Gradient weighted estimates at the natural exponent for Quasilinear Parabolic equations
aa r X i v : . [ m a t h . A P ] A p r Gradient weighted estimates at the natural exponent for Quasilinear Parabolicequations.
Karthik Adimurthi ✩ a, ∗ , Sun-Sig Byun , ✩✩ a,b a Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea. b Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea.
Abstract
In this paper, we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinearparabolic equations with weights in the Muckenhoupt class A qp ( R n +1 ) for q ≥ p on non-smooth domains. Herethe quasilinear nonlinearity is modelled after the standard p -Laplacian operator. Until now, all the weightedestimates for the gradient were obtained only for exponents q > p . The results for exponents q > p used the fullcomplicated machinery of the Calder´on-Zygmund theory developed over the past few decades, but the constantsblow up as q → p (essentially because the Maximal function is not bounded on L ).In order to prove the weighted estimates for the gradient at the natural exponent, i.e., q = p , we need toobtain improved a priori estimates below the natural exponent. To this end, we develop the technique of Lipschitztruncation based on [3, 26] and obtain significantly improved estimates below the natural exponent. Along theway, we also obtain improved, unweighted Calder´on-Zygmund type estimates below the natural exponent whichis new even for the linear equations. Keywords:
Quasilinear parabolic equations, Muckenhoupt weights, Lipschitz truncation
Contents1 Introduction 22 Preliminaries 3 ✩ Supported by the National Research Foundation of Korea grant NRF-2017R1A2B2003877. ✩✩ Supported by the National Research Foundation of Korea grant NRF-2015R1A4A1041675. ∗ Corresponding author
Email addresses: [email protected] and [email protected] (Karthik Adimurthi ✩ ), [email protected] (Sun-SigByun , ✩✩ ) .4 Estimates on the derivative of v λ,h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 Some more properties of v λ,h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.6 Proof of the Lipschitz continuity of v λ,h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.7 Two crucial estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 In this paper, we are interested in obtaining Calder´on-Zygmund type regularity estimates in weighted Lebesguespaces for equations of the form ( u t − div A ( x, t, ∇ u ) = div | f | p − f in Ω × ( − T, T ) ,u = 0 on ∂ p (Ω × ( − T, T )) , (1.1)where the nonlinearity A ( x, t, ∇ u ) is modelled after the well studied p -Laplacian operator given by |∇ u | p − ∇ u in a bounded domain Ω ⊂ R n with n ≥
2, potentially with non-smooth boundary ∂ Ω. The parabolic boundaryis given by ∂ p (Ω × ( − T, T )) := ∂ Ω × ( − T, T ) [ Ω × {− T } .Over the past decades, there have been a plethora of a priori estimates of the Calder´on-Zygmund type obtainedfor (1.1). We shall point out that all the estimates discussed in the introduction are quantitative, but in order tohighlight the novelty of the results in this paper, we shall only mention the qualitative nature of the estimatesexisting in the literature.The first extension of the Calder´on-Zygmund theory for (1.1) with A ( x, t, ∇ u ) = |∇ u | p − ∇ u for p > nn + 2(note that this restriction is natural for parabolic problems, see [19, Chapter 5]) was obtained in [1], where theyproved | f | ∈ L qloc = ⇒ |∇ u | ∈ L qloc for all q ≥ p. Since then, many extensions were obtained which generalized the estimates in [1] to more general nonlineari-ties, function spaces and up to the boundary (see [6, 7, 12, 13, 22, 30] and the references therein).
In this paper,the first result we will prove is an improved global a priori estimate of the form | f | ∈ L q (Ω T ) = ⇒ |∇ u | ∈ L q (Ω T ) for all q ∈ [ p − β , p ] , where β is a sufficiently small universal exponent. The improvement is two fold, firstly this estimate is obtainedbelow the natural exponent and secondly, the estimate assumes no regularity of the coefficients and hence isnon-perturbative. As a consequence, this result is new even for linear equations.The second result that we are interested in obtaining is global estimates in weighted Lebesgue spaces with theweight in Muckenhoupt class.
For general nonlinear structures with linear growth, i.e., A ( x, t, ∇ u ) ≈ ∇ u withthe coefficients satisfying a small bounded mean oscillation restriction, the following global weighted estimateswas obtained in [15]: | f | ∈ L qω (Ω T ) = ⇒ |∇ u | ∈ L qω (Ω T ) for all q > ω ∈ A q ( R n +1 ) . Note that in particular, they cannot consider q = 2 in [15].2ubsequently, in [16], they were able to prove analogous results for nonlinearities of the form A ( x, t, ∇ u ) ≈|∇ u | p − ∇ u with 2 nn + 2 < p < ∞ and more general Weighted Orlicz spaces, in particular, they prove | f | ∈ L qω (Ω T ) = ⇒ |∇ u | ∈ L qω (Ω T ) for all q > p and ω ∈ A qp ( R n +1 ) . Note that in particular, they cannot consider q = p in [16].The main obstacle in proving weighted estimates at q = p is due to the failure of strong L − L bounds for theHardy-Littlewood Maximal function. Therefore to reach the natural exponent, a different approach is needed.In this paper, we achieve this result by showing the weighted estimate holds with q = p , i.e., (1.2) holds. Toovercome this difficulty, we construct a suitable test function based on a modification of the techniques developedin [3, 26] and obtain estimates below the natural exponent, i.e., under suitable restrictions on the A ( x, t, ∇ u )and Ω (similar to those in [16]), we prove | f | ∈ L qω (Ω T ) = ⇒ |∇ u | ∈ L qω (Ω T ) for all q ≥ p and ω ∈ A qp ( R n +1 ) . (1.2)There are a few remarks to be made; firstly the estimate (1.2) represents an end point weighted estimate forquasilinear parabolic equations; secondly, the optimal weight class in the elliptic case is conjectured to be A qp − (see [5, Theorem 1.9] for more on this and the elliptic Iwaniec conjecture) and in the parabolic case too, theoptimal result is expected to be of the form | f | ∈ L qω (Ω T ) = ⇒ |∇ u | ∈ L qω (Ω T ) for all q > p − ω ∈ A qp − ( R n +1 ) , but this result seems to be far out of reach of current methods.The plan of the paper is as follows: In Section 2, we collect all the assumptions on the domain, nonlinearstructure and the weight class along with some preliminary well known results, in Section 3, we will describethe main theorem that will be proved, in Section 4, we will develop a general Lipschitz truncation techniqueand construct a suitable test function, in Section 5, we will define useful perturbations of (1.1) and prove crucialdifference estimates below the natural exponent, in Section 6, we will prove Theorem 3.1, in Section 7, we willuse standard covering arguments to prove the parabolic analogue of a good- λ estimate and finally use that inSection 8 to prove Theorem 3.3. Acknowledgement
The authors would like to thank the organisers of the conference
Recent developments in Nonlinear PartialDifferential Equations and Applications - NPDE2017 held at TIFR-CAM, Bangalore where part of this workwas done.
2. Preliminaries
The following restriction on the exponent p will always be enforced:2 nn + 2 < p < ∞ . (2.1) Remark 2.1.
The restriction in (2.1) is necessary when dealing with parabolic problems because, we invariablyhave to deal with the L -norm of the solution which comes from the time-derivative. On the other hand, thefollowing Sobolev embedding W ,p ֒ → L is true provided (2.1) holds. On the other hand, if we assume u ∈ L r (Ω T ) for some r ≥ such that Λ r := n ( p −
2) + rp > (see [19, Chapter 5] for more on this), then we can obtainanalogous result as to Theorem 3.3. This extension of Theorem 3.3 to the case < p ≤ nn + 2 requires only atechnical modification provided Λ r > and will be omitted.2.1. Assumptions on the Nonlinear structure We shall now collect the assumptions on the nonlinear structure in (1.1). We assume that A ( x, t, ∇ u ) is aCarath´eodory function, i.e., we have ( x, t )
7→ A ( x, t, ζ ) is measurable for every ζ ∈ R n and ζ
7→ A ( x, t, ζ ) iscontinuous for almost every ( x, t ) ∈ Ω × ( − T, T ). We also assume A ( x, t,
0) = 0 and A ( x, t, ζ ) is differentiable in ζ away from the origin, i.e., d ζ A ( x, t, ζ ) exists for a.e. ( x, t ) ∈ R n +1 .We further assume that for a.e. ( x, t ) ∈ Ω × ( − T, T ) and for any η, ζ ∈ R n , there exists two given positiveconstants Λ , Λ such that the following bounds are satisfied by the nonlinear structures : hA ( x, t, ζ − A ( x, t, η ) , ζ − η i ≥ Λ (cid:0) | ζ | + | η | (cid:1) p − | ζ − η | , (2.2) |A ( x, t, ζ ) − A ( x, t, η ) | ≤ Λ | ζ − η | (cid:0) | ζ | + | η | (cid:1) p − . (2.3)3ote that from the assumption A ( x, t,
0) = 0, we get for a.e. ( x, t ) ∈ R n +1 , there holds |A ( x, t, ζ ) | ≤ Λ | ζ | p − . ΩThe domain that we consider may be non-smooth but should satisfy some regularity condition. This conditionwould essentially say that at each boundary point and every scale, we require the boundary of the domain to bebetween two hyperplanes separated by a distance proportional to the scale.
Definition 2.2.
Given any γ ∈ (0 , and S > , we say that Ω is ( γ, S ) -Reifenberg flat domain if for every x ∈ ∂ Ω and every r ∈ (0 , S ] , there exists a system of coordinates { y , y , . . . , y n } (possibly depending on x and r ) such that in this coordinate system, x = 0 and B r (0) ∩ { y n > γr } ⊂ B r (0) ⊂ B r (0) ∩ { y n > − γr } . The class of Reifenberg flat domains is standard in obtaining Calder´on-Zygmund type estimates, in the ellipticcase, see [5, 11, 14, 17] and the references therein whereas for the parabolic case, see [10, 12, 13, 30] and thereferences therein.From the definition of ( γ, S )-Reifenberg flat domains, it is easy to see that the following property holds: Lemma 2.3.
Let γ > and S > be given and suppose that Ω is a ( γ, S ) -Reifenberg flat domain, then thereexists an m e = m e ( γ, S , n ) ∈ (0 , such that for every x ∈ Ω and every r > , there holds | Ω c ∩ B r ( x ) | ≥ m e | B r ( x ) | . (2.4) In order to prove the main results, we need to assume a smallness condition satisfied by ( A , Ω).
Definition 2.4.
Let γ ∈ (0 , and S > be given, we then say ( A , Ω) is ( γ, S ) -vanishing if the followingassumptions hold: (i) Assumption on A : For any parabolic cylinder Q ρ,s ( z ) centered at z := ( x , t ) ∈ R n +1 , let us define thefollowing: Θ( A , Q ρ,s ( z ))( x, t ) := sup ζ ∈ R n \{ } (cid:12)(cid:12) A ( x, t, ζ ) − A B ρ ( x ) ( t, ζ ) (cid:12)(cid:12) | ζ | p − , where we have used the notation A B ρ ( x ) ( t, ζ ) := B ρ ( x ) A ( x, t, ζ ) dx. (2.5) Then A is said to be ( γ, S ) vanishing if for some τ ∈ [1 , ∞ ) , there holds [ A ] τ,S := sup <ρ ≤ S
From (2.2) , we see that | Θ( A , Q ρ,s ( z ))( x, t ) | ≤ , thus combining this with the assumption (2.6) ,we see from standard interpolation inequality that for any ≤ t < ∞ , there holds Q ρ,s ( z ) | Θ( A , Q ρ,s ( z ))( z ) | t dz ≤ C ( γ, Λ ) , with C ( γ, Λ ) → whenever γ → .2.4. Muckenhoupt weights In this subsection, let us collect all the properties of the weights that will be considered in the paper. See[24, Chapter 9] for the details concerning this subsection.4 efinition 2.6 (Strong Muckenhoupt Weight) . A non negative, locally integrable function ω is a strong weight in A q ( R n +1 ) for some < q < ∞ if sup z ∈ R n +1 sup <ρ< ∞ ,
Lemma 2.7.
A parabolic weight w ∈ A q for < q < ∞ if and only if (cid:18) | Q | ¨ Q f ( x, t ) dx dt (cid:19) q ≤ cw ( Q ) ¨ Q | f ( x, t ) | q w ( x, t ) dx dt, holds for all non-negative, locally integrable functions f and all cylinders Q = Q ρ,s ( x, t ) . As a direct consequence of Lemma 2.7, the following Lemma holds:
Lemma 2.8.
Let ω ∈ A q ( R n +1 ) for some < q < ∞ , then there exists positive constants c = c ( n, q, [ ω ] q ) and τ = τ ( n, q, [ ω ] q ) ∈ (0 , such that c (cid:18) | E || Q | (cid:19) q ≤ ω ( E ) ω ( Q ) ≤ c (cid:18) | E || Q | (cid:19) τ , for all E ⊂ Q and all parabolic cylinders Q ρ,s ( z ) . Another important result regarding the strong Muckenhoupt weights that will be needed is the followingself-improvement property:
Lemma 2.9.
Let < q < ∞ and suppose ω ∈ A q be a given weight, then there exists an ε = ε ( n, q, [ ω ] q ) > such that ω ∈ A q − ε with the estimate [ ω ] q − ε ≤ C [ ω ] q where C = C ( q, n, [ ω ] q ) . We will now define the A ∞ class as follows: Definition 2.10.
A weight ω ∈ A ∞ if and only if there are constants τ , τ > such that for every paraboliccylinder Q = Q ρ,s ⊂ R n +1 and every measurable E ⊂ Q , there holds ω ( E ) ≤ τ (cid:18) | E || Q | (cid:19) τ ω ( Q ) . Moreover, if ω is an A q weight with [ ω ] q ≤ ω , then the constants τ and τ can be chosen such that max { τ , /τ } ≤ c ( ω, n ) . From the general theory of Muckenhoupt weights, we see that A ∞ = [ ≤ q< ∞ A q . Remark 2.11.
