Vagif S. Guliyev and Mehriban N. Omarova

Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

De Gruyter | Published online: February 13, 2016

Abstract

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space W ˙ 2 , 1 p , φ ( Q , ω ) .

MSC 2010: 35K20; 35D35; 35B45; 35R05

1 Introduction

We consider the regular oblique derivative problem in generalized weighted Sobolev- Morrey space W ˙ 2 , 1 p , φ ( Q , ω ) for linear non-divergence form parabolic equations in a cylinder

u t a i j ( x ) D i j u = f ( x )  a .e . in  Q , u ( x ' , 0 ) = 0 ,         on   Ω , u / l = l i ( x ) D i u = 0   on  S .

The unique strong solvability of this problem was proved in [38]. In [39] Softova studied the regularity of the solution in the Morrey spaces Lp,ƛ with p ∈ (1, ∞), ƛ ∈ (0, n + 2) and also its Hölder regularity. In [41] Softova extended these studies on generalized Morrey spaces Lp,ϕ with a Morrey function ϕ satisfying the doubling and integral conditions introduced in [27, 31]. The approach associated to the names of Calderón and Zygmund and developed by Chiarenza, Frasca and Longo in [7, 8] consists of obtaining of explicit representation formula for the higher order derivatives of the solution by singular and nonsingular integrals. Further the regularity properties of the solution follows by the continuity properties of these integrals in the corresponding spaces. In [39] and [40] the regularity of the corresponding operators in the Morrey and generalized Morrey spaces is studied, while in [38] we can find the corresponding results obtained in Lp by [9] and [5]. In recent works there have been studied the regularity of the solutions of elliptic and parabolic problems with Dirichlet data on the boundary in generalized Morrey spaces Mp,ϕ with a weight ϕ satisfying (10) with w ≡ 1 (cf. [18, 19]). Precisely, a boundedness in Mp,ϕ was obtained for sub-linear operators generated by singular integrals as the Calderon-Zygmund. More results concerning sub-linear operators in generalized Morrey spaces can be found in [3, 12, 40] see also the references therein.

After studying generalized Morrey spaces in detail, researchers passed to weighted Morrey spaces and generalized weighted Morrey spaces. Recently, Komori and Shirai [23] defined the weighted Morrey spaces and studied the boundedness of some classical operators, such as the Hardy-Littlewood maximal operator or the Calderón-Zygmund operator on these spaces. Also, Guliyev in [13] first introduced the generalized weighted Morrey spaces Mwp,ϕ and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [15, 17]). Note that, Guliyev [13] gave the concept of generalized weighted Morrey space which could be viewed as an extension of both Mp,ϕ and Lp,κ(w).

We call weight a positive measurable function defined on ℝn ×. ℝ+. In [29] Muckenhoupt shows that the maximal inequality holds in weighted Lebesgue spaces Lwq if and only if the weight w satisfies the following integral condition called parabolic Muckenhoupt condition or parabolic Aq-condition. We say that the measurable, nonnegative function w : ℝn → ℝ+ satisfies the parabolic Aq-condition for q ∈ (1, ∞) if

(1) sup I ( 1 | I | I w ( x , t ) d x d t ) ( 1 | I | I w ( x , t ) 1 q 1 d x d t ) q 1 A <
for all parabolic cylinders ℐ in ℝ n+1. Then w(ℐ) means the weighted measure of ℐ, that is
w ( I ) = I w ( x , t ) d x d t .

This measure satisfies strong and reverse doubling property. Precisely, for each ℐ and each measurable subset 𝒜 ⊂ ℐ, there exist positive constants c1 and τ1 ∈ (0, 1) such that

(2) 1 [ w ] q ( | A | | I | ) q w ( A ) w ( I ) c 1 ( | A | | I | ) τ 1 ,
where c 1 and τ 1 depend on n and q but not on ℐ and 𝒜.

Throughout this paper the following notations are to be used: x = ( x ' , t ) = ( x " , x n , t ) n + 1 , R + n + 1 = x ' n , t > 0 and D + n+1 = x " R n 1 , x n > 0 , t > 0 , D i u = u / x i , D i j u = 2 u / x i x j , D t u = u t = u / t stand for the corresponding derivatives while Du = (D1u, … , Dnu) and D 2 u = D i j u i , j = 1 n mean the spatial gradient and the Hessian matrix of u. For any measurable function f and A ⊂ ℝn+1 we write

f p , A = ( A | f ( y ) | p d y ) 1 / p , f A = 1 | A | A f ( y ) d y
where | A| is the Lebesgue measure of A. Through all the paper the standard summation convention on repeated upper and lower indexes is adopted. The letter C is used for various constants and may change from one occurrence to another.

