Estimates in generalized Morrey spaces for linear parabolic systems

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J. Math. Anal. Appl. ••• (••••) •••–•••

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Estimates in generalized Morrey spaces for linear parabolic systems Matt McBride Department of Mathematics and Statistics, Mississippi State University, 175 President’s Circle, Mississippi State, MS 39762, USA

a r t i c l e

i n f o

Article history: Received 8 June 2016 Available online xxxx Submitted by H.-M. Yin Keywords: Morrey spaces Regularity Parabolic systems

a b s t r a c t j i In this study, we consider the parabolic system uit − Dα (aαβ ij Dβ u ) = − div f in 2,λ the generalized Morrey Space Lϕ , where we aim to understand the regularity of 2,λ the solutions to this system. We show that: (1) if aαβ ij ∈ C(QT ), then Du ∈ Lϕ ; 2,λ and (2) if aαβ ij ∈ V M O(QT ), then Du ∈ Lϕ . Moreover, we obtain estimates of the gradient for the solutions to the system, thereby showing the regularity of the solutions. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The regularity of the solutions to differential equations is a vast and important topic in the theory of differential equations. Studying the regularity of the solutions to differential equations can facilitate estimates of the gradient for solutions in different function spaces. Thus, if u is a weak solution to a partial differential equation and the coefficients belong to some special class of function spaces, then the gradient Du belongs to a Morrey space and satisfies a certain type of integral estimate. Morrey spaces are a particular class of function spaces, which are of interest because of this reason. The standard textbooks by [1] and [3] explained some basic theory regarding these Morrey type estimates for second order elliptic equations and second order parabolic equations, respectively. Lieberman [4] described these types of estimates for elliptic and parabolic equations where the highest coefficient term belongs to the V M O space. Huang [2] provided gradient estimates for a system of elliptic equations with coefficients in the V M O and BM O spaces, thereby implying some regularity for the solutions. The main of the present study is to obtain these types of gradient estimates to deduce some regularity for the solutions of system (1.1). In this study, we will investigate linear parabolic systems with the following form   j uit − Dα aαβ (1.1) = − div f i (x, t), ij (x, t)Dβ u E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.03.021 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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where i, j = 1, . . . , N ; α, β = 1, . . . , n and the repeated indices denote summation such as i j

aij ξ ξ =

n  n 

aij ξ i ξ j .

i=1 j=1

Let Ω ⊂ Rn and QT = Ω × [0, T ]. Throughout this study, we assume a uniform ellipticity condition, i.e., for Λ > 0, ξ ∈ R(n+1)N , and (x, t) ∈ QT , we have: i j 2 Λ−1 |ξ|2 ≤ aαβ ij (x, t)ξα ξβ ≤ Λ|ξ| .

(1.2)

The main aim of this study is to demonstrate that we can obtain gradient estimates in generalized Morrey αβ spaces, L2,λ ϕ , for weak solutions of Equation (1.1), i.e., given a certain class of aij and weak solution u to (1.1), we have Du ∈ L2,λ ϕ and an integral estimate of Du. The types of integral estimate that we seek were discussed by [2] for elliptic systems. The remainder of this paper is organized as follows. In Section 2, we provide preliminary details, such as the parabolic cylinder, as well as stating the definitions for the Morrey space and both the BM O and V M O spaces. We also show some basic functional analysis properties of the generalized Morrey space. In Section 3, we consider the test functions for system (1.1). We also prove the main result if the coefficients for (1.1) are constant. This section is slightly technical but it provides the basis for the main technique used throughout the rest of this study. In Section 4, system (1.1) is considered as a homogeneous system and a nonhomogeneous system, where the coefficients are continuous, and then in the V M O space based on a series of lemmas and theorems. Finally the main result is shown, where we obtain gradient estimates for the nonhomogeneous system (1.1) with V M O coefficients. 2. Preliminaries In this section, we discuss some theorems and lemmas that are needed for the main results of this study. We also define some notations used throughout this study. We denote the n-dimensional ball centered at x0 with radius R as BR (x0 ) = {x ∈ Rn : x − x0  < R}. We also let z represent a n + 1 dimensional coordinate, i.e., z ∈ Rn × (0, T ], where z = (x, t), x ∈ Rn and t ∈ (0, T ]. Similarly, z0 = (x0 , t0 ). We denote the parabolic cylinder in Rn+1 with vertex at z0 by QR (z0 ) = BR (x0 ) × (t0 − R2 , t0 ]. The boundary of the parabolic cylinder comprises the lateral walls, lower boundary, and lower corners, but we use ∂p QR to denote the parabolic boundary of the parabolic cylinder. Next, we define the Morrey space for the parabolic setting. Definition. The parabolic Morrey space is defined as follows ⎧ ⎫ ⎛ ⎞ p1 ⎪ ⎪ ⎪ ⎪ ¨ ⎨ ⎬ 1 −λ ⎜ ⎟ p p,λ p ρ Lϕ (QT ) = f ∈ L (QT ) : ⎝ sup |f | dz ⎠ < ∞ ⎪ ⎪ z0 ∈QT ,0≤ρ≤d ϕ(ρ) ⎪ ⎪ ⎩ ⎭ QT ∩Qρ (z0 )