The weight class considered in Definition 2.6 is called Strong Muckenhoupt class because thecylinders are decoupled in space and time, i.e., ρ and s are not related when considering cylinders Q ρ,s . Whenconsidering linear equations (i.e., p = 2 ), the weight class is defined with respect to cylinders of the form Q ρ,ρ .This is possible because in the case p = 2 , there is an invariance property under normalization, which doesnot exist if p = 2 . It is an open question if one can obtain the results of this paper for Muckenhoupt weightsdefined with respect to cylinders belonging to a more restricted class (see the very nice thesis [32] for some resultsconcerning the weights arising in doubly nonlinear quasilinear equations).2.5. Function Spaces Let 1 ≤ ϑ < ∞ , then W ,ϑ (Ω) denotes the standard Sobolev space which is the completion of C ∞ c (Ω) underthe k · k W ,ϑ norm. 5he parabolic space L ϑ ( − T, T ; W ,ϑ (Ω)) for any ϑ ∈ [1 , ∞ ) is the collection of measurable functions φ ( x, t )such that for almost every t ∈ ( − T, T ), the function x φ ( x, t ) belongs to W ,ϑ (Ω) with the following normbeing finite: k φ k L ϑ ( − T,T ; W ,ϑ (Ω) := ˆ T − T k φ ( · , t ) k ϑW ,ϑ (Ω) dt ! ϑ < ∞ . Analogously, the parabolic space L ϑ ( − T, T ; W ,ϑ (Ω)) is the collection of measurable functions φ ( x, t ) suchthat for almost every t ∈ ( − T, T ), the function x φ ( x, t ) belongs to W ,ϑ (Ω).Given a weight ω ∈ A ϑ for some ϑ ∈ [1 , ∞ ), the weighted Lebesgue space L ϑ ( − T, T ; L ϑω (Ω)) is the set of allmeasurable functions φ : Ω T R satisfying ˆ T − T (cid:18) ˆ Ω | φ ( x, t ) | ϑ ω ( x, t ) dx (cid:19) dt < ∞ . Let us recall the following important characterization of Lebesgue spaces:
Lemma 2.12.
Let Ω be a bounded domain in R n and let w ∈ L (Ω T ) be any non-negative function, then for all β > α > and any non-negative measurable function g ( x, t ) : Ω T R , there holds ¨ Ω T g β w ( z ) dz = β ˆ ∞ λ β − w ( { z ∈ Ω T : g ( z ) > λ } ) dλ = ( β − α ) ˆ ∞ λ β − α − ¨ { z ∈ Ω T : g ( z ) >λ } g α w ( z ) dz ! dλ. Before we conclude this subsection, let us now recall the well known Poincar´e’s inequality (see [2, Corollary8.2.7] for the proof):
Theorem 2.13.
Let ≤ ϑ < ∞ and let f ∈ W ,ϑ ( ˜Ω) for some bounded domain ˜Ω and suppose that the followingmeasure density condition holds: (cid:12)(cid:12)(cid:12) { x ∈ ˜Ω : f ( x ) = 0 } (cid:12)(cid:12)(cid:12) ≥ m e > , then there holds ˆ ˜Ω (cid:12)(cid:12)(cid:12)(cid:12) f diam( ˜Ω) (cid:12)(cid:12)(cid:12)(cid:12) ϑ dx ≤ C ( n,ϑ,m e ) ˆ ˜Ω |∇ f | ϑ dx. Let us define the Parabolic metric on R n +1 that will be used throughout the paper: Definition 2.14.
We define the parabolic metric d p on R n +1 as follows: Let z = ( x , t ) and z = ( x , t ) beany two points on R n +1 , then d p ( z , z ) := max n | x − x | , p | t − t | o . For any f ∈ L ( R n +1 ), let us now define the strong maximal function in R n +1 as follows: M ( | f | )( x, t ) := sup ˜ Q ∋ ( x,t ) −− ¨ ˜ Q | f ( y, s ) | dy ds, (2.7)where the supremum is taken over all parabolic cylinders ˜ Q a,b with a, b ∈ R + such that ( x, t ) ∈ ˜ Q a,b . Anapplication of the Hardy-Littlewood maximal theorem in x − and t − directions shows that the Hardy-Littlewoodmaximal theorem still holds for this type of maximal function (see [28, Lemma 7.9] for details): Lemma 2.15. If f ∈ L ( R n +1 ) , then for any α > , there holds |{ z ∈ R n +1 : M ( | f | )( z ) > α }| ≤ n +2 α k f k L ( R n +1 ) , and if f ∈ L ϑ ( R n +1 ) for some < ϑ ≤ ∞ , then there holds kM ( | f | ) k L ϑ ( R n +1 ) ≤ C ( n,ϑ ) k f k L ϑ ( R n +1 ) . .8. Notation We shall clarify the notation that will be used throughout the paper:(i) We shall use ∇ to denote derivatives with respect the space variable x .(ii) We shall sometimes alternate between using dfdt , ∂ t f and f ′ to denote the time derivative of a function f .(iii) We shall use D to denote the derivative with respect to both the space variable x and time variable t in R n +1 .(iv) Let z = ( x , t ) ∈ R n +1 be a point and ρ, s > λ ∈ (0 , ∞ ). We shall usethe following notation to denote the following regions: ( Q λρ ( z ) := Q ρ,λ − p ρ ( z ) for p ≥ ,Q λρ ( z ) := Q λ p − ρ,ρ ( z ) for p ≤ ,I s ( t ) := ( t − s, t + s ) ⊂ R , Q ρ,s ( z ) := B ρ ( x ) × I s ( t ) ⊂ R n +1 ,αQ ρ,s ( z ) := B αρ ( x ) × I α s ( t ) ⊂ R n +1 , Q ρ ( z ) := B ρ ( x ) × I ρ ( t ) ⊂ R n +1 ,Q λ, + ρ ( z ) := Q λρ ( z ) ∩ { ( x, t ) : x n > } , Q + ρ ( z ) := Q ρ ( z ) ∩ { ( x, t ) : x n > } ,K λρ ( z ) := Q λρ ( z ) ∩ Ω T , K ρ ( z ) := Q ρ ( z ) ∩ Ω T . (2.8)(v) We shall use ˆ to denote the integral with respect to either space variable or time variable and use ¨ todenote the integral with respect to both space and time variables simultaneously.Analogously, we will use and −− ¨ to denote the average integrals as defined below: for any set A × B ⊂ R n × R , we define ( f ) A := A f ( x ) dx = 1 | A | ˆ A f ( x ) dx, ( f ) A × B := −− ¨ A × B f ( x, t ) dx dt = 1 | A × B | ¨ A × B f ( x, t ) dx dt. (vi) Given any positive function µ , we shall denote ( f ) µ := ˆ f µ k µ k L dm where the domain of integration isthe domain of definition of µ and dm denotes the associated measure. For this subsection, let us consider the following general problem: φ t − div A ( z, ∇ φ ) = − div | ~f | p − ~f in ˜Ω T ,φ = f on ∂ ˜Ω × ( − T, T ) ,φ ( · , − T ) = f on ˜Ω . (2.9)Now let us define the Steklov average as follows: let h ∈ (0 , T ) be any positive number, then we define φ h ( · , t ) := −− ˆ t + ht φ ( · , τ ) dτ t ∈ ( − T, T − h ) , . (2.10) Definition 2.16 (Weak solution) . We then say φ ∈ L ( − T, T ; L (Ω)) ∩ L p ( − T, T ; W ,p (Ω)) is a weak solutionof (1.1) if the following holds for any ψ ∈ W ,p (Ω) ∩ L ∞ (Ω) : ˆ Ω ×{ t } d [ φ ] h dt ψ + h [ A ( x, t, ∇ φ )] h , ∇ ψ i dx = ˆ Ω ×{ t } h| ~f | p − ~f , ∇ ψ i dx for a.e. − T < t < T − h. (2.11) Moreover, the initial datum is taken in the sense of L (Ω) , i.e., ˆ Ω | φ h ( x, − T ) − f ( x ) | dx h ց −−−→ .
7e have the following well known existence result (for example, see [33, Chapter III, Section 6] for the details):
Proposition 2.17.
Let Ω be any bounded domain satisfying a uniform measure density condition, i.e., thereexists a constant m e > such that | B r ( y ) ∩ Ω | ≥ m e | B r ( y ) | holds for every r > and y ∈ ∂ Ω and suppose that ~f ∈ L p (Ω T ) , ∇ f ∈ L p (Ω T ) with dfdt ∈ (cid:0) W ,p (Ω T ) (cid:1) ′ and f ∈ L (Ω) are given. Then there exists a unique weaksolution φ ∈ C (cid:0) − T, T ; L (Ω) (cid:1) ∩ L p (cid:0) − T, T ; W ,p (Ω) (cid:1) solving (2.9) .Moreover if f = 0 , then we have the following energy estimate sup − T ≤ t ≤ T k φ ( · , t ) k L (Ω) + ¨ Ω T |∇ φ | p dz ≤ C ( n,p, Λ , Λ ) (cid:18) ¨ Ω T | ~f | p dz + k f k L (Ω) (cid:19) . In this subsection, let us collect a few important higher integrability results that will be used throughoutthe paper. In order to state the general theorems, let φ ∈ L ( − T, T ; L (Ω)) ∩ L p (cid:16) − T, T ; W ,p (Ω) (cid:17) be a weaksolution of ( φ t − div A ( x, t, ∇ φ ) = − div( | ~f | p − ~f ) in Ω × ( − T, T ) ,φ = 0 on ∂ Ω × ( − T, T ) , (2.12)where the nonlinearity is assumed to satisfy (2.2) and (2.3). Here the domain is assumed to satisfy a uniformmeasure density condition with constant m e as in Lemma 2.3The first one is the higher integrability above the natural exponent . In the interior case, this was proved in[25] whereas in the boundary case, using the measure density condition satisfied by Ω, the result was proved in[29, 31]. Lemma 2.18 ([29, 31]) . Let ˜ σ > be given, then there exists a ˜ β = ˜ β ( n, p, Λ , Λ , m e ) ∈ (0 , ˜ σ ] such that if ~f ∈ L p (1+˜ σ ) (Ω T ) and φ ∈ L p (cid:16) − T, T ; W ,p (Ω) (cid:17) is a weak solution to (2.12) , then |∇ φ | ∈ L p (1+ β ) (Ω T ) for all β ∈ (0 , ˜ β ] . Moreover, for any z ∈ Ω × ( − T, T ) , there holds −− ¨ K ρ ( z ) |∇ φ | p + β dz ≤ ( n,p, Λ , Λ ,m e ) −− ¨ K ρ ( z ) (cid:16) |∇ φ | + | ~f | (cid:17) p dz ! β ˜ ϑ + −− ¨ K ρ ( z ) (cid:16) | ~f | + 1 (cid:17) p (1+ β ) dz. Here the constant ˜ ϑ := p if p ≥ , pp ( n + 2) − n if nn + 2 < p < . We will also need an improved higher integrability result below the natural exponent. The following theoremwas proved for a weaker class of solutions called very weak solutions , but also holds true for weak solutions asconsidered in this paper. The interior higher integrability result was proved in the seminal paper [26] whereasthe boundary analogue was proved in [3].
Lemma 2.19 ([26, 3]) . Let ~f ∈ L p (Ω T ) and φ ∈ L p (cid:16) − T, T ; W ,p (Ω) (cid:17) be the unique weak solution to (2.12) .There exists ˜ β = ˜ β ( n, Λ , Λ , p, m e ) ∈ (0 , / such that for any z ∈ Ω × ( − T, T ) , there holds −− ¨ K ρ ( z ) |∇ φ | p dz ≤ ( n, Λ , Λ ,p,m e ) −− ¨ K ρ ( z ) (cid:16) |∇ φ | + | ~f | (cid:17) p − β dz ! β ˜ ϑ + −− ¨ K ρ ( z ) (cid:16) | ~f | + 1 (cid:17) p dz. Here the constant ˜ ϑ := − β if p ≥ ,p − β − (2 − p ) n if nn + 2 < p < .
3. Main Results
In this section, let us describe the main theorem that will be proved. The first is unweighted a priori estimatesbelow the natural exponent. 8 heorem 3.1.
Let Ω be a bounded domain satisfying (2.4) , then there exists an β = β ( p, n, Λ , Λ ) ∈ (0 , such for any β ∈ (0 , β ) , the following holds: For any f ∈ L p (Ω T ) , let u ∈ C ( − T, T ; L (Ω)) ∩ L p ( − T, T ; W ,p (Ω)) be the unique weak solution of (1.1) , then there holds −− ¨ Ω T |∇ u | p − β dz > ( n,p,β, Λ , Λ ) −− ¨ Ω T | f | p − β dz. Remark 3.2.
As a corollary, we can extend the results of [21, Theorem 1.6] to obtain Lorentz space estimatesbelow the natural exponent. The techniques that we develop to prove Theorem 3.1 can be used to obtain theparabolic analogue of [4, Theorem 1.2] for weak solutions . In a forthcoming paper, we obtain these results formore general solutions called very weak solutionsThe second theorem we will prove is the end point weighted estimate. As mentioned in the introduction, themain contribution is the case q = p . Theorem 3.3.
Let q ∈ [ p, ∞ ) and w ∈ A qp be a Muckenhoupt weight, then there exists a positive constants ϑ = ϑ (Λ , Λ , n, p, Ω) and γ = γ ( n, Λ , Λ , p, q ) such that the following holds: Suppose ( A , Ω) is ( γ, S ) vanishing forsome fixed S > , then the problem (1.1) has a unique weak solution u satisfying the estimate ¨ Ω T |∇ u | q w ( z ) dz > ( n, Λ , Λ ,p,q, [ w ] qp , Ω) (cid:18) ¨ Ω T | f | q w ( z ) dz + 1 (cid:19) ϑ .
4. Construction of test function via Lipschitz truncation
In this section, we will consider the following two problems: Let ~f ∈ L p (Ω T ) be given and suppose that ϕ ∈ L ( − T, T ; L (Ω)) ∩ L p ( − T, T ; W ,p (Ω)) is a weak solution of n ϕ t − div A ( x, t, ∇ ϕ ) = div | ~f | p − ~f in Ω × ( − T, T ) . (4.1)We will extend ϕ = 0 on Ω c × ( − T, T ), then for any fixed cylinder Q ρ,s ( z ) ⊂ R n × ( − T, T ), we see from Proposition2.17 that for any ~g ∈ L p ( Q ρ,s ( z )), there exists a unique weak solution φ ∈ L p ( I s ( t ); W ,p ( B ρ ( x ))) solving ( φ t − div A ( x, t, ∇ φ ) = div | ~g | p − ~g in Q ρ,s ( z ) ,φ = ϕ on ∂ p Q ρ,s ( z ) . (4.2)From (4.1), we see that the condition φ = ϕ on ∂ p Q ρ,s ( z ) makes sense.In Section 5, we obtain difference estimates below the natural exponent between equations of the form (4.1)and (4.2). In order to do this, we need to construct a suitable test function which will be done in this section. We shall recall the following well known lemmas that will be used throughout this section. The first one is astandard lemma regarding integral averages (for a proof in this setting, see for example [8, Chapter 8.2] for thedetails).