2 Definitions and statement of the problem

Let Ω ⊂ ℝn, n ≥ 1 be a bounded C1,1-domain, Q = Ω × (0, T) be a cylinder in + n + 1 , and S = ∂Ω × (0, T) stands for the lateral boundary of Q. We consider the problem

(3) B u : = u t a i j ( x ) D i j u = f  a .e . in    Q , F u : = u ( x ' , 0 ) = 0 ,              on     Ω , B u : = u / l = l i ( x ) D i u = 0   on  S ,
under the following conditions:
  1. (i)

    The operator 𝔅 is supposed to be uniformly parabolic, i.e. there exists a constant Λ > 0 such that for almost all xQ

    (4) Λ 1 | ξ | 2 a i j ( x ) ξ i ξ j Λ | ξ | 2 , ξ n , a i j ( x ) = a j i ( x ) ,         i , j = 1 , , n .

    The symmetry of the coefficient matrix a = a i j i , j = 1 n implies essential boundedness of aij’s and we set a , Q = i , j = 1 n a i j , Q .

  2. (ii)

    The boundary operator 𝔅 is prescribed in terms of a directional derivative with respect to the unit vector field l(x) = (l1(x), … , ln(x)), xS. We suppose that 𝔅 is a regular oblique derivative operator, i.e., the field l is never tangential to S:

    (5) l ( x ) n ( x ) = l i ( x ) n i ( x ) > 0   on  S , l i L i p ( S ¯ ) .
    Here Lip( S) is the class of uniformly Lipschitz continuous functions on S and n( x) stands for the unit outward normal to Ω.

In the following, besides the parabolic metric Q(x) = max(|x′|, |t|1/2) and the defined by it parabolic cylinders

I r ( x ) = y n + 1 : | x ' y ' | < r , | t τ | < r 2 ,     | I r | = C r n + 2 .
we use the equivalent one ρ ( x ) = ( | x ' | 2 + | x ' | 4 + 4 t 2 2 ) 1 2 (see [ 9]). The balls with respect to this metric are ellipsoids
ε r ( x ) = y n + 1 : | x ' y ' | 2 r 2 + | t τ | 2 r 4 < 1 , | ε r | = C r n + 2 .

Because of the equivalence of the metrics all estimates obtained over ellipsoids hold true also over parabolic cylinders and in the following we shall use this without explicit references.

Definition 2.1

([20, 37]). Let a L I o c 1 ( n + 1 ) , denoted by

η a ( R ) = sup ε r , r R 1 | ε r | ε r | f ( y ) f ε r | d y ,    f o r e v e r y R > 0
wherer ranges over all ellipsoids inn+1. The Banach space BMO (bounded mean oscillation) consists of functions for which the following norm is finite a * = sup R > 0 η a ( R ) < .

A function a belongs to VMO (vanishing mean oscillation) with VMO- modulus ηa(R) provided lim R 0 η a ( R ) = 0.

For any bounded cylinder Q we define BMO(Q) and VMO(Q) taking aL1(Q) and Qr = Q ⋂ ℐrinstead ofrin the definition above.

According to [1, 21] having a function aBMO/VMO(Q) it is possible to extend it in the whole ℝn+1 preserving its BMO-norm or VMO-modulus, respectively. In the following we use this property without explicit references.

For this goal we recall some well known properties of the BMO functions.

Lemma 2.2

(John-Nirenberg lemma, [20]). Let aBMO and p ∈ (1, ∞). Then for any ballthere holds

( 1 | B | B | a ( y ) a B | p d y ) 1 p C ( p ) a * .

As an immediate consequence of Lemma 2.2 we get the following property.

Corollary 2.3.

Let aBMO then for all 0 < 2r < t holds

(6) | a B r + a B t + | C a * ln t r
where the constant is independent of a, x, t and r.

As mentioned before, we call weight a positive measurable function defined on ℝn × ℝ+. Given a weight w and a measurable set 𝕊 we denote by

w ( S ) = S w ( x ) d x
the w-measure of 𝕊. A weight w belongs to the Muckenhoupt class Ap, 1 < p < ∞, if
(7) [ w ] A p : = sup B ( 1 | B | B w ( x ) d x ) ( 1 | B | B w ( x ) p ' / p d x ) p / p ' < ,
where 1 p + 1 p ' = 1. Note that, for any ball we have (see [ 11])
(8) [ w ] A p ( B ) 1 / p = | B | 1 w L 1 ( B ) 1 / p w 1 / p L p ' ( B ) 1.

