ /2m r i /64] when p<2 such that equality holds In the following, we denote briefly Q = Q z ϱ, θϱ 2m ),B= B x ϱ), 4Q = Q z 4ϱ, θ4ϱ) 2m ), 4B = B x 4ϱ), 4Λ = Λθ4ϱ) 2m ), etc Note that 64Q Q 0 by the choice of ϱ Hence, from the previous reasoning we have Q Q Ω T D m u p + G p) dz = c p αλ)p c p αλ) p 64) and D m u p + 64Q G p) dz c p αλ)p 65) 64Q Ω T Now we set q = max{p,γ, p }, where γ = γn, p, µ) is the constant from Lemma 44 From64) and 65), we see that after enlarging the domain of integration if necessary) conditions 45) and 58) are satisfied with λ replaced by αλ and, further, s = θϱ 2m =αλ) 2 p ϱ 2m We now distinguish severalcasesif8b \ Ω, we can apply Lemma 44 with 8Q instead of Q then the condition B x0 4ϱ/3)\Ω has to be replaced by B x0 32ϱ/3)\Ω which is fulfilled) In this case, the first inequality in 45) is satisfied on 8Q with 8 n+2m c instead of κ after enlarging the domain of integration in 64) from Q to 8Q Weget D m u p dz 8Q 8Q Ω T ) p/q c D m u q dz + c 32Q 32Q Ω T 32Q Gp dz 66) 32Q Ω T On the other hand, if 8B Ω, we still have to distinguish the cases when the cylinder lies near, respectively far from the initial boundary If 8B Ω and 2Q intersects the initial boundary, ie 0 2Λ, then we are in a position to apply Lemma 56 note that the second inequality in 58) is satisfied on 8Q with 8 n+2m c instead of κ due to 65) after enlarging the domain of integration from 8Q to 64Q) Therefore the application of the lemma implies ) p/q D m u p c dz D m u q dz 4Q 4Q Ω T + c 4Q Gp dz + c αλ) p p D m g p 0 dx p/ p 67) 4Q Ω T In the remaining case 8B Ωand2Λ 0,T), we have 2Q Ω T In that case we can use the interior analogue of Lemma 56 to obtain p/q D m u p c dz D m u dz) q + G p dz 68) Q 2Q Q 2Q 4B 2Q