Lemma 4.1.
Let λ > be any fixed number and suppose [ ψ ] h ( x, t ) := −− ˆ t + λh t − λh ψ ( x, τ ) dτ for some ψ ∈ L loc .Then we have the following properties:(i) [ ψ ] h → ψ a.e ( x, t ) ∈ R n +1 as h ց .(ii) [ ψ ] h ( x, · ) is continuous and bounded in time for a.e. x ∈ R n .(iii) For any cylinder Q r,λr ⊂ R n +1 with r > , there holds −− ¨ Q r,γr [ ψ ] h ( x, t ) dx dt > n −− ¨ Q r,λ ( r + h )2 ψ ( x, t ) dx dt. (iv) The function [ ψ ] h ( x, t ) is differentiable with respect to t ∈ R , moreover [ ψ ] h ( x, · ) ∈ C ( R ) for a.e. x ∈ R n . Let us now prove a time localized version of the Parabolic Poincar´e inequality.9 emma 4.2.
Let ψ ∈ L ϑ ( − T, T ; W ,ϑ (Ω)) with ϑ ∈ [1 , ∞ ) and suppose that B r ⋐ Ω be compactly contained ballof radius r > . Let I ⊂ ( − T, T ) be a time interval and ρ ( x, t ) ∈ L ( B r × I ) be any positive function such that k ρ k L ∞ ( B r × I ) > n | B r × I |k ρ k L ( B r × I ) , and µ ( x ) ∈ C ∞ c ( B r ) be such that ˆ B r µ ( x ) dx = 1 with | µ | > r n and |∇ µ | > r n +1 , then there holds: −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( z ) χ J − (cid:16) ψχ J (cid:17) ρ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ dz > ( n,ϑ ) −− ¨ B r × I |∇ ψ | ϑ χ J dz + sup t ,t ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ψχ J (cid:17) µ ( t ) − (cid:16) ψχ J (cid:17) µ ( t ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ , where ( ψ ) ρ := ˆ B r × I ψ ( z ) ρ ( z ) k ρ k L ( B r × I ) χ J dz , (cid:16) ψχ J (cid:17) µ ( t i ) := ˆ B r ψ ( x, t i ) µ ( x ) χ J dx and J ⋐ ( −∞ , ∞ ) be somefixed time-interval.Proof. Let us first consider the case of ρ ( x, t ) = µ ( x ) χ I ( t ). In this case, we get −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( z ) χ J − (cid:16) ψχ J (cid:17) µ × χ I r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ dz > −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( z ) χ J − (cid:16) ψχ J (cid:17) µ ( t ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ψχ J (cid:17) µ ( t ) − (cid:16) ψχ J (cid:17) µ × I r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ dz ( a ) > −− ¨ B r × I |∇ ψ | ϑ χ J dz + sup t ,t ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ψχ J (cid:17) µ ( t ) − (cid:16) ψχ J (cid:17) µ ( t ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ = −− ¨ B r × I |∇ ψ | ϑ χ J dz + sup t ,t ∈ I ∩ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ψ ) µ ( t ) − ( ψ ) µ ( t ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ . To obtain ( a ) above, we made us of the standard Poincar´e’s inequality in the spatial direction which only needsto be applied over a.e. t ∈ I ∩ J . Note that the derivative is only in the spatial direction and hence the term χ J does not cause any problem when applying Poincar´e’s inequality.For the general case, we observe that −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψχ J − (cid:16) ψχ J (cid:17) ρ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ dz > −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψχ J − (cid:16) ψχ J (cid:17) µ × χ I r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ dz + −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ψχ J (cid:17) ρ − (cid:16) ψχ J (cid:17) µ × χ I r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ dz. (4.3)The first term of (4.3) can be controlled as in (4.1) and to control the second term, we observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ψχ J (cid:17) ρ − (cid:16) ψχ J (cid:17) µ × χ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ρ k L ∞ ( B r × I ) k ρ k L ( B r × I ) ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψχ J − (cid:16) ψχ J (cid:17) µ × χ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz > −− ¨ B r × I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψχ J − (cid:16) ψχ J (cid:17) µ × χ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz. This completes the proof of the Lemma.
Remark 4.3.
In Lemma 4.2, we can take any bounded region ˜Ω instead of B r such that ˜Ω admits the ϑ -Poincar´einequality. For example, if ˜Ω satisfies the measure density condition as defined in Definition 2.3 for some m e > ,then Lemma 4.2 is applicable. We will use the following result which can be found in [23, Theorem 3.1] (see also [18]) for proving theLipschitz regularity for the constructed test function. This very useful simplification of the original techniquefrom [26] first appeared in [9, Chapter 3].
Lemma 4.4.
Let γ > and D ⊂ R n +1 be given. For any z ∈ D and r > , let Q r,γr ( z ) be the parabolic cylindercentered at z with radius r . Suppose there exists a constant C > independent of z and r such that the followingbound holds: | Q r,γr ( z ) ∩ D| ¨ Q r,γr ( z ) ∩D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x, t ) − ( f ) Q r,γr ( z ) ∩D r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx dt ≤ C ∀ z ∈ D and r > , then f is Lipschitz with respect to the metric d ( z , z ) := max {| x − x | , p γ − | t − t |} . .2. Construction of test function Let us denote the following functions: v ( z ) := ϕ ( z ) − φ ( z ) and v h ( z ) := [ ϕ − φ ] h ( z ) . where [ · ] h denotes the usual Steklov average. From Lemma 4.1, we see that v h h ց −−−→ v . It is easy to see from(4.2) that v ( z ) = 0 for z ∈ ∂ p Q ρ,s ( z ).Let us fix the following exponents for this Section:1 < q < p − β < p − β < p, (4.4)for some β ∈ (0 , β = β ( n, p, Λ , Λ , m e ) such that all the estimateshold for any β ∈ (0 , β ).Let us now define the following function: g ( z ) := M (cid:16)h |∇ v | q + |∇ ϕ | q + |∇ φ | q + | ~f | q + | ~g | q i χ Q ρ,s ( z ) (cid:17) q ( z ) , where M is as defined in (2.7).For a fixed λ >
0, let us define the good set by E λ := { ( x, t ) ∈ R n +1 : g ( x, t ) ≤ λ } . We now have the following parabolic Whitney type decomposition of E cλ (see [20, Lemma 3.1] or [9, Chapter3] for details): Lemma 4.5.
Let κ := λ − p , then there exists an κ -parabolic Whitney covering { Q i ( z i ) } of E cλ in the followingsense: (W1) Q j ( z j ) = B j ( x j ) × I j ( t j ) where B j ( x j ) = B r j ( x j ) and I j ( t j ) = ( t j − κr j , t j + κr j ) . (W2) we have d λ ( z j , E λ ) = 16 r j . (W3) [ j Q j ( z j ) = E cλ . (W4) for all j ∈ N , we have Q j ⊂ E cλ and Q j ∩ E λ = ∅ . (W5) if Q j ∩ Q k = ∅ , then r k ≤ r j ≤ r k . (W6) Q j ∩ Q k = ∅ for all j = k . (W7) X j χ Q j ( z ) ≤ c ( n ) for all z ∈ E cλ .Subject to this Whitney covering, we have an associated partition of unity denoted by { Ψ j } ∈ C ∞ c ( R n +1 ) suchthat the following holds: (W8) χ Q j ≤ Ψ j ≤ χ Q j . (W9) k Ψ j k ∞ + r j k∇ Ψ j k ∞ + r j k∇ Ψ j k ∞ + λr j k ∂ t Ψ j k ∞ ≤ C .For a fixed k ∈ N , let us define A k := (cid:26) j ∈ N : 34 Q k ∩ Q j = ∅ (cid:27) , then we have (W10) Let i ∈ N be given, then X j ∈ A i Ψ j ( z ) = 1 for all z ∈ Q i . (W11) Let i ∈ N be given and let j ∈ A i , then max {| Q j | , | Q i |} ≤ C ( n ) | Q j ∩ Q i | . (W12) Let i ∈ N be given and let j ∈ A i , then max {| Q j | , | Q i |} ≤ (cid:12)(cid:12)(cid:12)(cid:12) Q j ∩ Q i (cid:12)(cid:12)(cid:12)(cid:12) . W13)
For any i ∈ N , we have A i ≤ c ( n ) . (W14) Let i ∈ N be given, then for any j ∈ A i , we have Q j ⊂ Q i . Now we define the following Lipschitz extension function as follows: v λ,h ( z ) := v h ( z ) − X i Ψ i ( z )( v h ( z ) − v ih ) , (4.5)where v ih := k Ψ i k L ( Q i ) ¨ Q i v h ( z )Ψ i ( z ) χ [ t − s, t + s ] dz if 34 Q i ⊂ B ρ ( x ) × [ t − s, ∞ ) , . (4.6)Since ϕ − φ = 0 on ∂B ρ ( x ) × [ t − s, t + s ], we can switch between χ [ t − s, t + s ] and χ Q ρ,s ( z ) without affecting thecalculations. Remark 4.6.
Note that even though v h ( x, t − s ) = 0 in general, nevertheless the following initial boundary valuesare satisfied: • The initial condition ( ϕ − φ )( x, t − s ) = 0 is to be understood in the sense [ ϕ − φ ] h ( · , t − s ) h ց −−−→ in L ( B ρ ( x )) . • For ( x, t − s ) ∈ E λ , we have v λ,h ( x, t − s ) = v h ( x, t − s ) . • For ( x, t − s ) / ∈ E λ , we have v λ,h ( x, t − s ) = 0 by using (4.6) . Remark 4.7.
From Lemma 4.1, we see that v λ,h ( z ) h ց −−−→ v λ ( z ) almost everywhere. We now have the following useful lemma that can be proved just by using the definition of the weak formulation(see for example [3, Lemma 3.5] for details):
Lemma 4.8.
Let ϕ, φ, ~f , ~g be as in (4.1) and (4.2) and h ∈ (0 , s ) . Let α ( x ) ∈ C ∞ c ( B ρ ( x )) and β ( t ) ∈ C ∞ ( t − s, t + s ) with β ( t − s ) = 0 be a non-negative function and [ · ] h be the Steklov average as defined in (4.1) . Then thefollowing estimate holds for any time interval ( t , t ) ⊂ ( t − s, t + s ) : | ( v h β ) α ( t ) − ( v h β ) α ( t ) | ≤ C (Λ , p ) k∇ α k L ∞ ( B ρ ( x )) k β k L ∞ ( t ,t ) ¨ B ρ ( x ) × ( t ,t ) [ |∇ φ | p − + |∇ ϕ | p − ] h dz + k∇ α k L ∞ ( B ρ ( x )) k β k L ∞ ( t ,t ) ¨ B ρ ( x ) × ( t ,t ) [ | ~f | p − + | ~g | p − ] h dz + k φ k L ∞ ( B ρ ( x )) k ϕ ′ k L ∞ ( t ,t ) ¨ B ρ ( x ) × ( t ,t ) | [ ϕ − φ ] h | dz. Lemma 4.9.
For any z ∈ E cλ , we have | v λ,h ( z ) | > ( n,p,q, Λ , Λ ,b ) ρλ. (4.7) Proof.