In case p = 1, we say that wA1 if

1 | B | B w ( x ) d x A  ess B inf w ( x )
and [ w ] A 1 is the smallest A for which the above inequality holds. It is an immediate consequence of (7) that whenever wAp than it satisfies the doubling property, precisely
(9) w ( 2 B r ) C ( n , p ) w ( B r ) .

The following lemma collects some of the most important properties of the Muckenhoupt weights.

Lemma 2.4

([11]). We have the following:

  1. (1)

    If wApfor some 1 ≤ p < ∞, then for all ƛ > 1 we have

    w ( λ B ) λ n p [ w ] A p w ( B ) .

  2. (2)

    The following equality is valid: A = ⋃1≤p<∞Ap.

  3. (3)

    If wA, then for all ƛ > 1 we have w ( λ B ) 2 λ n [ w ] A λ n w ( B ) .

  4. (4)

    If wApfor some 1 ≤ p ≤ ∞, then there exist C > 0 and δ > 0 such that for any balland a measurable set 𝕊 ⊂ ℬ,

    1 [ w ] A p ( | S | | B | ) w ( S ) w ( B ) C ( | S | | B | ) δ .

Lemma 2.5

([30, Theorem 5]). Let wA. Then the norm of BMO(w) is equivalent to the norm of BMO(ℝn), where

B M O ( w ) = { a : a * , w = sup x n , r > 0 1 w ( B r ( x ) ) B r ( x ) | a ( y ) a B r ( x ) , w | w ( y ) d y < }
and
a B r ( x ) , w = 1 w ( B r ( x ) ) B r ( x ) a ( y ) w ( y ) d y .

Lemma 2.6

(The John-Nirenberg inequality). Let aBMO,

  1. (1)

    there exist constants C1, C2 > 0, such that for all β > 0

    | x B : | a ( x ) a B | > β | C 1 | B | e C 2 β / a * ,    B n ;

  2. (2)

    for all p ∈ (1, ∞)

    a * = C sup B ( 1 | B | B | a ( y ) a B | p d y ) 1 / p ;

  3. (3)

    for all p ∈ [1, ∞) and wA a * = C sup B ( 1 w ( B ) B | a ( y ) a B | p w ( y ) d y ) 1 / p .

Definition 2.7.

Let ϕ(x, r) be weight in ϕ: ℝn × ℝ+ → ℝ+and ωAp, p ∈ [1, ∞). The generalized weighted Morrey space Mp,ϕ(ℝn, ω) or Mp,ϕ(ω) consists of all functions f L p , ω l o c ( n ) such that f p , φ ; ω = sup x n , r > 0 φ ( x , r ) 1 ( ω ( ε r ( x ) ) 1 ε r ( x ) | f ( y ) | p ω ( y ) d y ) 1 / p < .

The space Mp,ϕ(Q, ω) consists of L ω p ( Q ) functions provided the following norm is finite

f p , φ , w ; Q = sup x n , r > 0 φ ( x , r ) 1 ( ω ( Q r ( x ) ) 1 Q r ( x ) | f ( y ) | p ω ( y ) d y ) 1 / p .

The generalized Sobolev-Morrey spaceconsist of all Sobolev functions u W 2 , 1 p ( Q , ω ) with distributional derivatives D t l D x s u M p , φ ( Q , ω ) , 0 2 l + | s | 2 , endowed by the norm

u W 2 , 1 p , φ ( Q , ω ) = u t p , φ , ω ; Q + | s | 2 D s u p , φ , ω ; Q
and
W ˙ 2 , 1 p , φ ( Q , ω ) = { u W 2 , 1 p , φ ( Q , ω ) : u ( x ) = 0 , x Q } , u W ˙ 2 , 1 p , φ ( Q , ω ) = u W 2 , 1 p , φ ( Q , ω )
where ∂Q means the parabolic boundary Ω ∪ ﹛ Ω × (0, T)﹜.

Theorem 2.8.

(Main result) Let (i) and (ii) hold,aVMO(Q, ω) and u W ˙ 2 , 1 p ( Q , ω ) , ω), p(1, ∞), ωApbe a strong solution of (3). If fMp,ϕ (Q, ω) with ϕ(x, r) being measurable positive function satisfying

(10) r ( 1 + ln s r ) ess inf φ ( x , ς ) ω ( Q ς ( x ) ) 1 p s < ς < ω ( Q s ( x ) ) 1 p d s s C
for each ( x, r) ∈ Q × ℝ +, then u W ˙ 2 , 1 p , φ ( Q , ω ) and
(11) u W ˙ 2 , 1 p , φ ( Q , ω ) C f p , φ , ω ; Q
with C = C ( n , p , [ ω ] A p , Λ , Ω , T , a ; Q , η a ) and η a = i , j = 1 n η a i j .