(2.1)

with 1 ≤ p < ∞, 0 ≤ λ ≤ n + 2, and ϕ is a continuous function on [0, d], ϕ > 0 on (0, d], and d is the diameter of QT = Ω × (0, T ] with Ω ⊂ Rn .

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We now provide the first lemma, which states that the parabolic Morrey space is a Banach space. Lemma 2.1. The space Lp,λ ϕ (QT ) is a Banach space under the following norm ⎛ ⎜ f Lp,λ =⎝ ϕ

⎞ p1

¨

1 −λ ρ 0≤ρ≤d ϕ(ρ)

⎟ p |f | dz ⎠ .

sup

z0 ∈QT ,

QT ∩Qρ (z0 )

Proof. First, we must show that  · Lp,λ is indeed a norm. Then, we must show that the space is comϕ plete. Showing that the condition that f Lp,λ is positive definite and the homogeneity condition is trivial. ϕ However, the triangle inequality is not obvious. For this calculation, to minimize the notation, the interval in which the supremum is taken is suppressed. Using Minkowski’s inequality, the fact that ϕ > 0 on (0, d], and the fact that the supremum is p independent, we obtain the following: ⎛ 1 ⎜ 1 · f + gLp,λ = sup ⎝ ϕ ϕ(ρ) ρλ 

1 1 · λ ϕ(ρ) ρ ⎛

≤ sup

1 ⎜ 1 · ≤ sup ⎝ ϕ(ρ) ρλ

⎞ p1

¨

⎟ |f + g|p dz ⎠ = sup



QT ∩Qρ (z0 )

 p1

 p1

1 1 · ϕ(ρ) ρλ

f + gLp

(f Lp + gLp ) ⎞ p1

¨ QT ∩Qρ (z0 )

⎛

1 ⎟ ⎜ 1 · |f |p dz ⎠ + sup ⎝ ϕ(ρ) ρλ

¨

⎞ p1 ⎟ |g|p dz ⎠

QT ∩Qρ (z0 )

= f Lp,λ + gLp,λ . ϕ ϕ This string of inequalities shows that the triangle inequality is satisfied, and thus  · Lp,λ defines a norm. ϕ We only need to show that the space is complete under the norm. Thus, we must show that every Cauchy p,λ ∞ p,λ sequence from Lp,λ ϕ (QT ) converges to an element in Lϕ (QT ). Let {fk }k=1 be a Cauchy sequence in Lϕ . Tschebyshev’s inequality implies that ¨

m {z ∈ QT : |fk (z) − fm (z)| > ε} ≤ ε−p

p

|fk − fm | dz,

QT ∩Qρ (z0 )

where m is the standard Lebesgue measure. Therefore, a subsequence {fkj } and f exist such that {fkj } converges to f a.e. in QT . Then, for every ε > 0, K exists such that fkj − fk Lp,λ < ε if kj , k > K. ϕ If we let kj → ∞, then Fatou’s lemma implies that f − fk Lp,λ < ε for k > K. Thus, f ∈ Lp,λ by ϕ ϕ p,λ f Lp,λ ≤ f − fk Lp,λ + fk Lp,λ < ∞ and f − fk Lp,λ → 0 as k → ∞. Therefore Lϕ is complete, and ϕ ϕ ϕ ϕ hence it is a Banach space. This completes the proof. 2 In the following, we set p = 2 in the Morrey space Lp,λ ϕ . We state this next for convenience.