By construction of the extension in (4.5), for z ∈ E cλ , we see that v λ,h ( z ) = X j Ψ j ( z ) v jh with v jh = 0whenever 34 Q j * B ρ ( x ) × [ t − s, ∞ ).In order to prove the Lemma, making use of ( W8 ), we see that (4.7) follows if the following holds: | v jh | > ( n,p,q, Λ , Λ ,b ) ρλ. (4.8)We shall now proceed with proving (4.8). Since we only have to consider the case 34 Q j ⊂ B ρ ( x ) × [ t − s, ∞ ),which automatically implies r j > ρ . We now proceed as follows:12 ase r j ≥ ρ : In this case, we observe that B ρ ( x ) ⊂ B j which gives the following sequence of estimates: | v jh | > r j −− ¨ Q j (cid:12)(cid:12)(cid:12)(cid:12) [ ϕ − φ ] h ( z ) r j (cid:12)(cid:12)(cid:12)(cid:12) χ [ t − s, t + s ] dz ( a ) > ρ | I j | ˆ I j ∩ [ t − s, t + s ] −− ˆ B j (cid:12)(cid:12)(cid:12)(cid:12) [ ϕ − φ ] h ( x, t ) r j (cid:12)(cid:12)(cid:12)(cid:12) q dx ! q dt ( b ) > ρ | I j | ˆ I j ∩ [ t − s, t + s ] −− ˆ B j |∇ v h ( x, t ) | q χ [ t − s, t + s ] dx ! q dt ( c ) > ρλ. To obtain (a), we used the fact that r j > ρ along with H¨older’s inequality, to obtain (b), we made use ofPoincar´e’s inequality and finally to obtain (c), we made use of ( W4 ). Case r j ≤ ρ : In this case, we gradually enlarge 34 Q i until it goes outside B ρ ( x ) × [ − s, ∞ ). As a consequence,we have to further consider two subcases, the first where 2 ˜ k Q j crosses the lateral boundary first, and the secondwhen 2 ˜ k Q j crosses the initial boundary first.Let us define the following constant k := min { ˜ k , ˜ k } where ˜ k and ˜ k satisfy2 ˜ k − r j < ρ ≤ ˜ k r j , ˜ k − Q j ⊂ B ρ ( x ) × [ t − s, ∞ ) but 2 ˜ k Q j * B ρ ( x ) × [ t − s, ∞ ) . (4.9)Note that k denotes the first scaling exponent under which either we end up in the situation r j ≥ k ρ or2 k Q j goes outside B ρ ( x ) × [ t − s, ∞ ).Since we only consider the case 34 Q i ⊂ B ρ ( x ) × [ t − s, ∞ ), using triangle inequality, we get | v jh | > k − X m =0 (cid:18)(cid:16) [ ϕ − φ ] h χ Q ρ,s ( z ) (cid:17) m Q j − (cid:16) [ ϕ − φ ] h χ Q ρ,s ( z ) (cid:17) m +1 Q j (cid:19) + (cid:16) [ ϕ − φ ] h χ Q ρ,s ( z ) (cid:17) k − Q j := k − X m =0 S m + S . (4.10)We shall estimate S m and S separately as follows: Estimate for S m : In this case, we see that 2 m +1 Q j ⊂ B ρ ( x ) × [ t − s, ∞ ). Thus applying Lemma 4.2 for any µ ∈ C ∞ c ( B m +1 r j ( x j )) satisfying | µ ( x ) | ≤ C ( n )(2 m +1 r j ) n and |∇ µ ( x ) | ≤ C ( n )(2 m +1 r j ) n +1 , we get S m > (2 m +1 r j ) −− ¨ m +1 Q j | [ ∇ ( ϕ − φ )] h | q χ Q ρ,s ( z ) dz ! q +(2 m +1 r j ) sup t ,t ∈ m +1 I j ∩ [ t − s, t + s ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ([ ϕ − φ ] h ) µ ( t ) − ([ ϕ − φ ] h ) µ ( t )2 m +1 r j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ! q ( W4 ) > (2 m +1 r j ) λ + (2 m +1 r j ) sup t ,t ∈ m +1 I j ∩ [ t − s, t + s ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ([ ϕ − φ ] h ) µ ( t ) − ([ ϕ − φ ] h ) µ ( t )2 m +1 r j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ! q . (4.11)To estimate the second term on the right of (4.11), using B m +1 r j ( x j ) ⊂ B ρ ( x ) × [ t − s, ∞ ), we can applyLemma 4.8 with the test function α ( x ) = µ ( x ) and β ( t ) = 1, which gives for any t , t ∈ I j ∩ [ t − s, t + s ], theestimate | ([ ϕ − φ ] h ) µ ( t ) − ([ ϕ − φ ] h ) µ ( t ) | ( a ) > m +1 r j (cid:0) κλ p − (cid:1) = 2 m +1 r j λ. (4.12)To obtain (a), we first applied Lemma 4.8 along with ( W1 ), ( W4 ) and the definition κ = λ − p .Substituting (4.12) into (4.11), we get S m > m +1 r j λ. (4.13)13 stimate for S : For this term, we know that 2 k − Q j / ∈ B ρ ( x ) × [ t − s, ∞ ), which implies 2 k − Q j crosseseither the lateral boundary ∂B ρ ( x ) × [ t − s, ∞ ) or crosses the initial boundary B ρ ( x ) × { t − s } first. We willconsider both the cases separately and estimate S as follows: In the case k − Q j crosses the lateral boundary ∂B ρ ( x ) × [ t − s, ∞ ) first , we can directly apply Theorem 2.13to obtain −− ¨ k − Q j [ ϕ − φ ] h χ Q ρ,s ( z ) dz > (2 k r j ) −− ¨ k Q j |∇ [ ϕ − φ ] h | q χ Q ρ,s ( z ) dz ! /q ( a ) > ρλ. (4.14)To obtain (a), we made use of ( W4 ) along with 2 k − r j ≤ ρ given by (4.9). In the case k Q j crosses the initial boundary B ρ ( x ) × { t − s } first , by enlarging the cylinder to 2 k +1 Q j , wecan find a cut-off function θ ( x, t ) such that spt θ ( x, t ) ⊂ k +1 Q j ∩ R n × ( −∞ , t − s ), which combined with thefact v h ( z ) χ [ t − s, t + s ] = 0 on R n × ( −∞ , t − s ), we get (cid:16) v h χ [ t − s, t + s ] (cid:17) θ = 0. Thus applying Lemma 4.2, we get −− ¨ k Q j | v h ( z ) | χ [ t − s, t + s ] dz = −− ¨ k Q j (cid:12)(cid:12)(cid:12)(cid:12) v h ( z ) χ [ t − s, t + s ] − (cid:16) v h χ [ t − s, t + s ] (cid:17) θ (cid:12)(cid:12)(cid:12)(cid:12) dz > (2 k +1 r j ) −− ¨ k Q j | [ ∇ ( ϕ − φ )] h | q χ [ t − s, t + s ] dz ! q +(2 k +1 r j ) sup t ,t ∈ k I j ∩ [ t − s, t + s ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ([ ϕ − φ ] h ) µ ( t ) − ([ ϕ − φ ] h ) µ ( t )2 k +1 r j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ! q ( a ) > k +1 r j λ ( b ) > ρλ. (4.15)To obtain (a), we made use of ( W1 ),( W4 ) along with an application of Lemma 4.8 and to obtain (b), we used(4.9).Combining (4.14) and (4.15), we get S > ρλ. (4.16)Thus combining (4.13) and (4.16) into (4.10), we get | v jh | ≤ k − X m =0 S m + S > λ k − X m =0 m +1 r j + ρ ! (4.9) > ρλ. This completes the proof of the Lemma.Now we prove a sharper estimate.
Lemma 4.10.
For any j ∈ A i , there holds | v ih − v jh | > ( n,p,q, Λ , Λ ,m e ) min { ρ, r i } λ. Proof.
We only have to consider the case r i ≤ ρ because if ρ ≤ r i , we can directly use Lemma 4.9 to get therequired conclusion. If either v ih = 0 or v jh = 0 , then 34 Q i must necessarily intersect the lateral or initial boundary. Initial Boundary Case Q i ⊂ B ρ ( x ) × R : Without loss of generality, we can assume 2 Q i ⊂ B ρ ( x ) × R . We nowpick θ ( x, t ) ∈ C ∞ c ( R n +1 ) such that spt( θ ) ⊂ B i × ( −∞ , t − s ). Since ϕ − φ = 0 on 2 B i × ( −∞ , t − s ), we seethat (cid:16) v h χ [ t − s, t + s ] (cid:17) θ = (cid:16) [ ϕ − φ ] h χ [ t − s, t + s ] (cid:17) θ = 0. Thus we get | v ih | > −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12) [ ϕ − φ ] h χ [ t − s, t + s ] − (cid:16) [ ϕ − φ ] h χ [ t − s, t + s ] (cid:17) θ (cid:12)(cid:12)(cid:12)(cid:12) dz ( a ) > r i −− ¨ Q i |∇ v h | q χ [ t − s, t + s ] dz + sup t ,t ∈ I i ∩ [ t − s, t + s ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) v h χ [ t − s, t + s ] (cid:17) µ ( t ) − (cid:16) v h χ [ t − s, t + s ] (cid:17) µ ( t ) r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q ( b ) > r i λ.
14o obtain (a), we made use of Lemma 4.2 and to obtain (b), we proceed similarly to how (4.12) was estimated.
Lateral Boundary Case Q i ∩ ( B ρ ( x ) × R ) c = ∅ : In this case, using Theorem 2.13 and ( W4 ), we get | v ih | > r i −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ ϕ − φ ] h χ [ t − s, t + s ] r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dz ! q > r i (cid:18) −− ¨ Q i |∇ [ ϕ − φ ] h | q χ [ t − s, t + s ] dz (cid:19) q > r i λ. (4.17)From (4.17) and (4.8), we see that the lemma is proved provided v jh = 0. Now let us consider the case v ih = 0 and v jh = 0 , which implies 34 Q i ⊂ B ρ ( x ) × [ − s, ∞ ) and 34 Q j ⊂ B × [ t − s, ∞ ). From the definition of v ih in (4.6), triangle inequality and ( W12 ), we get | v ih − v jh | > | Q i || Q i ∩ Q j | −− ¨ Q i (cid:12)(cid:12)(cid:12) v h ( z ) χ [ t − s, t + s ] − v ih (cid:12)(cid:12)(cid:12) dz + | Q j || Q i ∩ Q j | −− ¨ Q j (cid:12)(cid:12)(cid:12) v h ( z ) χ [ t − s, t + s ] − v jh (cid:12)(cid:12)(cid:12) dz > −− ¨ Q i (cid:12)(cid:12)(cid:12) v h ( z ) χ [ t − s, t + s ] − v ih (cid:12)(cid:12)(cid:12) dz + −− ¨ Q j (cid:12)(cid:12)(cid:12) v h ( z ) χ [ t − s, t + s ] − v jh (cid:12)(cid:12)(cid:12) dz. (4.18)Let us now estimate each of the terms in (4.18) as follows: we apply H¨older’s inequality followed by Lemma 4.2with α ∈ C ∞ c (cid:18) B i (cid:19) with | α ( x ) | > r ni and |∇ α ( x ) | > r n +1 i to get −− ¨ Q i | v h ( z ) χ [ t − s, t + s ] − v ih | dz = r i −− ¨ Q i |∇ v h | q χ [ t − s, t + s ] dz ! q + r i sup t ,t ∈ I i ∩ [ t − s, t + s ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ([ ϕ − φ ] h ) µ ( t ) − ([ ϕ − φ ] h ) µ ( t ) r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ! q . (4.19)The first term on the right of (4.19) can be controlled using ( W4 ) and the second term can be controlledsimilarly as (4.12). Thus we get −− ¨ Q i | v h ( z ) χ [ t − s, t + s ] − v ih | dz > r i λ. This completes the proof of the Lemma.Once we have the bounds in Lemma 4.9 and Lemma 4.10, we can obtain the following important estimates:
Lemma 4.11.
Given any z ∈ E cλ , we have z ∈ Q i for some i ∈ N . Then there holds |∇ v λ,h ( z ) | ≤ C ( n,p,q, Λ , Λ ,m e ) λ. (4.20) Proof.
We observe that X j Ψ j ( z ) = X j : j ∈ A i Ψ j ( z ) = 1 for any z ∈ E cλ , which implies X j ∇ Ψ j ( z ) = 0 for all z ∈ E cλ .Thus using (4.5) along with ( W9 ), ( W13 ) and Lemma 4.10, we get |∇ v λ,h ( z ) | ≤ X j : j ∈ A i |∇ Ψ j ( z ) | (cid:12)(cid:12)(cid:12) v jh − v ih (cid:12)(cid:12)(cid:12) > λ. This completes the proof of the Lemma. v λ,h We will now mention some improved estimates which can be proved using H¨older’s inequality along with thetechniques from Lemma 4.11.
Lemma 4.12.
Let z ∈ E cλ and ε ∈ (0 , be any number, then z ∈ Q i for some i ∈ N from ( W1 ) . There existsa constant C = C ( n,p,q, Λ , Λ ,m e ) such that the following holds: | v λ,h ( z ) | ≤ C −− ¨ Q i | v h (˜ z ) | χ [ t − s, t + s ] d ˜ z ≤ Cr i λε + Cελr i −− ¨ Q i | v h (˜ z ) | χ [ t − s, t + s ] d ˜ z, |∇ v λ,h ( z ) | ≤ C r i −− ¨ Q i | v h (˜ z ) | χ [ t − s, t + s ] d ˜ z ≤ Cλε + Cελr i −− ¨ Q i | v h (˜ z ) | χ [ t − s, t + s ] d ˜ z. emma 4.13. Let z ∈ E cλ and ε ∈ (0 , be any number, then z ∈ Q i for some i ∈ N from ( W1 ) . There existsa constant C = C ( n,p,q, Λ , Λ ,m e ) such that the following holds: | v λ,h ( z ) | ≤ C (cid:0) min { ρ, r i } λ + | v ih | (cid:1) ≤ C (cid:18) r i λε + εr i λ | v ih | (cid:19) , (4.21) |∇ v λ,h ( z ) | ≤ C λε , (4.22) | ∂ t v λ,h ( z ) | ≤ C λ − p r i −− ¨ Q i | v h (˜ z ) | χ [ t − s, t + s ] d ˜ z, | ∂ t v λ,h ( z ) | ≤ C λ − p r i min { r i , ρ } λ. (4.23) v λ,h Lemma 4.14.
For any ϑ ≥ , we have the following bound: ¨ Q ρ,s ( z ) \ E λ | v λ,h ( z ) | ϑ dz > ( n,p,q, Λ , Λ ,m e ) ¨ Q ρ,s ( z ) \ E λ | v h ( z ) | ϑ χ [ t − s, t + s ] dz. Proof.
Since E cλ is covered by Whitney cylinders (see Lemma 4.5), let us pick some i ∈ N and consider thecorresponding parabolic Whitney cylinder. Using the construction from (4.5) along with ( W5 ), ( W9 ) and( W13 ), we get ¨ Q i | v λ,h ( z ) | ϑ dz > X j : j ∈ A i ¨ Q i Ψ j ( z ) ϑ | v jh | ϑ dz > ¨ Q i | v h ( z ) | ϑ χ [ t − s, t + s ] dz. (4.24)Summing (4.24) over all i ∈ N and making use of ( W4 ) and ( W7 ), we get ¨ Q ρ,s ( z ) \ E λ | v λ,h ( z ) | ϑ dz > X i ¨ Q i | v h ( z ) | ϑ χ [ t − s, t + s ] dz > ¨ Q ρ,s ( z ) \ E λ | v h ( z ) | ϑ χ [ t − s, t + s ] dz. This proves the Lemma.
Lemma 4.15.
For any < ϑ ≤ q with q defined as in 4.4, there holds ¨ Q ρ,s ( z ) \ E λ | ∂ t v λ,h ( z )( v λ,h ( z ) − v h ( z )) | ϑ dz > ( n,p,q, Λ , Λ ,m e ,ϑ ) λ ϑp | R n +1 \ E λ | . Proof.
From ( W3 ), we see that Q ρ,s ( z ) \ E λ ⊂ [ i ∈ Z Q i , thus, for a given i ∈ N , let us define the following: J i := ¨ Q i | ∂ t v λ,h ( z )( v λ,h ( z ) − v h ( z )) | ϑ χ Q ρ,s ( z ) dz. Making use of (4.23) and H¨older’s inequality (recall γ = λ − p ), we get J i > (cid:18) λ − p r i r i λ (cid:19) ϑ ¨ Q i | v λ,h ( z ) χ Q ρ,s ( z ) − v h ( z ) χ Q ρ,s ( z ) | ϑ dz ( a ) > (cid:18) λ − p r i r i λ (cid:19) ϑ X j ∈ A i ¨ Q i | v h ( z ) χ Q ρ,s ( z ) − v jh | ϑ dz ( b ) > λ ϑp | Q i | . (4.25)To obtain (a), we made use of (4.5), ( W9 ) and ( W10 ) and to obtain (b), we applied Theorem 2.13 along with( W4 ).Summing (4.25) over all i ∈ N and making use of ( W7 ) completes the proof of the lemma. v λ,h We shall now prove the Lipschitz continuity of v λ,h on H := R n × [ t − s, t + s ]. Lemma 4.16.