If ϕ(x, r) = r(ƛ−n−2)/p, then Mp,ϕLp,ƛ and the condition (10) holds with a constant depending on n, p and ƛ. If ϕ(x, r) = ω(x, r)1/pr−(n+2)/p with ω : ℝn+1 × ℝ+ → ℝ+ satisfying the conditions

k 1 ω ( x 0 , s ) ω ( x 0 , r ) k 2    x 0 n + 1 , r s 2 r r ω ( x 0 , s ) s d s k 3 ω ( x 0 , r ) k i > i = 1 , 2 , 3
than we obtain the spaces Lp,ω studied in [ 27, 31]. The following results are obtained in [ 19] and treat continuity in Mp,ϕ(ℝ n+1, ω) of certain singular and nonsingular integrals.

Definition 2.9.

A measurable function 𝔎(x; ζ) : ℝn+1 × ℝn+1 \ ﹛0﹜ → ℝ is called variable parabolic Calderon-Zygmund kernel (PCZK) if:

  1. i)

    𝔎(x; ·) is a PCZK for a.a. xn+1:

    1. a)

      𝔎(x; ·)C(ℝn+1 \ ﹛0﹜),

    2. b)

      𝔎(x; μζ) = μ−(n+2) 𝔎(x; ζ) ∀μ > 0,

    3. c)

      S n K ( x ; ξ ) d σ ξ = 0 ,     S n | K ( x ; ξ ) | d σ ξ < + .

  2. ii)

    D ξ β K ; n + 1 × S n M ( β ) < for each multi-index β.

    Consider the singular integrals

    (12) K f ( x ) = P . V . n + 1 K ( x ; x y ) f ( y ) d y , [ a , f ] ( x ) = P . V . n + 1 K ( x ; x y ) [ a ( y ) a ( x ) ] f ( y ) d y .

Theorem 2.10.

For any fMp,ϕ(ℝn+1, ω) with (p, ϕ) as in Theorem 2.8 and aBMO there exist constants depending on n, p, ϕ, ω and the kernel such that

(13) K f p , φ , ω ; n + 1 C [ ω ] A p 1 p f p , φ , ω ; n + 1 [ a , f ] p , φ , ω ; n + 1 C [ ω ] A p 1 p a * f p , φ , ω ; n + 1 .

Corollary 2.11.

Let Q be a cylinder in + n + 1 , f M p , φ ( Q , ω ) , a B M O ( Q ) and K ( x , ξ ) : Q × + n + 1 \ 0 . Then the operators (12) are bounded in Mp,ϕ (Q, ω) with p, ϕ, and ω as in Theorem 2.10. Then

(14) K f p , φ , ω ; Q C [ ω ] A p 1 p f p , φ , ω ; Q , [ a , f ] p , φ , ω ; Q C [ ω ] A p 1 p a * f p , φ , ω ; Q
with C = C ( n , p , φ , [ ω ] A p , | Ω | , K ) .

Corollary 2.12.

Let aVMO and (p, ϕ) be as in Theorem 2.8. Then for any ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for anyr(x0) with a radius r(0, r0) and all fMp,ϕ(ℰr(x0), ω)

(15) [ a , f ] p , φ , ω ; ε r ( x 0 ) C ε f p , φ , ω ; ε r ( x 0 )
where C is independent of ϵ, f, r and x 0.

For any x ' + n and any fixed t > 0 define the generalized reflection

(16) T ( x ) = ( T ' ( x ) , t ) ,    T ( x ) = x ' 2 x n a n ( x ' , t ) a n n ( x ' , t )
where a n( x) is the last row of the coefficients matrix a( x) of (3). The function 𝒯′( x) maps + n into n and the kernel 𝔎( x; 𝒯( x) − y) = 𝒦( x; 𝒯′( x) − y′, t − τ) is a nonsingular one for any x , y D + n + 1 . Taking x = ( ʺ, −xn, t) there exist positive constants k 1 and k 2 such that
(17) k 1 ρ ( x ˜ y ) ρ ( T ( x ) y ) k 2 ρ ( x ˜ y )

For any f M p , φ ( D + n + 1 , ω ) with a norm

f p , φ , ω ; D + n + 1 = sup x D + n + 1 , r > 0 φ ( x , r ) 1 ( ω ( ε r ( x ) ) 1 ε r ( x ) | f ( y ) | p ω ( y ) d y ) 1 / p
and a B M O ( D + n + 1 , ω ) defines the nonsingular integral operators
(18) K ˜ f ( x ) = D + n + 1 K ( x ; T ( x ) y ) f ( y ) d y ˜ [ a , f ] ( x ) = D + n + 1 K ( x ; T ( x ) y ) [ a ( x ) a ( y ) f ( y ) d y . ]

Theorem 2.13.