L2,λ ϕ (QT ) =

⎧ ⎪ ⎪ ⎨

⎛

⎪ ⎪ ⎩

⎜ f ∈ L2 (QT ) : ⎝

1 −λ ρ z0 ∈QT ,0≤ρ≤d ϕ(ρ)

¨

sup

QT ∩Qρ (z0 )

⎞ 12 ⎟ 2 |f | dz ⎠ < ∞

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

The next definition was provided by [2] and the subsidiary lemma was also proved in the same study. The following definition defines when a function is said to be “almost” increasing, where we place almost in

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quotations because we would naturally expect that the function is increasing everywhere except on a set of measure zero, but this is not the case. Definition. A function h : [0, d] → [0, ∞) is said to be almost increasing if Kh ≥ 1 exists such that h(s) ≤ Kh h(t) for 0 ≤ s ≤ t ≤ d. Now we state the lemma proved by [2]. Lemma 2.2. Let H be a non-negative almost increasing function in [0, R0 ] and F is a positive function on (0, R0 ]. We assume the following.   β (1) A, B, ε, β > 0 exists such that H(ρ) ≤ A (ρ/R) + ε H(R) + BF (R) for 0 ≤ ρ ≤ R ≤ R0 . (2) γ ∈ (0, β) exists such that

ργ F (ρ)

is almost increasing in (0, R0 ].

(ρ) Thus, ε0 = ε0 (A, β, γ) and C = C (A, β, γ, KH , K) exist such that if ε < ε0 , then H(ρ) ≤ C FF(R) H(R) + CBF (ρ).

Next we define the bounded mean oscillation and vanishing mean oscillation spaces in the parabolic setting. This is already well understood for an n-dimensional ball or on some bounded domain Ω in Rn . We define the bounded mean oscillation, BM O(QT ), in the following manner. Definition. Let ψ ∈ C[0, d] and ψ > 0 on [0, d], so ψ is a positive continuous function on the interval [0, d]. BM O(QT ) is defined by BM Oψ (QT ) = ⎧ ⎫ ⎛ ⎞ 12 ⎪ ⎪ ⎪ ⎪ ˆ ˆ ⎨ ⎬   1 2 ⎜ ⎟ 2   f (z) − fQT ∩Qρ (z0 ) (z0 ) dz ⎠ < ∞ , = f ∈ L (QT ) : ⎝ sup −− ⎪ ⎪ z0 ∈Q,0≤ρ≤d ψ(ρ) ⎪ ⎪ ⎩ ⎭ QT ∩Qρ (z0 ) ˆˆ where fA = −− f (z)dz = A

1 m(A)

¨ f (z)dz and A ⊂ Rn+1 . A

Next, we define the vanishing mean oscillation space for the parabolic setting in a similar manner. Definition. If ψ = 1, where ψ is the continuous function defined in the definition of the bounded mean oscillation space, then V M O(QT ) is defined by   V M O(QT ) = f ∈ BM O(QT ) : [f ]BM O(QT :σ) → 0 as σ → 0 , where ⎛ ⎜ [f ]BM O(QT :σ) = ⎝

sup

z0 ∈Q,0≤ρ≤σ

ˆˆ −− QT ∩Qρ (z0 )

 f (z) − fQ

T ∩Qρ (z0 )

⎞ 12

2 ⎟ (z0 ) dz ⎠ .

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3. Weak solutions In this section, we discuss the weak solutions to a variation of the system of parabolic partial differential equations that we consider in this study. Let aαβ ij be constant and consider the following system in QT :   j uit − Dα aαβ D u = 0. β ij

(3.1)

For QR (z0 ) ⊂ QT , let ui ξ 2 (x)η(t) be a test function with ξ ∈ C0∞ (BR (x0 )), the space of smooth functions C vanishing at infinity, 0 ≤ ξ ≤ 1, and |Dξ| ≤ R−ρ , with Bρ (x0 ) ⊂ BR (x0 ) ⊂ Ω and η(t) defined in the following manner  η(t) =

t−(t0 −R2 ) R2 −ρ2

t ∈ (t0 − R2 , t0 − ρ2 ) . t ∈ [t0 − ρ2 , t0 )

1

After multiplying Equation (3.1) by the test function, applying integration by parts, and noting that the boundary term is zero by the definition of η and ξ, we obtain ¨



0=

  j D u uit − Dα aαβ ui ξ 2 η dxdt β ij

BR (x0 )×(t0 −R2 ,t]

¨

 i 2  j uit ui ξ 2 η + aαβ ij Dβ u Dα u ξ η dxdt

= BR (x0 )×(t0 −R2 ,t]

¨



= BR (x0 )×(t0 −R2 ,t]

1 2 |u| 2

¨

 ξ 2 η dxdt t

 2  j i i aαβ ij Dβ u ξ Dα u + 2ξu Dα ξ η dxdt

+ BR (x0 )×(t0 −R2 ,t]



¨

= BR (x0 )×(t0 −R2 ,t]

1 2 |u| η 2

¨

 ξ2 − t

1 2 2 |u| ξ ηt dxdt 2

  2 i j i aαβ ij Dβ u ξ u + 2ξu Dα ξ η dxdt.