The function v λ,h from (4.5) is C , ( H ) with respect to the parabolic metric given in Definition (2.14) . roof. Let us consider a parabolic cylinder Q r ( z ) := Q r,κr ( z ) := Q for some z ∈ H and r > κ = λ − p ).To prove the Lemma, we make use of Lemma 4.4 and prove the following bound: I r ( z ) := −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v λ,h (˜ z ) − (cid:16) v λ,h (cid:17) Q ∩H r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z ≤ o (1) , where o (1) denotes a constant independent of z ∈ H and r > Case Q ⊂ E cλ : In this case, from ( W3 ), we see that z ∈ Q i for some i ∈ N . From the construction in (4.5),we see that v λ,h ∈ C ∞ ( E cλ ) which combined with the mean value theorem gives I r ( z ) > r −− ¨ Q ∩H −− ¨ Q ∩H | v λ,h (˜ z ) − v λ,h (˜ z ) | d ˜ z d ˜ z > sup ˜ z ∈ Q ∩H (cid:16) |∇ v λ,h (˜ z ) | + λ − p r | ∂ t v λ,h (˜ z ) | (cid:17) . Let us pick some ˜ z ∈ Q ⊂ E cλ , then ˜ z ∈ Q j for some j ∈ N . Thus we can make use of (4.20) and (4.23)to get |∇ v λ,h (˜ z ) | + λ − p r | ∂ t v λ,h (˜ z ) | > λ + λ − p r λ − p r j r j λ. (4.26)In (4.26), we need to understand the relation between r j and r . To this end, from 2 Q ⊂ E cλ , we see that r ≤ d λ (˜ z , E λ ) ≤ d λ (˜ z , z j ) + d λ ( z j , E λ ) ≤ r j + 16 r j = 17 r i . (4.27)Combining (4.26) and (4.27), we get |∇ v λ,h (˜ z ) | + λ − p r | ∂ t v λ,h (˜ z ) | > λ. Case Q * E cλ : In this case, we shall split the proof into three subcases:
Subcase Q ⊂ R n × ( −∞ , t + s ] or Q ⊂ R n × [ t − s, ∞ ) : In this situation, it is easy to see that the fol-lowing holds: | Q ∩ H| ? | Q | . (4.28)We apply triangle inequality and estimate I r ( z ) by I r ( z ) ≤ −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v λ,h (˜ z ) − v h (˜ z ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) − ( v h ) Q ∩H r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( v h ) Q ∩H − (cid:16) v λ,h (cid:17) Q ∩H r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z ≤ J + J , (4.29)where we have set J := −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v λ,h (˜ z ) − v h (˜ z ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z and J := −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) − ( v h ) Q ∩H r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z. (4.30)We now estimate each of the terms of (4.30) as follows: Estimate for J : From (4.5), we get J > X i ∈ N | Q ∩ H| ¨ Q ∩H∩ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) χ [ t − s, t + s ] − v ih r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z. (4.31)Let us fix an i ∈ N and take two points ˜ z ∈ Q ∩ Q i and ˜ z ∈ E λ ∩ Q . Let z i denote the center of 34 Q i ,making use of ( W2 ) along with the trivial bound d λ (˜ z , ˜ z ) ≤ r and d λ ( z i , ˜ z ) ≤ r i , we get16 r i = d λ ( z i , E λ ) ≤ d λ ( z i , ˜ z ) + d λ (˜ z , ˜ z ) ≤ r i + 4 r = ⇒ r i ≤ r. (4.32)Note that (4.28) holds and thus summing over all i ∈ N such that Q ∩ H ∩ Q i = ∅ in (4.31) and making17se of (4.32), we get J > X i ∈ N Q ∩H∩ Q i = ∅ | Q i || Q ∩ H| −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) χ [ t − s, t + s ] − v ih r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z ( a ) > X i ∈ N −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) χ [ t − s, t + s ] − v ih r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z ( b ) > λ. To obtain (a), we made use of (4.28) and (4.32), to obtain (b), we follow the calculation from bounding(4.19).
Estimate for J : Note that Q ∩ H is another cylinder. If Q ⊂ B ρ ( x ) × R , then choose a cut-off function µ ∈ C ∞ c ( B ρ ( x )) and apply Lemma 4.2 to get J > −− ¨ Q ∩H |∇ v h | q χ Q ρ,s ( z ) + sup t ,t ∈ [ t − s, t + s ] ∩ Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) v h χ Q ρ,s ( z ) (cid:17) µ ( t ) − (cid:16) v h χ Q ρ,s ( z ) (cid:17) µ ( t ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q . Recall that we are in the case 2 Q ∩ E λ = ∅ and 2 Q ∩ E cλ = ∅ . Further applying Lemma 4.8 and proceedingas in (4.11), we get J > λ. (4.33) On the other hand, if Q * B ρ ( x ) × R , then we can apply Poincar´e’s inequality from Theorem 2.13 directlyand make use of the fact that 2 Q ∩ E λ = ∅ to get J > (cid:18) −− ¨ Q ∩H (cid:12)(cid:12)(cid:12) ∇ v h (˜ z ) χ [ t − s, t + s ] (cid:12)(cid:12)(cid:12) q d ˜ z (cid:19) q > λ. Subcase Q ∩ R n × ( −∞ , t − s ] = ∅ and Q ∩ R n × [ t + s, ∞ ) = ∅ AND κr ≤ s : In this case, we see that | Q ∩ H| = | B | r n × s. We apply triangle inequality and estimate I r ( z ) as we did in (4.29) to get I r ( z ) ≤ J + J , where we have set J := −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v λ,h (˜ z ) − v h (˜ z ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z and J := −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) − ( v h ) Q ∩H r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z. We estimate J as follows J > X i ∈ N | Q i || Q ∩ H| −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) χ [ t − s, t + s ] − v ih r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z (4.32) > r n +2 i κr n s X i ∈ N −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) χ [ t − s, t + s ] − v ih r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z (4.32) > r n +2 κr n s X i ∈ N −− ¨ Q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v h (˜ z ) χ [ t − s, t + s ] − v ih r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z ( a ) > r κs λ ( b ) > λ. To obtain (a), we proceed similarly to (4.19) and to obtain (b), we made use of κr ≤ s .The estimate for J is already obtained in (4.33) which shows J > λ. Subcase Q ∩ R n × ( −∞ , t − s ] = ∅ and Q ∩ R n × [ t + s, ∞ ) = ∅ AND κr > s : Using triangle inequal-18ty and the bound | Q ∩ H| = | B | r n × s , we get −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v λ,h (˜ z ) − (cid:16) v λ,h (cid:17) Q ∩H r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z > | Q ∩ H| ¨ Q ∩H | v λ,h (˜ z ) | d ˜ z > | Q ∩ H| ¨ Q ∩H∩ E λ | v λ,h (˜ z ) | d ˜ z + 1 | Q ∩ H| ¨ Q ∩H\ E λ | v λ,h (˜ z ) | d ˜ z. By construction of v λ,h in (4.5), we have v λ,h = v h on E λ . On Q ρ,s ( z ) \ E λ , we can apply Lemma 4.9 toobtain the following bound: −− ¨ Q ∩H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v λ,h (˜ z ) − ( Q ∩ H ) v λ,h r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ z > r n s ¨ Q ρ,s ( z ) | v h (˜ z ) | d ˜ z + 1 | Q ∩ H| ¨ Q ∩H\ E λ ρλ d ˜ z > (cid:16) κs (cid:17) n s k v h k L ( Q ρ,s ( z )) + ρλ > o (1) . This completes the proof of the Lipschitz regularity.
We shall now prove the first crucial estimate which holds on each time slice.
Lemma 4.17.
For any i ∈ N and any < ε ≤ , there exists a positive constant C ( n, p, q, Λ , Λ , m e ) such thatfor almost every t ∈ [ t − s, t + s ] , there holds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ ( x ) ( v ( x, t ) − v i ) v λ ( x, t )Ψ i ( x, t ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) λ p ε | Q i | + ε | B i || v i | (cid:19) . (4.34) Proof.
Let us fix any t ∈ [ t − s, t + s ], i ∈ N and take Ψ i ( y, τ ) v λ,h ( y, τ ) as a test function in (4.1) and (4.2).Further integrating the resulting expression over t i − κ (cid:18) r i (cid:19) , t ! or ( t − s, t ) depending on the location of34 Q i , along with making use of the fact that Ψ i ( y, t i − κ (3 r i / ) = 0 or v λ,h ( y, t − s ) = 0, we get for any a ∈ R ,the equality ˆ B ρ ( x ) (cid:16) ( v h − a )Ψ i v λ,h (cid:17) ( y, t ) dy = ˆ t max n t i − κ ( r i ) , t − s o ˆ B ρ ( x ) ∂ t (cid:16) ( v h − a )Ψ i v λ,h (cid:17) ( y, τ ) dy dτ = ˆ t max n t i − κ ( r i ) , t − s o ˆ B ρ ( x ) ∂ t (cid:16) [ ϕ − φ ] h Ψ i v λ,h − a Ψ i v λ,h (cid:17) ( y, τ ) dy dτ = ˆ t max n t i − κ ( r i ) , t − s o ˆ B ρ ( x ) h [ A ( y, τ, ∇ φ )] h − [ A ( y, τ, ∇ ϕ )] h , ∇ (Ψ i v λ,h ) i dy dτ + ˆ t max n t i − κ ( r i ) , t − s o ˆ B ρ ( x ) h [ | ~f | p − ~f + | ~g | p − ~g ] h , ∇ (Ψ i v λ,h ) i dy dτ − ˆ t max n t i − κ ( r i ) , t − s o ˆ B ρ ( x ) a∂ t (cid:16) Ψ i v λ,h (cid:17) dy dτ. (4.35)We can estimate |∇ (Ψ i v λ ) | using the chain rule and ( W9 ), to get |∇ (Ψ i v λ,h ) | > r i | v λ | + |∇ v λ | . (4.36)Similarly, we can estimate (cid:12)(cid:12) ∂ t (cid:0) Ψ i v λ (cid:1)(cid:12)(cid:12) using the chain rule and ( W9 ), to get (cid:12)(cid:12) ∂ t (cid:0) Ψ i v λ (cid:1)(cid:12)(cid:12) > κr i | v λ | + | ∂ t v λ | . (4.37)Let us take a = v ih in the (4.35) followed by letting h ց (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ ( x ) (cid:0) ( v − v i )Ψ i v λ (cid:1) ( y, t ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > J + J + J , (4.38)19here we have set J := 1 r i ¨ Q ρ,s ( z ) (cid:16) |∇ ϕ | p − + |∇ φ | p − + | ~f | p − + | ~g | p − (cid:17) | v λ | χ Q i ∩ Q ρ,s ( z ) dy dτ,J := ¨ Q ρ,s ( z ) (cid:16) |∇ ϕ | p − + |∇ φ | p − + | ~f | p − + | ~g | p − (cid:17) |∇ v λ | χ Q i ∩ Q ρ,s ( z ) dy dτ,J := ¨ Q ρ,s ( z ) | v − v i || ∂ t (Ψ i v λ ) | χ Q i ∩ Q ρ,s ( z ) dy dτ. (4.39)Let us now estimate each of the terms as follows: Bound for J : If ρ ≤ r i , we can directly use H¨older’s inequality, Lemma 4.9 and ( W4 ), to find that for any ε ∈ (0 , J > λ | Q i | (cid:18) −− ¨ Q i (cid:16) |∇ ϕ | q + |∇ φ | q + | ~f | q + | ~g | q (cid:17) χ Q ρ,s ( z ) dy dτ (cid:19) p − q > λ p ε | Q i | . (4.40)In the case r i ≤ ρ , we make use of (4.21), ( W4 ) along with the fact | Q i | = | B i | × λ − p r i , to get J > r i (cid:18) r i λε + ελr i | v i | (cid:19) | Q i | (cid:18) −− ¨ Q i (cid:16) |∇ ϕ | q + |∇ φ | q + | ~f | q + | ~g | q (cid:17) χ Q ρ,s ( z ) dy dτ (cid:19) p − q > r i (cid:18) r i λε + ελr i | v i | (cid:19) | Q i | λ p − > λ p ε | Q i | + ε | B i || v i | . (4.41)Thus combining (4.41) and (4.40), we get J > λ p ε | Q i | + χ r i ≤ ρ ε | B i || v i | , (4.42)where we have set χ r i ≤ ρ = 1 if r i ≤ ρ and χ r i ≤ ρ = 0 else. Bound for J : In this case, we can directly use Lemma 4.11 and ( W4 ) to get for any ε ∈ (0 , J > λ | Q i | ε (cid:18) −− ¨ Q i (cid:16) |∇ ϕ | q + |∇ φ | q + | ~f | q + | ~g | q (cid:17) χ Q ρ,s ( z ) dy dτ (cid:19) p − q > λ p ε | Q i | . (4.43) Bound for J : Substituting (4.22), (4.23) and ( W9 ) into (4.37), for any ε ∈ (0 , | ∂ t (Ψ i v λ )( z ) | > κr i (cid:18) r i λε + εr i λ | v i | (cid:19) + 1 κr i min { r i , ρ } λ ≈ κr i (cid:18) r i λε + εr i λ | v i | (cid:19) . (4.44)Making use of (4.44) in the expression for J in (4.39), we get J > κr i (cid:18) r i λε + εr i λ | v i | (cid:19) ¨ Q i | v − v i | χ Q ρ,s ( z ) dy dτ. We can now proceed similarly to (4.19) to get J > κr i (cid:18) r i λε + εr i λ | v i | (cid:19) r i λ | Q i | > λ p ε | Q i | + ε | B i || v i | . (4.45)Substituting the estimates (4.42), (4.43) and (4.45) into (4.38) gives the proof of (4.34).We now come to essentially the most important estimate which will be needed to prove the difference estimate: Lemma 4.18.
There exists a positive constant C ( n, p, q, Λ , Λ , m e ) such that the following estimate holds forevery t ∈ [ − s, s ] : ˆ B ρ ( x ) \ E tλ ( | v | − | v − v λ | )( x, t ) dx ≥ − Cλ p | R n +1 \ E λ | . (4.46) Proof.