Let a B M O ( D + n + 1 ) , ω A p and f M p , φ ( D + n + 1 , ω ) with (p, ϕ) as in Theorem 2.8. Then the operators K ˜ f and ˜ [ a , f ] are continuous in M p , φ ( D + n + 1 , ω ) and

(19)      K ˜ f p , φ , ω ; D + n + 1 C f p , φ , ω ; D + n + 1 , ˜ [ a , f ] p , φ , ω ; D + n + 1 C [ ω ] A p 1 p a * f p , φ , ω ; D + n + 1 .
with a constant independend of a and f.

Corollary 2.14.

Let a2 ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for any ε r + ( x 0 ) = ε r ( x 0 ) D + n + 1 with a radius r(0, r0) and center x0 = (ʺ, 0, 0) and for all f M p , φ ( ε r + ( x 0 ) , ω ) holds

(20) ˜ [ a , f ] p , φ , ω ; ε r + ( x 0 ) C ε f p , φ , ω ; ε r + ( x 0 ) ,
where C is independent of ϵ, f, r and x 0.

3 Proof of the main result

As it follows by [39], the problem (3) is uniquely solvable in W ˙ 2 , 1 p ( Q , ω ) .

We are going to show that fMp,ϕ(Q, ω) implies u W ˙ 2 , 1 p , φ ( Q , ω ) . For this goal we obtain an a priori estimate of u. Following the method used by Chiarenza, Frasca and Longo in [7] and [8], we prove the results considering two steps.

Interior estimate. For any x 0 + n + 1 consider the parabolic semi-cylinders 𝒞r(x0) = ℬr(x′0) × (t0r2, t0). Let ν𝒞0(𝒞r) and suppose that ν(x, t) = 0 for t ≤ 0. According to [[5], Theorem 1.4] for any xsupp ν the following representation formula for the second derivatives of ν holds true

(21) D i j υ ( x ) = P . V . n + 1 Γ i j ( x ; x y ) [ a h k ( y ) a h k ( x ) ] D h k υ ( y ) d y + P . V . n + 1 Γ i j ( x ; x y ) B υ ( y ) d y + B υ ( x ) S n Γ j ( x ; y ) v i d σ y ,

where ν(ν1, … , νn+1) is the outward normal to 𝕊n. Here Γ(x; ζ) is the fundamental solution of the operator 𝔅 and Γ(x; ζ) = ∂2Γ(x; ζ)/∂ζiζj.

Because of density arguments the representation formula (21) still holds for any νW2, 1p(𝒞r(x0), ω). The properties of the fundamental solution (cf. [5, 25, 38]) imply Γij are Calderon-Zygmund kernels in the sense of Definition 2.9. We denote by 𝒦ij and Cij the singular integrals defined in (12) with kernels 𝔎(x; x − y) = Γij (x; x − y). Then we can write that

(22) D i j υ ( x ) = i j [ a h k , D h k υ ] ( x ) + K i j ( B υ ) ( x ) + B υ ( x ) S n Γ j ( x ; y ) v i d σ y .

Because of Corollaries 2.11 and 2.12 and the equivalence of the metrics we get

D 2 υ p , φ , ω ; C r ( x 0 ) C ( ε D 2 υ p , φ , ω ; C r ( x 0 ) + B u p , φ , ω ; C r ( x 0 ) )

for some r small enough. Moving the norm of D2ν on the left-hand side we get

D 2 υ p , φ , ω ; C r ( x 0 ) C B υ p , φ , ω ; C r ( x 0 )

with a constant depending on n, p, ηa(r), ║a║∞,Q and ║DΓ║∞,Q. Define a cut-off function ϕ(x) = ϕ1(x′)ϕ2(t), with ϕ1C0(ℬr(x′0)), ϕ2C0(ℝ) such that