+ BR (x0 )×(t0 −R2 ,t]

Then, by the uniform ellipticity condition (1.2) and the Cauchy–Schwarz inequality, we have ˆ

1 2 |u(x, t)| ξ 2 (x) dx + C 2

BR (x0 )

1 ≤ 2

ˆt

2

ξ 2 (x) |Du| dxdt t0 −R2 BR (x0 )

ˆ

ˆt 2

|u| ξ ηt dxdt + C t0 −R2 BR (x0 )

ˆ

t0 −R2 BR (x0 )

t0 −R2

  1 2 2 |u| |Dξ| η + ξ ηt dxdt. 2

and by a simple computation, ηt ≤

2

|Dξ| |u| η dt

2

≤C

C R−ρ

ˆ

2 2

ˆt

Then, since |Dξ| ≤

ˆt

C R2 −ρ2 ,

we have

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ˆ

ˆt

1 2 |u(x, t)| ξ 2 dx + 2

ˆ 2

|Du| ξ 2 η dxdt

t0 −R2 BR (x0 )

BR (x0 )

ˆt



ˆ 2

≤C

|u| t0

−R2

1 1 + 2 2 (R − ρ) R − ρ2

 dxdt.

BR (x0 )

This last inequality also implies the following inequality ¨

ˆ

C |Du| dz ≤ (R − ρ)2

|u| dx + 2

sup t0 −ρ2 ≤t≤t0 Bρ (x0 )

¨ |u|2 dz.

2

Qρ (z0 )

(3.2)

QR (z0 )

Inequality (3.2) is called the energy estimate for the system of partial differential equations stated in Equation (3.1). We derive this energy estimate because it has applications in obtaining the proofs of the main results and it also lets us define a Sobolev space counterpart for parabolic equations. Consider the following definition of a space   V2 (QT ) = u : u ∈ L∞ (0, T ; L2 (QT )), Du ∈ L2 (QT ) . V2 (QT ) is said to be the Sobolev space counterpart for parabolic equations. We use this definition because it is very similar to the definition of the Sobolev space with p = q = 2. Using these energy estimates and the Sobolev embedding theorem, we can obtain the Morrey estimate for the system of partial differential equations stated in (3.1) with constant coefficients. The Morrey estimate for a system of homogeneous parabolic partial differential equations with constant coefficients is ¨ |Du|2 dz ≤ C

 ρ n+2 ¨ R

Qρ (z0 )

|Du|2 dz

(3.3)

QR (z0 )

for QR (z0 ) ⊂ QT and 0 ≤ ρ ≤ R. We end this section with a formal statement of this and a proof. Lemma 3.1. Let u ∈ V2 (QT ) be a solution to the system of partial differential equations defined in Equation (3.1) in QT = Ω × (0, T ]. Then, for QR (z0 ) ⊂ QT and 0 ≤ ρ ≤ R, the following inequality holds ¨ |Du|2 dz ≤ C Qρ (z0 )

 ρ n+2 ¨ R

|Du|2 dz.

QR (z0 )

Proof. Recall the system of partial differential equations stated in Equation (3.1):   j uit − Dα aαβ = 0. ij Dβ u

(3.4)

The coefficients, aαβ ij , are constant, so differentiating the equation above with respect to x shows that Dα u is still a solution to the system of differential equations. By [5], we have ˆˆ  ρ 2 ˆˆ 2 −− |u| dz ≤ C −− |u|2 dz, R Qρ (z0 )

QR (z0 )

where u is a solution to the system of equations given above. Using this inequality and the fact that Dα u is still a solution, we obtain

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¨ |Du| dz ≤ C 2

 ρ n+2 ¨

|Du|2 dz

R

Qρ (z0 )