Let us fix any t ∈ [ t − s, t + s ] and any point x ∈ B ρ ( x ) \ E tλ . Now defineΥ := (cid:8) i ∈ N : spt(Ψ i ) ∩ B ρ ( x ) × { t } 6 = ∅ and | v | + | v λ | 6 = 0 on spt(Ψ i ) ∩ ( B ρ ( x ) × { t } ) (cid:9) . Hence we only need to consider i ∈ Υ. Note that X i ∈ Υ Ψ i ( · , t ) ≡ R n ∩ E tλ , we can rewrite the left-hand20ide of (4.46) as ˆ B ρ ( x ) \ E tλ ( | v | − | v − v λ | )( x, t ) dx = X i ∈ Υ ˆ B ρ ( x ) Ψ i ( | v | − | v − v λ | ) dx = X i ∈ Υ ˆ B ρ ( x ) Ψ i ( z ) (cid:0) | v i | + 2 v λ ( v − v i ) (cid:1) dx − X i ∈ Υ ˆ B ρ ( x ) Ψ i ( z ) | v λ − v i | dx := J + J . Estimate of J : Using (4.34), we get J ? X i ∈ Υ ˆ B ρ ( x ) Ψ i ( z ) | v i | dz − ε X i ∈ Υ | B i || v i | − X i ∈ Υ λ p ε | Q i | . (4.47)From (4.6), we have v i = 0 whenever spt(Ψ i ) ∩ B ρ ( x ) c = ∅ . Hence we only have to sum over all those i ∈ Υ forwhich spt(Ψ i ) ⊂ B ρ ( x ) × [ t − s, ∞ ). In this case, we make use of a suitable choice for ε ∈ (0 , W7 )along with ( W8 ), to estimate (4.47) from below to get J ? − λ p | R n +1 \ E λ | . (4.48) Estimate of J : For any x ∈ B ρ ( x ) \ E tλ , we have from ( W10 ) that X j Ψ j ( x, t ) = 1, which givesΨ i ( z ) | v λ ( z ) − v i | > Ψ i ( z ) X j ∈ A i | Ψ j ( z ) | (cid:0) v j − v i (cid:1) a ) > min { ρ, r i } λ . (4.49)To obtain (a) above, we made use of Lemma 4.10 along with ( W13 ). Substituting (4.49) into the expressionfor J and using | Q i | = | B i | × κr i , we get J > X i ∈ Υ | B i | κr i κ λ > λ p | R n +1 \ E λ | . (4.50)Substituting (4.48) and (4.50) into (4.7), we get J ? − λ p | R n +1 \ E λ | . This completes the proof of the lemma.
5. Comparison estimates
Before we state the main difference estimates, let us define the the approximations that we will make andrecall some useful results in existing literature. Let us fix the point z = ( x , t ) ∈ Ω × ( − T, T ) . Let u be a weak solution of (1.1) and consider the unique weak solution w ∈ C (cid:0) I ρ ( t ); L (Ω ρ ( x ) (cid:1) ∩ L p (cid:0) I ρ ( t ); W ,p (Ω ρ ( x ) (cid:1) solving ( w t − div A ( x, t, ∇ w ) = 0 in K ρ ( z ) ,w = u on ∂ p K ρ ( z ) . (5.1)This is possible, since (1.1) shows u ∈ L p (cid:0) I ρ ( t ); W ,p (Ω ρ ( x ) (cid:1) and dudt ∈ (cid:0) W ,p ( K ρ ( z )) (cid:1) ′ in the sense of distri-bution.Recalling the notation from (2.5), we will need to make another approximation to (5.1): ( v t − div A B ρ ( x ) ( ∇ v, t ) = 0 in K ρ ( z ) ,v = w on ∂ p K ρ ( z ) , (5.2)which admits a unique weak solution v ∈ C (cid:0) I ρ ( t ); L (Ω ρ ( x ) (cid:1) ∩ L p (cid:0) I ρ ( t ); W ,p (Ω ρ ( x ) (cid:1) since Proposition 2.17is applicable. 21 .2. Interior Lipschitz regularity In the case K ρ ( z ) = Q ρ ( z ), i.e., we are in the interior case, then we have the following interior Lipschitzregularity for (5.2) (see [19, Theorem 5.1 and Theorem 5.2]): Lemma 5.1.
There exists a weak solution v ∈ C (cid:0) I ρ ( t ); L ( B ρ ( x ) (cid:1) ∩ L p (cid:0) I ρ ( t ); W ,p (Ω +2 ρ ( x ) (cid:1) solving (5.2) .Furthermore, there holds sup Q ρ ( z ) |∇ v | ≤ C ( n,p, Λ , Λ ) −− ¨ Q ρ ( z ) |∇ v | p dz ! p . (5.3) In the boundary case, we may not have Lipschitz regularity for solutions of (5.2) up to the boundary ingeneral. In order to overcome this difficulty, we need to make one further approximation in which we consider aweak solution V ∈ C (cid:0) I ρ ( t ); L (Ω +2 ρ ( x ) (cid:1) ∩ L p (cid:0) I ρ ( t ); W ,p (Ω +2 ρ ( x ) (cid:1) solving ( V t − div A B ρ ( x ) ( ∇ V , t ) = 0 in Q +2 ρ ( z ) ,V = 0 on T ρ ( z ) . From [27, Theorem 1.6], the following important lemma holds:
Lemma 5.2.
There exists a weak solution V ∈ C (cid:0) I ρ ( t ); L (Ω +2 ρ ( x ) (cid:1) ∩ L p (cid:0) I ρ ( t ); W ,p (Ω +2 ρ ( x ) (cid:1) solving (5.3) .Furthermore, there holds sup Q + ρ ( z ) |∇ V | ≤ C ( n, p, Λ , Λ ) −− ¨ Q +2 ρ ( z ) |∇ V | p dz ! p . In this subsection, we will prove a improved difference estimate between solutions of (1.1) and (5.1).
Theorem 5.3.
Let δ > be given and Let u be a weak solution of (1.1) and w be the unique weak solution of (5.1) , then there exists an β = β (Λ , Λ , p, n, m e , δ ) ∈ (0 , such that for any β ∈ (0 , β ) , the following estimateholds: −− ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz ≤ δ −− ¨ K ρ ( z ) |∇ u | p − β dz + C ( n, p, β, Λ , Λ , δ ) −− ¨ K ρ ( z ) | f | p − β . Proof.
Consider the following cut-off function ζ ε ∈ C ∞ ( t − (4 ρ ) , ∞ ) such that 0 ≤ ζ ε ( t ) ≤ ζ ε ( t ) = ( t ∈ ( t − (4 ρ ) + ε, t + (4 ρ ) − ε ) , t ∈ ( −∞ , t − (4 ρ ) ) ∪ ( t + (4 ρ ) , ∞ ) . It is easy to see that ζ ′ ε ( t ) = 0 for t ∈ ( −∞ , t − (4 ρ ) ) ∪ ( t − (4 ρ ) + ε, t + (4 ρ ) − ε ) ∪ ( t + (4 ρ ) , ∞ ) , | ζ ′ ε ( t ) | ≤ cε for t ∈ ( t − (4 ρ ) , t − (4 ρ ) + ε ) ∪ ( t + (4 ρ ) − ε, t − (4 ρ ) ) . Without loss of generality, we shall always take 2 h ≤ ε , since we will take limits in the following order lim ε → lim h → .Let us apply the results of Section 4 with ϕ = u , φ = w , ~f = f and ~g = 0 over Q ρ,s ( z ) = K ρ ( z ) to get aLipschitz test function v λ,h satisfying Lemma 4.18. From Lemma 4.16, we have v λ,h ∈ C , ( K ρ ( z )) and thus weshall use v λ,h ( z ) ζ ε ( t ) as a test function to get L + L := ¨ K ρ ( z ) d [ u − w ] h dt v λ,h ζ ε dx dt + ¨ K ρ ( z ) h [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h , ∇ v λ,h i ζ ε dx dt = ¨ K ρ ( z ) h [ | f | p − f ] h , ∇ v λ,h i ζ ε dx dt =: L . Let us recall from Section 4 the following: for a fixed 1 < q < p − β , we have g ( z ) := M (cid:16) [ |∇ u − ∇ w | q + |∇ u | q + |∇ w | q + | f | q ] χ K ρ ( z ) (cid:17) q , where M is as defined in (2.7) and E λ = { z ∈ R n +1 : g ( z ) ≤ λ } . Note that at this point, we have not really madeany choice of β . k g k L p − β ( R n +1 ) > k∇ u k L p − β ( K ρ ( z )) + k∇ w k L p − β ( K ρ ( z )) + k f k L p − β ( K ρ ( z )) + k∇ u − ∇ w k L p − β ( K ρ ( z )) > k∇ u k L p − β ( K ρ ( z )) + k f k L p − β ( K ρ ( z )) + k∇ u − ∇ w k L p − β ( K ρ ( z )) . (5.4) Estimate for L : L = ˆ t +(4 ρ ) t − (4 ρ ) ˆ Ω ρ ( x ) dv h ( y, s ) ds v λ,h ( y, s ) ζ ε ( s ) dy ds + ˆ t +(4 ρ ) t − (4 ρ ) ˆ Ω ρ ( x ) d (cid:16)h ( v h ) − ( v λ,h − v h ) i ζ ε ( s ) (cid:17) ds dy ds − ˆ t +(4 ρ ) ˆ Ω ρ ( x ) dζ ε ds (cid:16) v h − ( v λ,h − v h ) (cid:17) dy ds := J + J ( t + (4 ρ ) ) − J ( t − (4 ρ ) ) − J , (5.5)where we have set J ( s ) := 12 ˆ Ω ρ ( x ) (( v h ) − ( v λ,h − v h ) )( y, s ) ζ ε ( s ) dy. Note that J ( t − (4 ρ ) ) = J ( t + (4 ρ ) ) = 0 since ζ ε ( t − (4 ρ ) ) = ζ ε ( t + (4 ρ ) ) = 0.Form Lemma 4.15 applied with ϑ = 1, we have the bound | J | > ¨ K ρ ( z ) \ E λ (cid:12)(cid:12)(cid:12)(cid:12) dv λ,h ds ( v λ,h − v h ) (cid:12)(cid:12)(cid:12)(cid:12) dy ds > λ p | R n +1 \ E λ | . (5.6) Estimate for L : We split L and make use of the fact that v λ,h ( z ) = v h ( z ) for all z ∈ E λ ∩ K ρ ( z ) .L = ¨ K ρ ( z ) ∩ E λ h [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h , ∇ v λ,h i ζ ε dz + ¨ K ρ ( z ) \ E λ h [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h , ∇ v λ,h i ζ ε dz = ¨ K ρ ( z ) ∩ E λ h [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h , ∇ [ u − w ] h i ζ ε dz + ¨ K ρ ( z ) \ E λ h [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h , ∇ v λ,h i ζ ε dz = L + L . (5.7) Estimate for L : Using ellipticity, we get L = ¨ K ρ ( z ) ∩ E λ h [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h , ∇ [ u − w ] h i ζ ε dz ? ¨ K ρ ( z ) ∩ E λ (cid:2) |∇ u − w | (cid:0) |∇ u | + |∇ w | (cid:1)(cid:3) p − h ζ ε dz. (5.8) Estimate for L : Using the bound from Lemma 4.11, we get L > ¨ K ρ ( z ) \ E λ | [ A ( x, t, ∇ u ) − A ( x, t, ∇ w )] h | |∇ v λ,h | dz > λ ¨ K ρ ( z ) \ E λ |∇ [ u ] h | p − + |∇ [ w ] h | p − dz. (5.9) Estimate for L : Analogous to estimate for L , we split L into integrals over E λ and E cλ followed by makinguse of (4.5) and Lemma 4.11 to get L > ¨ K ρ ( z ) ∩ E λ [ | f | p − ] h |∇ [ u − w ] h | dz + λ ¨ K ρ ( z ) \ E λ | [ f ] h | p − dz. (5.10)Combining (5.6) into (5.5) followed by (5.8) and (5.9) into (5.7) and making use of (5.5), (5.6) and (5.10),23e get − ˆ t +(4 ρ ) t − (4 ρ ) ˆ Ω ρ ( x ) dζ ε ds (cid:16) v h − ( v λ,h − v h ) (cid:17) dy ds + ¨ K ρ ( z ) ∩ E λ |∇ [ u − w ] h | (cid:0) |∇ [ u ] h | + |∇ [ w ] h | (cid:1) p − ζ ε dz > ¨ K ρ ( z ) ∩ E λ [ | f | p − ] h |∇ [ u − w ] h | dz + λ ¨ K ρ ( z ) \ E λ |∇ [ u ] h | p − + |∇ [ w ] h | p − + | [ f ] h | p − dz + λ p | R n +1 \ E λ | . (5.11)In order to estimate − ˆ t +(4 ρ ) t − (4 ρ ) ˆ Ω ρ ( x ) dζ ε ds (cid:16) v h − ( v λ,h − v h ) (cid:17) dy ds , we take limits first in h ց ε ց − ˆ t +(4 ρ ) t − (4 ρ ) ˆ Ω ρ ( x ) dζ ε ds (cid:16) v h − ( v λ,h − v h ) (cid:17) dy ds lim ε ց lim h ց −−−−−→ ˆ Ω ρ ( x ) ( v − ( v λ − v ) )( x, t + (4 ρ ) ) dx − ˆ Ω ρ ( x ) ( v − ( v λ − v ) )( x, t − (4 ρ ) ) dx. (5.12)For the second term on the right of (5.12), we observe that on E λ , we have v λ = v and on E cλ and we alsohave v λ ( · , t − (4 ρ ) ) = v ( · , t − (4 ρ ) ) = 0. Thus, the second term vanishes because on E λ , we can use the initialboundary condition and on E cλ , it is zero by construction. Thus we get − ˆ t +(4 ρ ) t − (4 ρ ) ˆ Ω ρ ( x ) dζ ε ds (cid:16) v h − ( v λ,h − v h ) (cid:17) dy ds lim ε ց lim h ց −−−−−→ ˆ Ω ρ ( x ) ( v − ( v λ − v ) )( x, t + (4 ρ ) ) dx. (5.13)Thus using (5.13) into (5.11) gives