ϕ 1 ( x ' ) = 1 x ' B θ r ( x 0 ' ) 0 x ' B θ ' r ( x 0 ' ) , ϕ 2 ( t ) = 1    t ( t 0 ( θ r ) 2 , t 0 ] 0 t < t 0 ( θ ' r ) 2

with θ(0, 1), θ′ = θ(3 − θ)/2 > θ and |Dsϕ| ≤ C ∈ [θ(1 − θ)r]−s, s = 0, 1, 2, |ϕt| ∼ |D2ϕ|. For any solution uW2,1p(Q, ω) of (3) define ν(x) = ϕ(x)u(x)W2,1p(𝒞r, ω). Then we get

D 2 u p , φ , ω ; C θ r ( x 0 ) D 2 υ p , φ , ω ; C θ ' , r ( x 0 ) C B υ p , φ , ω ; C θ ' r ( x 0 ) C ( f p , φ , ω ; C θ ' r ( x 0 ) + D u p , φ , ω ; C θ ' r ( x 0 ) θ ( 1 θ ) r + u p , φ , ω ; C θ ' r ( x 0 ) [ θ ( 1 θ ) r ] 2 ) .

By the choice of θ′ it holds θ(1 − θ) ≤ 2θ′(1 − θ′) which leads to

[ θ ( 1 θ ) r ] 2 D 2 u p , φ , ω ; C θ r ( x 0 ) C ( r 2 f p , φ , ω ; Q + θ ( 1 θ ) r D u p , φ , ω ; C θ r ( x 0 ) + u p , φ , ω ; C θ r ( x 0 ) ) .

Introducing the semi-norms

Θ s = sup 0 < θ < 1 [ θ ( 1 θ ) r ] s D 2 u p , φ , ω ; C θ r ( x 0 ) s = 0 , 1 , 2

and taking the supremo with respect to θ and θ′ we get

(23) Θ 2 C ( r 2 f p , φ , ω ; Q + Θ 1 + Θ 0 ) .

The interpolation inequality [26, Lemma 4.2] gives that there exists a positive constant C independent of r such that

Θ 1 ε Θ 2 + C ε Θ 0  for any ε ( 0 , 2 ) .

Thus (23) becomes

[ θ ( 1 θ ) r ] 2 D 2 u p , φ , ω ; C θ r ( x 0 ) Θ 2 C ( r 2 f p , φ , ω ; Q + Θ 0 )

for each θ(0, 1). Taking θ = 1/2 we get the Caccioppoli-type estimate

D 2 u p , φ , ω ; C r / 2 ( x 0 ) C ( f p , φ , ω ; Q + 1 r 2 u p , φ , ω ; C r ( x 0 ) ) .

To estimate ut we exploit the parabolic structure of the equation and the boundedness of the coefficients

u t p , φ , ω ; C r / 2 ( x 0 ) a , Q D 2 u p , φ , ω ; C r / 2 ( x 0 ) + f p , φ , ω ; C r / 2 ( x 0 ) C ( f p , φ , ω ; Q + 1 r 2 u p , φ , ω ; C r ( x 0 ) ) .

Consider cylinders Q′ = Ω′ × (0, T) and = Ωʺ × (0, T) with Ω′ ⋐ Ωʺ ⋐ Ω, by standard covering procedure and partition of the unity we get

(24) u W 2 , 1 p , φ ( Q ' , ω ) C ( f p , φ , ω ; Q + u p , φ , ω ; Q " ) .

where C depends on n , p , [ ω ] A p 1 2 Λ , T D Γ ; Q , η a ( r ) , a , Q and dist(Ω′, ∂Ωʺ).

Boundary estimates. For any fixed R > 0 and x0 = (ʺ, 0, 0) define the semi-cylinders

C R + ( x 0 ) = C R ( x 0 ) D + n + 1 .

Without lost of generality we can take x0 = (0, 0, 0). Define ℬR+ = ﹛|x′| < R, xn > 0﹜, SR+ = ﹛|ʺ| < R, xn = 0, t(0, R2)﹜ and consider the problem

(25) B u : = u t a i j ( x ) D i j u = f ( x )    a .e . in  C R + , u : = u ( x ' , 0 ) = 0 ,             on    B R + , B u : = l i ( x ) D i u = 0        on    S R + .