ˆ Ω ρ ( x ) ( v − ( v λ − v ) )( x, t + (4 ρ ) ) dx + ¨ K ρ ( z ) ∩ E λ |∇ [ u − w ] h | (cid:0) |∇ [ u ] h | + |∇ [ w ] h | (cid:1) p − ζ ε dz > ¨ K ρ ( z ) ∩ E λ [ | f | p − ] h |∇ [ u − w ] h | dz + λ ¨ K ρ ( z ) \ E λ |∇ [ u ] h | p − + |∇ [ w ] h | p − + | [ f ] h | p − dz + λ p | R n +1 \ E λ | . (5.14)In fact, if we consider a cut-off function ζ t ε ( · ) for some t ∈ ( t − (4 ρ ) , t + (4 ρ ) ), where ζ t ε ( t ) = ( t ∈ ( − t + ε, t − ε ) , t ∈ ( −∞ , − t ) ∪ ( t , ∞ ) . we get the following analogue of (5.14) ˆ Ω ρ ( x ) ( v − ( v λ − v ) )( x, t ) dx + ˆ t − t ˆ Ω ρ ( x ) ∩ E tλ |∇ ( u − w ) | (cid:0) |∇ u | + |∇ w | (cid:1) p − dz > ¨ K ρ ( z ) ∩ E λ | f | p − |∇ ( u − w ) | dz + λ ¨ K ρ ( z ) \ E λ |∇ u | p − + |∇ w | p − + | f | p − dz + λ p | R n +1 \ E λ | . (5.15)Using Lemma 4.18, we get for any t ∈ ( t − (4 ρ ) , t + (4 ρ ) ), the estimate ˆ Ω ρ ( x ) | ( v ) − ( v λ − v ) | ( y, t ) dy ? ˆ E tλ | v ( x, t ) | dx − λ p | R n +1 \ E λ | . (5.16)Since ˆ E tλ | v ( x, t ) | dx occurs on the left hand side and is positive, we can ignore this term. Thus combining(5.16) with (5.15), we get ¨ K ρ ( z ) ∩ E λ |∇ ( u − w ) | (cid:0) |∇ u | + |∇ w | (cid:1) p − dx dt > ¨ K ρ ( z ) ∩ E λ | f | p − |∇ ( u − w ) | dz + λ ¨ K ρ ( z ) \ E λ |∇ u | p − + |∇ w | p − + | f | p − dz + λ p | R n +1 \ E λ | . (5.17)Let us now multiply (5.17) with λ − − β and integrating over (0 , ∞ ) with respect to λ , we get K + K > K + K , (5.18)24here we have set K := ˆ ∞ λ − − β ¨ K ρ ( z ) ∩ E λ |∇ ( u − w ) | (cid:0) |∇ u | + |∇ u | (cid:1) p − dz dλ,K := ˆ ∞ λ − − β ¨ K ρ ( z ) ∩ E λ | f | p − |∇ ( u − w ) | dz dλ,K := ˆ ∞ λ − β ¨ K ρ ( z ) \ E λ |∇ u | p − + |∇ w | p − + | f | p − dz dλ,K := ˆ ∞ λ − − β λ p | R n +1 \ E λ | dλ. Estimate for K : Applying Fubini, we get K ? β ¨ K ρ ( z ) g ( z ) − β |∇ ( u − w ) | (cid:0) |∇ u | + |∇ u | (cid:1) p − dz. Using Young’s inequality along with (2.2) and (5.4), we get for any ǫ >
0, the estimate ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz > C ( ǫ ) βK + ǫ ¨ K ρ ( z ) |∇ u − ∇ w | p − β + |∇ u | p − β dz + C ( ǫ ) ¨ K ρ ( z ) | f | p − β dz. (5.19) Estimate for K : Again by Fubini, we get K = 1 β ¨ K ρ ( z ) g ( z ) − β h| f | p − f , ∇ u − ∇ w i dz. From the definition of g ( z ), we see that for z ∈ K ρ ( z ), we have g ( z ) ≥ |∇ u − ∇ w | ( z ), which implies g ( z ) − β ≤ |∇ u − ∇ w | − β ( z ). Now we apply Young’s inequality, for any ǫ >
0, we get K ≤ β ¨ K ρ ( z ) |∇ u − ∇ w | − β | f | p − dz > C ( ǫ ) β ¨ K ρ ( z ) | f | p − β dz + ǫ β ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz. (5.20) Estimate for K : Again applying Fubini, we get K = 1 p − β ¨ K ρ ( z ) g ( z ) − β (cid:0) |∇ u | p − + |∇ w | p − + | f | p − (cid:1) dz. Applying Young’s inequality followed by making use of (5.4), we get K > ¨ K ρ ( z ) |∇ u − ∇ w | p − β + |∇ u | p − β + | f | p − β dz. (5.21) Estimate for K : Applying the layer cake representation followed by using (5.4), we get K = 1 p − β ¨ R n +1 g ( z ) p − β dz > ¨ K ρ ( z ) |∇ u − ∇ w | p − β + |∇ u | p − β + | f | p − β dz. (5.22)We now combine (5.19), (5.20), (5.21) and (5.22) into (5.18), we get ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz > [ ǫ + C ( ǫ )( ǫ + β )] ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz + C ( ǫ , ǫ , β ) ¨ K ρ ( z ) | f | p − β dz + [ ǫ + C ( ǫ ) β ] ¨ K ρ ( z ) |∇ u | p − β dz. Choosing ǫ small followed by ǫ and β , for any δ >
0, we get a β = β ( n, p, Λ , Λ , δ ) such that for any β ∈ (0 , β ), there holds −− ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz ≤ δ −− ¨ K ρ ( z ) |∇ u | p − β dz + C ( n, p, β, Λ , Λ , δ ) −− ¨ K ρ ( z ) | f | p − β . This completes the proof of the theorem. 25 .5. Second comparison estimate
In this subsection, we will prove an improved comparison estimate between solutions of (5.1) and (5.2).
Theorem 5.4.
Let w be a weak solution of (5.1) and v be the unique weak solution of (5.2) , then there exists an β = β (Λ , Λ , p, n, m e , ε ) ∈ (0 , such that for any β ∈ (0 , β ) and ε > , there exists a C = C ( n, Λ , Λ , p, ε ) > and σ = σ (Λ , Λ , p ) such that the following estimate holds: −− ¨ K ρ ( z ) |∇ w − ∇ v | p − β dz ! pp − β ≤ ε −− ¨ K ρ ( z ) |∇ w | p − β dz ! β ˜ ϑ + C [ A ] σ ,R −− ¨ K ρ ( z ) |∇ w | p − β dz ! (1+ β ˜ ϑ β ˜ ϑ pp + β . Here ˜ ϑ and ˜ ϑ are from Lemma 2.18 and Lemma 2.19.Proof. Since we are in the setting of weak solutions, from [21, Lemma 2.8], we have the following estimate: forany ε >
0, there exists a C = C ( n, Λ , Λ , p, ε ) > σ = σ (Λ , Λ , p ) such that −− ¨ K ρ ( z ) |∇ w − ∇ v | p dz ≤ ε −− ¨ K ρ ( z ) |∇ w | p dz + C [ A ] σ ,R −− ¨ K ρ ( z ) |∇ w | p + β dz ! pp + β . (5.23)Since w solves the homogeneous equation (5.1), we can control the right hand side of (5.23) by using Lemma2.18 and Lemma 2.19. For β := min { ˜ β , ˜ β } , for any β ∈ (0 , β ), there holds −− ¨ K ρ ( z ) |∇ w | p + β dz > −− ¨ K ρ ( z ) |∇ w | p dz β ˜ ϑ > −− ¨ K ρ ( z ) |∇ w | p − β dz ! ( β ˜ ϑ )( β ˜ ϑ ) , (5.24)where ˜ ϑ and ˜ ϑ are from Lemma 2.18 and Lemma 2.19 respectively.We now combine (5.24) and (5.23) to get −− ¨ K ρ ( z ) |∇ w − ∇ v | p dz ≤ ε −− ¨ K ρ ( z ) |∇ w | p − β dz ! β ˜ ϑ + C [ A ] σ ,R −− ¨ K ρ ( z ) |∇ w | p − β dz ! (1+ β ˜ ϑ β ˜ ϑ pp + β . A simple application of H¨older’s inequality now gives −− ¨ K ρ ( z ) |∇ w − ∇ v | p − β dz ! pp − β ≤ ε −− ¨ K ρ ( z ) |∇ w | p − β dz ! β ˜ ϑ + C [ A ] σ ,R −− ¨ K ρ ( z ) |∇ w | p − β dz ! (1+ β ˜ ϑ β ˜ ϑ pp + β . In this subsection, we will prove the interior approximation lemma:
Lemma 5.5.
Let β ∈ (0 , β ) for β = min { β , β } where β is from Theorem 5.3 and β is from Theorem 5.4.For each ε > , there exists a δ > (possibly depending on ε ) such that the following holds true: Assume u is aweak solution of (1.1) satisfying −− ¨ K ρ ( z ) |∇ u | p − β dz ≤ , (5.25) then under the condition −− ¨ K ρ ( z ) | f | p − β dz ≤ δ p − β , (5.26) there exists a weak solution v to (5.2) satisfying k∇ v k L ∞ ( K ρ ( z )) > , and −− ¨ K ρ ( z ) |∇ u − ∇ v | p − β dz ≤ ε p − β . (5.27) Proof.
Let us prove each of the assertions of (5.27) as follows:
First estimate in (5.27) : From Lemma 5.1, we have existence of a weak solution v solving (5.2) satisfying theestimate k∇ v k pL ∞ ( K ρ ( z )) ≤ C ( n, p, Λ , Λ ) −− ¨ Q ρ ( z ) |∇ v | p dz.
26e now estimate the right hand side as follows: −− ¨ Q ρ ( z ) |∇ v | p dz ( a ) > −− ¨ Q ρ ( z ) |∇ v − ∇ w | p dz + −− ¨ Q ρ ( z ) |∇ w | p dz ( b ) > (1 + ε ) −− ¨ K ρ ( z ) |∇ w | p − β dz ! β ˜ ϑ + C [ A ] σ ,S −− ¨ K ρ ( z ) |∇ w | p − β dz ! (1+ β ˜ ϑ β ˜ ϑ pp + β ( c ) > (1 + ε ) −− ¨ K ρ ( z ) |∇ w − ∇ u | p − β dz ! β ˜ ϑ + C [ A ] σ ,S −− ¨ K ρ ( z ) |∇ w − ∇ u | p − β dz ! (1+ β ˜ ϑ β ˜ ϑ pp + β +(1 + ε ) −− ¨ K ρ ( z ) |∇ u | p − β dz ! β ˜ ϑ + C [ A ] σ ,S −− ¨ K ρ ( z ) |∇ u | p − β dz ! (1+ β ˜ ϑ β ˜ ϑ pp + β . In order to obtain (a), we made use of triangle inequality, to obtain (b), we made use of Theorem 5.4 alongwith Lemma 2.19 and finally to obtain (c), we applied triangle inequality.We can control −− ¨ K ρ ( z ) |∇ w − ∇ u | p − β dz using Theorem 5.3 along with making use of (5.25) and (5.26) andobserving that [ A ] ,S ≤ C ( p, Λ , Λ ), we get −− ¨ Q ρ ( z ) |∇ v | p dz ≤ C (Λ , Λ , n, p, δ ) . This proves the first assertion of (5.27).
Second estimate in (5.27) : Using triangle inequality, we get −− ¨ K ρ ( z ) |∇ u − ∇ v | p − β dz > −− ¨ K ρ ( z ) |∇ u − ∇ w | p − β dz + −− ¨ K ρ ( z ) |∇ w − ∇ v | p − β dz. Each of the above terms can be controlled using Theorem 5.3 and Theorem 5.4 along with (5.25) followed bychoosing δ sufficiently small (depending on ε ) and γ sufficiently small such that ( A , Ω) is ( γ, S )-vanishing toget the desired conclusion. In this subsection, we will prove the boundary approximation lemma:
Lemma 5.6.
Let β ∈ (0 , β ) be fixed and let w be a weak solution of (5.1) satisfying −− ¨ K ρ ( z ) |∇ w | p − β dz ≤ , then for any ε > , there exists a small γ = γ (Λ , Λ , n, p, ε ) > such that if ( A , Ω) is ( γ, S ) vanishing, thenthere exists a weak solution V of (5.3) whose zero extension to Q ρ ( z ) satisfies −− ¨ Q +2 ρ ( z ) |∇ V | p dz ≤ , and −− ¨ K ρ ( z ) |∇ w − ∇ V | p dz ≤ ε p . (5.28) Proof.
From Lemma 2.19, we see that the −− ¨ K ρ ( z ) |∇ w | p − β dz ≤ ⇒ −− ¨ K ρ ( z ) |∇ w | p dz > . Hence we can apply [13, Lemma 3.8] to get an γ = γ (Λ , Λ , n, p, ε ) > A , Ω) is ( γ, S ) vanishing,then there exists a weak solution V of (5.3) whose zero extension to Q ρ ( z ) satisfies −− ¨ Q +2 ρ ( z ) |∇ V | p dz ≤ , and −− ¨ K ρ ( z ) |∇ w − ∇ V | p dz ≤ ε p . This completes the proof of the lemma. 27 orollary 5.7.
Let β ∈ (0 , β ) be fixed and let w be a weak solution of (5.1) satisfying −− ¨ K ρ ( z ) |∇ w | p − β dz ≤ , then for any ε > , there exists a small γ = γ (Λ , Λ , n, p, ε ) > such that if ( A , Ω) is ( γ, S ) vanishing, thenthere exists a weak solution V of (5.3) whose zero extension to Q ρ ( z ) satisfies k∇ V k L ∞ ( Q ρ ( z )) ≤ C ( n, p, Λ , Λ ) , and −− ¨ K ρ ( z ) |∇ w − ∇ V | p − β dz ≤ ε p − β . Proof.
All the hypothesis of Lemma 5.6 is satisfied. Thus the first conclusion follows directly by combining(5.28) along with Lemma 5.1 and the second conclusion follows by a simple application of H¨older’s inequality to(5.28).