Let uW2, 1p (CR+, ω) with u = 0 for t ≤ 0 and xn ≤ 0, then the following representation formula holds (see [26, 38])

D i j u ( x ) = I i j ( x ) J i j ( x ) + H i j ( x ) ,

where

I i j ( x ) = P . V . C R + Γ i j ( x ; x y ) F ( x ; y ) d y + f ( x ) S n Γ j ( x ; y ) v i d σ y ,    i , j = 1 , , n ; J i j ( x ) = C R + Γ i j ( x ; T ( x ) y ) F ( x ; y ) d y ; J i n ( x ) = J n i ( x ) = C R + Γ i l ( x ; T ( x ) y ) ( T ( x ) x n ) l F ( x ; y ) d y , i , j = 1 , , n 1 J n n ( x ) = C R + Γ l s ( x ; T ( x ) y ) ( T ( x ) x n ) l ( T ( x ) x n ) s F ( x ; y ) d y ; F ( x ; y ) d y = f ( y ) + [ a h k ( y ) a h k ( x ) ] D h k u ( y ) , H i j ( x ) = ( G i j * 2 g ) ( x ) + g ( x " , t ) S n G j ( x ; y " , x n , τ ) n i d σ ( y " , τ ) ,      i , j = 1 , , n , T ( x ) x n = ( 2 a n 1 ( x ) a n n ( x ) , , 2 a n n 1 ( x ) a n n ( x ) , 1 ) .

Here the kernel G = Γ 𝒬, is a byproduct of the fundamental solution and a bounded regular function 𝒬. Hence its derivatives Gij behave as Γij and the convolution that appears in Hij is defined as follows

( G i j * 2 g ) ( x ) = P . V . S R + G i j ( x ; x " y " , x n , t τ ) g ( y " , 0 , τ ) d y " d τ , g ( x " , 0 , t ) = [ ( l k ( 0 ) l k ( x " , 0 , t ) ) D k u l k ( 0 ) ( Γ k * F ) ] | x n = 0 ( x " , 0 , t ) , ( Γ k * F ) ( x ) = C R + Γ k ( x ; x y ) F ( x ; y ) d y .

Here Iij are a sum of singular integrals and bounded surface integrals hence the estimates obtained in Corollaries 2.11 and 2.12 hold true. On the nonsingular integrals Jij we apply the estimates obtained in Theorem 2.13 and Corollary 2.14 that give

(26) I i j p , φ , ω ; C R + + J i j p , φ , ω ; C R + C ( f p , φ , ω ; C R + + η a ( R ) D 2 u p , φ , ω ; C R + )

for all i, j = 1, … , n. To estimate the norm of Hij we suppose that the vector field l is extended in 𝒞R+ preserving its Lipschitz regularity and the norm. This automatically leads to extension of the function g in 𝒞R+ that is

(27) g ( x ) = ( l k ( 0 ) l k ( x ) ) D k u ( x ) l k ( 0 ) ( Γ k * F ) ( x ) .

Applying the estimates for the heat potentials [[25], Chapter 4] and the trace theorems in Lp [[2], Theorems 7.48, 7.53] (see also [[38], Theorem 1]) we get

C R + | ( G i j * 2 g ) ( y ) | p w ( y ) d y C ( C R + | g ( y ) | p w ( y ) d y + C R + | D g ( y ) | p w ( y ) d y ) .

Taking a parabolic cylinder ℐr(x) centered in some point x𝒞R+ we have

C R + I r ( x ) | G i j * 2 g ( y ) | p w ( y ) d y C ω ( I r ( x ) ) φ ( x , r ) p ( φ ( x , r ) p ω ( I r ( x ) ) C R + I r ( x ) | g ( y ) | p w ( y ) d y + φ ( x , r ) p ω ( I r ( x ) ) C R + I r ( x ) | D g ( y ) | p w ( y ) d y ) C ω ( I r ( x ) ) φ ( x , r ) p ( g p , φ , ω ; C R + p + D g p , φ , ω ; C R + p ) .

Moving φ ( x , r ) p ω ( I r ( x ) ) on the left-hand side and taking the supremo with respect to (x, r)𝒞R+ × ℝ+ we get

G i j * 2 g p , φ , ω ; C R + p C ( g p , φ , ω ; C R + p + D g p , φ , ω ; C R + p ) .

An immediate consequence of (27) is the estimate

g p , φ , ω ; C R + [ l k ( 0 ) l k ( ) ] D k u p , φ , ω ; C R + + l k ( 0 ) ( Γ k * F ) p , φ , ω ; C R + C R l L i p ( S ¯ ) D u p , φ , ω ; C R + + Γ k * f p , φ , ω ; C R + + Γ k * [ a h k ( ) a h k ( x ) ] D h k u p , φ , ω ; C R + .