6. Proof of Theorem 3.1.
Consider the following cut-off function ζ ε ∈ C ∞ ( − T, ∞ ) such that 0 ≤ ζ ε ( t ) ≤ ζ ε ( t ) = ( t ∈ ( − T + ε, T − ε ) , t ∈ ( −∞ , − T ) ∪ ( T, ∞ ) . It is easy to see that ζ ′ ε ( t ) = 0 for t ∈ ( −∞ , − T ) ∪ ( − T + ε, T − ε ) ∪ ( T, ∞ ) , | ζ ′ ε ( t ) | ≤ cε for t ∈ ( − T, − T + ε ) ∪ ( T − ε, − T ) . Without loss of generality, we shall always take 2 h ≤ ε , since we will take limits in the following order lim ε → lim h → .Since u = 0 on ∂ Ω × ( − T, T ), we can apply the results of Section 4 with ϕ = u , φ = 0, ~f = f and ~g = 0 over Q ρ,s = Ω × ( − T, T ) to get a Lipschitz test function v λ,h satisfying Lemma 4.18. Thus we shall use v λ,h ( z ) ζ ε ( t ) asa test function in (1.1) to get ¨ Ω T d [ u ] h dt v λ,h ζ ε dx dt + ¨ Ω T h [ A ( x, t, ∇ u )] h , ∇ v λ,h i ζ ε dx dt = ¨ Ω T h [ | f | p − f ] h , ∇ v λ,h i ζ ε dx dt, which we write as L + L = L . Let us recall from Section 4 the following: for a fixed 1 < q < p − β , we have g ( z ) := M (cid:16) [ |∇ u | q + | f | q ] χ Ω T (cid:17) q , where M is as defined in (2.7) and E λ = { z ∈ R n +1 : g ( z ) ≤ λ } . Note that at this point in the proof, we have notreally made any choice of β . From the strong Maximal function estimates (see [28, Lemma 7.9] for the proof), we have k g k L p − β ( R n +1 ) ≤ C ( n ) (cid:16) k|∇ u | χ Ω T k L p − β ( R n +1 ) + k| f | χ Ω T k L p − β ( R n +1 ) (cid:17) . (6.1) Estimate for L : L = ˆ T − T ˆ Ω du h ( y, s ) ds v λ,h ( y, s ) ζ ε ( s ) dy ds = ˆ T − T ˆ Ω \ E sλ dv λ,h ds ( v λ,h − u h ) ζ ε ( s ) dy ds + ˆ T − T ˆ Ω d (cid:16)h ( u h ) − ( v λ,h − u h ) i ζ ε ( s ) (cid:17) ds dy ds − ˆ T ˆ Ω dζ ε ds (cid:16) u h − ( v λ,h − u h ) (cid:17) dy ds := J + J ( T ) − J ( − T ) − J , (6.2)where we have set J ( s ) := 12 ˆ Ω (( u h ) − ( v λ,h − u h ) )( y, s ) ζ ε ( s ) dy. Note that J ( − T ) = J ( T ) = 0 since ζ ε ( − T ) = ζ ε ( T ) = 0.Form Lemma 4.15 applied with ϑ = 1, we have the bound | J | > ¨ Ω T \ E λ (cid:12)(cid:12)(cid:12)(cid:12) dv λ,h ds ( v λ,h − u h ) (cid:12)(cid:12)(cid:12)(cid:12) dy ds > λ p | R n +1 \ E λ | . (6.3)28 stimate for L : We split L and make use of the fact that v λ,h ( z ) = u h ( z ) for all z ∈ E λ ∩ Ω T .L = ¨ Ω T ∩ E λ h [ A ( x, t, ∇ u )] h , ∇ v λ,h i ζ ε dz + ¨ Ω T \ E λ h [ A ( x, t, ∇ u )] h , ∇ v λ,h i ζ ε dz = ¨ Ω T ∩ E λ h [ A ( x, t, ∇ u )] h , ∇ [ u ] h i ζ ε dz + ¨ Ω T \ E λ h [ A ( x, t, ∇ u )] h , ∇ v λ,h i ζ ε dz = L + L . (6.4) Estimate for L : Using ellipticity from (2.2), we get L ? ¨ Ω T ∩ E λ [ |∇ u | p ] h ζ ε dz. (6.5) Estimate for L : Using the bound from Lemma 4.11, we get L > λ ¨ Ω T \ E λ |∇ [ u ] h | p − dz. (6.6) Estimate for L : Analogous to estimate for L , we split L into integrals over E λ and E cλ followed by makinguse of (4.5) and Lemma 4.11 to get L > ¨ Ω T ∩ E λ [ | f | p − ] h |∇ [ u ] h | dz + λ ¨ Ω T \ E λ | [ f ] h | p − dz. (6.7)Combining (6.3) into (6.2) followed by (6.5) and (6.6) into (6.4) and making use of (6.2), (6.3) and (6.7), weget − ˆ T − T ˆ Ω dζ ε ds (cid:16) u h − ( v λ,h − u h ) (cid:17) dy ds + ¨ Ω T ∩ E λ |∇ [ u ] h | p ζ ε dz > ¨ Ω T ∩ E λ [ | f | p − ] h |∇ [ u ] h | dz + λ ¨ Ω T \ E λ | [ f ] h | p − dz + λ p | R n +1 \ E λ | . (6.8)In order to estimate − ˆ T − T ˆ Ω dζ ε ds (cid:16) u h − ( v λ,h − u h ) (cid:17) dy ds , we take limits first in h ց ε ց − ˆ T − T ˆ Ω dζ ε ds (cid:16) u h − ( v λ,h − u h ) (cid:17) dy ds lim ε ց lim h ց −−−−−→ ˆ Ω ( u − ( v λ − u ) )( x, T ) dx − ˆ Ω ( u − ( v λ − u ) )( x, − T ) dx. (6.9)For the second term on the right of (6.9), we observe that on E λ , we have v λ = u and on E cλ and we also have v λ ( · , − T ) = u ( · , − T ) = 0 using the initial condition. Thus, the second term on the right of (6.9) vanishes fromwhich we get − ˆ T − T ˆ Ω dζ ε ds (cid:16) u h − ( v λ,h − u h ) (cid:17) dy ds lim ε ց lim h ց −−−−−→ ˆ Ω ( u − ( v λ − u ) )( x, T ) dx. Thus using (5.13) into (6.8) gives ˆ Ω ( u − ( v λ − u ) )( x, T ) dx + ¨ Ω T ∩ E λ |∇ u | p dz > ¨ Ω T ∩ E λ | f | p − |∇ u | dz + λ ¨ Ω T \ E λ | f | p − dz + λ p | R n +1 \ E λ | . (6.10)In fact, if we consider a cut-off function ζ t ε ( · ) for some t ∈ ( − T, T ), where ζ t ε ( t ) = ( t ∈ ( − t + ε, t − ε ) , t ∈ ( −∞ , − t ) ∪ ( t , ∞ ) . we get the following analogue of (6.10) ˆ Ω ( v − ( v λ − v ) )( x, t ) dx + ˆ t − t ˆ Ω ∩ E tλ |∇ u ) | p dz > ¨ Ω T ∩ E λ | f | p − |∇ u | dz + λ ¨ Ω T \ E λ | f | p − dz + λ p | R n +1 \ E λ | . (6.11)29sing Lemma 4.18, we get for any t ∈ ( − T, T ), the estimate ˆ Ω | ( u ) − ( v λ − u ) | ( y, t ) dy ? ˆ E tλ | u ( x, t ) | dx − λ p | R n +1 \ E λ | . (6.12)Since ˆ E tλ | u ( x, t ) | dx occurs on the left hand side and is positive, we can ignore this term. Thus combining(6.12) with (6.11), we get ¨ Ω T ∩ E λ |∇ u | p dz > ¨ Ω T ∩ E λ | f | p − |∇ u | dz + λ ¨ Ω T \ E λ | f | p − dz + λ p | R n +1 \ E λ | . (6.13)Let us now multiply (6.13) with λ − − β and integrating over (0 , ∞ ) with respect to λ , we get K > K + K + K , (6.14)where we have set K := ˆ ∞ λ − − β ¨ Ω T ∩ E λ |∇ ( u − w ) | (cid:0) |∇ u | + |∇ u | (cid:1) p − dz dλ,K := ˆ ∞ λ − − β ¨ Ω T ∩ E λ | f | p − |∇ ( u − w ) | dz dλ,K := ˆ ∞ λ − β ¨ Ω T \ E λ |∇ u | p − + |∇ w | p − + | f | p − dz dλ,K := ˆ ∞ λ − − β λ p | R n +1 \ E λ | dλ. Estimate for K : Applying Fubini, we get K ? β ¨ Ω T g ( z ) − β |∇ u | p dz. Using Young’s inequality along with (6.1), we get for any ǫ >
0, the estimate ¨ Ω T |∇ u | p − β dz > C ( ǫ ) βK + ǫ ¨ Ω T |∇ u | p − β dz + C ( ǫ ) ¨ Ω T | f | p − β dz. (6.15) Estimate for K : Again by Fubini, we get K = 1 β ¨ Ω T g ( z ) − β | f | p − |∇ u | dz. From the definition of g ( z ), we see that for z ∈ Ω T , we have g ( z ) ≥ |∇ u | ( z ) which implies g ( z ) − β ≤|∇ u | − β ( z ). Now we apply Young’s inequality, for any ǫ >
0, we get K > C ( ǫ ) β ¨ Ω T | f | p − β dz + ǫ β ¨ Ω T |∇ u | p − β dz. (6.16) Estimate for K : Again applying Fubini, we get K = 1 p − β ¨ Ω T g ( z ) − β | f | p − dz ( a ) > ¨ Ω T |∇ u | p − β + | f | p − β dz. (6.17)To obtain (a), we made use of Young’s inequality followed by (6.1). Estimate for K : Applying the layer cake representation followed by using (6.1), we get K = 1 p − β ¨ R n +1 g ( z ) p − β dz > ¨ Ω T |∇ u | p − β + | f | p − β dz. (6.18)We now combine (6.15), (6.16), (6.17) and (6.18) into (6.14), we get ¨ Ω T |∇ u | p − β dz > [ ǫ + C ( ǫ )( ǫ + β )] ¨ Ω T |∇ u | p − β dz + C ( ǫ , ǫ , β ) ¨ Ω T | f | p − β dz + [ ǫ + C ( ǫ ) β ] ¨ Ω T |∇ u | p − β dz. Choosing ǫ small followed by ǫ and β , we get a β = β ( n, p, Λ , Λ , ε ) such that for any β ∈ (0 , β ), there holds −− ¨ Ω T |∇ u | p − β dz > ( n,p,β, Λ , Λ ) −− ¨ Ω T | f | p − β dz. This completes the proof of the theorem. 30 . Covering arguments
Once we have the estimates in Section 5 and Section 6, the covering arguments can be proved in the standardway. We will only provide a brief sketch of the estimates.
Remark 7.1.
In this section and following section, let β be that such that for all β ∈ (0 , β ] , the results inSection 5 and Section 6 are applicable. We will now fix an β with β ≤ β . Let us define α p − βd := −− ¨ Ω T " |∇ u | p − β + (cid:18) | f | γ (cid:19) p − β dx dt. (7.1)where d is defined to be d := p − β − β if p ≥ , p − β )2 p − β + np − n if 2 nn + 2 < p < . (7.2)Furthermore, define c e := (cid:20)(cid:18) (cid:19) n | Ω T || B | S n +20 (cid:21) dp − β , and let λ ≥ c e α . (7.3)We will also need to consider the following superlevel set: E λ := { z ∈ Ω T : |∇ u ( z ) | > λ } . The first lemma that we need is the following:
Lemma 7.2.
Let γ ∈ (0 , be any constant, then for any λ satisfying (7.3) , there exists a family of disjointcylinders { K λr i ( z i ) } i ∈ N with z i ∈ E λ and r i ∈ (0 , S ) such that −− ¨ K λri ( z i ) " |∇ u | p − β + (cid:18) | f | γ (cid:19) p − β dx dt = λ p − β , −− ¨ K λri ( z i ) " |∇ u | p − β + (cid:18) | f | γ (cid:19) p − β dx dt < λ p − β for every r > r i ,E λ ⊂ [ i ∈ N K λ r i ( z i ) . The proof following using standard techniques from Measure theory and Lemma 2.3 (see [13, Pages 4311-4313]for the details).Using Lemma 2.7 the following lemma follows:
Lemma 7.3.
Let w ∈ A s for any s ≥ p − βp − β , then it is automatically in A . Then there exists a constant c ∗ = c ∗ ([ w ] , n, p ) > such that there holds w ( K λr i ( z i )) ≤ Cλ p − β ¨ K λri ( z i ) ∩{|∇ u | > λ c ∗ } |∇ u | p − β w ( z ) dz + ¨ K λri ( z i ) ∩{| f u | > γλ c ∗ } (cid:12)(cid:12)(cid:12)(cid:12) f γ (cid:12)(cid:12)(cid:12)(cid:12) p − β w ( z ) dz . The proof of the above Lemma is standard and we refer to [16, Page 4114 - (3.8)] for the necessary details.Now making use of the a priori estimates in Section 5 and Section 6 and combining with the techniques of[13, Lemma 4.3], the following Lemma holds:
Lemma 7.4.
There exists a constant N = N (Λ , Λ , n, p ) > such that for any ε ∈ (0 , , there exists a small γ = γ (Λ , Λ , ε, n, p ) such that if ( A , Ω) is ( γ, S ) vanishing for such a small γ and some fixed S > , then thereholds (cid:12)(cid:12) { z ∈ K λ r i ( z i ) : |∇ u ( z ) | > N λ } (cid:12)(cid:12) | K λr i ( z i ) ≤ c (Λ , Λ ,n,p ) ε p − β . We can now combine Lemma 7.3 and Lemma 7.4 to prove the following weighted estimate on the level sets(again the proof follows exactly as in [16, STEP 4 on Page 4115] and will be omitted).31 emma 7.5.
Let β be as in Remark 7.1, then there holds w ( E (2 N λ )) > ε ( p − β ) τ λ p − β ¨ Ω T ∩{|∇ u | > λ c ∗ } |∇ u | p − β w ( z ) dz + ¨ Ω T ∩{| f u | > γλ c ∗ } (cid:12)(cid:12)(cid:12)(cid:12) f γ (cid:12)(cid:12)(cid:12)(cid:12) p − β w ( z ) dz . Here τ is defined as in Definition 2.10.
8. Proof of Theorem 3.3.
From Lemma 2.12, we get ¨ Ω T |∇ u | q w ( z ) dz = q ˆ c e α (2 N λ ) q − w ( { z ∈ Ω T : |∇ u | > N λ } ) d (2 N λ )+ q ˆ ∞ c e α (2 N λ ) q − w ( { z ∈ Ω T : |∇ u | > N λ } ) d (2 N λ )=: II + II . (8.1) Estimate for II : With d as defined in (7.2), we get II > w (Ω T ) α q = w (Ω T ) −− ¨ Ω T " |∇ u | p − β + (cid:18) | f | γ (cid:19) p − β dx dt ! qdp − β Theorem 3.1 > w (Ω T ) (cid:18) −− ¨ Ω T | f | p − β dx dt (cid:19) qdp − β Lemma 2.7 > w (Ω T ) (cid:18) w (Ω T ) ¨ Ω T | f | q w ( x, t ) dx dt (cid:19) d > (cid:18) ¨ Ω T | f | q w ( x, t ) dx dt (cid:19) d . (8.2) Estimate for II : Since we have q ≥ p > p − β > p − β , we proceed as follows: II > ε ( p − β ) τ ˆ ∞ λ q − λ p − β ¨ Ω T ∩{|∇ u | > λ c ∗ } |∇ u | p − β w ( z ) dz dλ + ε ( p − β ) τ γ ( p − β ) τ ˆ ∞ λ q − λ p − β ¨ Ω T ∩{| f | > γλ c ∗ } | f | p − β w ( z ) dz dλ Lemma 2.12 > ε p − β ¨ Ω T |∇ u | q w ( z ) dz + C ( γ, ε ) ¨ Ω T | f | q w ( z ) dz. (8.3)Combining (8.2) and (8.3) into (8.1) following by choosing ε sufficiently small, the proof follows. References [1] Emilio Acerbi and Giuseppe Mingione. Gradient estimates for a class of parabolic systems.
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