The convolution Γk * f is a Riesz potential. On the other hand

| ( Γ k * f ) ( x ) | C C R + | f ( y ) | ρ ( x y ) n + 1 d y C R C R + | f ( y ) | ρ ( x y ) n + 2 d y C C R + | f ( y ) | ρ ( x y ) n + 2 d y

with a constant depending on T and diam Ω. Apply [[16], Theorem 4.8, [13], Theorem 3.1] that gives

Γ k * f p , φ , ω , C R + C f p , φ , ω , C R + .

Analogously

| Γ k * [ a h k ( ) a h k ( x ) ] D h k u ( ) | C C R + | a h k ( y ) a h k ( x ) | | D h k u ( y ) | ρ ( x y ) n + 2 d y

with a constant depending on diam Ω and T. The kernel ρ(x − y)−(n+2) is a nonnegative singular one and applying again the results for sub-linear integrals [[13], Theorem 3.7] we get

| Γ k * [ a h k ( ) a h k ( x ) ] D h k u ( ) | ρ , ω , φ , C R + C a * D 2 u ρ , ω , φ , C R + .

Hence

(28) g ρ , ω , φ , C R + C ( R l L i p ( S ¯ ) D u p , φ , ω ; C R + + f p , φ , ω ; C R + + R η a ( R ) D 2 u p , φ , ω ; C R + ) .

Further, the Rademacher theorem asserts existence almost everywhere of the derivatives DhlkL, thus

D h g ( x ) = D h l k ( x ) D k u ( x ) + [ l k ( 0 ) l k ( x ) ] D k h u l k ( 0 ) ( Γ k h * F ) ( x ) .

The Mp,ϕ(ω) norm of the last term is estimated as above and

(29) D g p , φ , ω ; C R + C ( D l : S D u p , φ , ω ; C R + + R l L i p ( S ¯ ) D 2 u p , φ , ω ; C R + + f p , φ , ω ; C R + + η a ( R ) D 2 u p , φ , ω ; C R + ) .

Finally unifying (26), (28) and (29) we get

D 2 u p , φ , ω ; C R + C ( f p , φ , ω ; Q + ( 1 + R ) D u p , φ , ω ; C R + + ( R + η a ( R ) + R η a ( R ) ) D 2 u p , φ , ω ; C R + )

with a constant depending on known quantities and ║lLip(S) and ║Dl∞;𝒮. Direct calculations lead to an interpolation inequality in Mp,ϕ(ω) analogous to [[25], Lemma 3.3] (cf. [41])

D u p , φ , ω ; C R + δ D 2 u p , φ , ω ; C R + + C δ u p , φ , ω ; C R + , δ ( 0 , R ) .

Taking 0 < δ = R R + 1 < R we get

D 2 u p , φ , ω ; C R + C ( f p , φ , ω ; Q + R D 2 u p , φ , ω ; C R + + C R u p , φ , ω ; C R + + ( R + η a ( R ) + R η a ( R ) ) D 2 u p , φ , ω ; C R + ) .

Choosing R small enough and moving the terms containing the norm of D2u on the left-hand side we get

D 2 u p , φ , ω ; C R + C ( f p , φ , ω ; C R + + 1 R u p , φ , ω ; C R + ) .

Because of the parabolic structure of the equation analogous estimate holds also for ut. Further the Jensen inequality applied to gives

u p , φ , ω ; C R + C R 2 u t p , φ , ω ; C R + C ( R 2 f p , φ , ω ; C R + + R u p , φ , ω ; C R + ) .

Choosing R smaller, if necessary, we get u p , φ , ω ; C R + C f p , φ , ω ; C R + and therefore

(30) u W 2 , 1 p , φ ( C R + , ω ) C f p , φ , ω ; C R + C f p , φ , ω ; C R + .

Making a covering ﹛Cα+﹜, α𝒜 such that Q \ Q α A C α + , considering a partition of unity subordinated to that covering and applying (30) for each 𝒞α+ we get

(31) u W 2 , 1 p , φ ( Q \ Q ' , ω ) C f p , φ , ω ; Q

with a constant depending on n , p , [ ω ] A p 1 2 , Λ , T , d i a m Ω , and ║Dl∞,𝒮. The estimate (11) follows from (24) and (31).

Acknowledgement

The authors are thankful to the referee for very valuable comments. The research of V.S. Guliyev and M.N. Omarova is partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015.

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Received: 2015-11-15
Accepted: 2016-1-14
Published Online: 2016-2-13
Published in Print: 2016-1-1

© 2016 Guliyev and Omarova, published by De Gruyter Open.

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