7

QR (z0 )

and the result follows. Thus, this completes the proof. 2 We now consider the main results of this study. 4. Main results In the previous section, the Morrey estimate of interest was obtained when the coefficients, aαβ ij , are constant, as mainly described by [5]. In this section, we extend this result to the system of partial differential equations defined in Equation (1.1). First, we establish the Morrey estimate for the case where aαβ ij ∈ C(QT ), αβ the space of continuous functions on the closure of QT , and the second case where aij ∈ L∞ (QT ) ∩ V M O(QT ), the space of all bounded functions with vanishing mean oscillation. Theorem 4.1. Let u ∈ V2 (QT ) be a weak solution in QT to the following system of partial differential equations   j uit − Dα aαβ = −div f i ij (z)Dβ u for i = 1, . . . , N . Let aαβ ij ∈ C(QT ) and assume that they satisfy the uniform ellipticity condition with γ−λ

r f i ∈ L2,λ ϕ (QT ). Suppose that λ and γ exist such that λ < γ < n + 2, and that the function ϕ(r) is almost   increasing, then Du ∈ L2,λ ϕ (Q ) for any Q ⊂⊂ QT , for QR (z0 ) ⊂ QT and ρ ≤ R. Moreover, the following inequality holds

¨ |Du|2 dz ≤ C

ρλ ϕ2 (ρ) Rλ ϕ2 (R)

Qρ (z0 )

¨ |Du|2 dz + Cϕ2 (ρ)ρλ f 2L2,λ . ϕ

QR (z0 )

Proof. Let w satisfy the following system   ⎧ j ⎨ wti − Dα aαβ (z )D w =0 0 β ij ⎩

w=u

in QR (z0 )

(4.1)

on ∂p QR (z0 )

where z0 is a fixed point. Then, v = u − w will satisfy the following system      ⎧ αβ αβ j j ⎨ vti − Dα aαβ (z )D v (z) − a (z ) D u = D a − div f i 0 β α 0 β ij ij ij ⎩

in QR (z0 )

.

(4.2)

on ∂p QR (z0 )

v=0

Since v = u − w, we have Dv = Du − Dw. Then, by Young’s inequality, we obtain ¨

¨ |Du| dz ≤ 2 2

Qρ (z0 )

  |Dw|2 + |Dv|2 dz.

Qρ (z0 )

In addition, since z0 is a fixed point, this means that aαβ ij (z0 ) is constant in z such that w solves the equation given by Equation (3.1); therefore, by Lemma 3.1, we have

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¨ |Dw| dz ≤ const 2

 ρ n+2 ¨

|Dw|2 dz.

R

Qρ (z0 )

QR (z0 )

From the two inequalities above, it follows that ¨  ρ n+2 ¨ |Du|2 ≤ const R Qρ (z0 )

¨ |Dw|2 +

QR (z0 )

≤ const

 ρ n+2 ¨

|Dv|2

Qρ (z0 )

|Dw| + const

R QR (z0 )

≤ const

¨ |Dv|2

2

 ρ n+2 ¨

QR (z0 )

¨

|Du|2 + const

R QR (z0 )

|Dv|2 .

QR (z0 )

After multiplying Equation (4.2) by v, applying integration by parts, and noting that v = 0 on ∂p QR (z0 ), we obtain the following ¨  ¨  αβ αβ j i aαβ (z )D v D v dz = (z ) − a (z) Dβ uj Dα v i dz a 0 β α 0 ij ij ij QR (z0 )

QR (z0 )

¨

¨ f Dα v dz − i

+

i

QR (z0 )

vti v i dz.

QR (z0 )

The equality above and the uniform ellipticity condition (1.2) imply that ¨

¨

    αβ  αβ aij (z) − aij (z0 ) |Du||Dv| + |f ||Dv| dz.

|Dv|2 dz ≤ const QR (z0 )

QR (z0 )

   αβ  αβ Since aαβ ∈ C(Q ), then for a sufficiently small R, we have (z) − a (z ) a  < ε for any ε > 0. Therefore, T 0 ij ij ij from the inequality above and using the Cauchy–Schwarz inequality, we obtain ¨ ¨ |Dv|2 dz ≤ const (ε|Du| + |f |) |Dv|dz QR (z0 )

QR (z0 )

⎛ ⎜ ≤ const ⎝

¨

⎞ 12 ⎛ ⎟ ⎜ (ε|Du| + |f |)2 dz ⎠ ⎝

QR (z0 )

¨

⎞ 12 ⎟ |Dv|2 dz ⎠ .

QR (z0 )

This last inequality yields ⎛ ⎜ ⎝

¨

QR (z0 )

⎞ 12

⎛

⎟ ⎜ |Dv|2 dz ⎠ ≤ const ⎝

¨

⎞ 12 ⎟ (ε|Du| + |f |)2 dz ⎠ .

QR (z0 )

We can assume that the integral used for dividing by is positive because we assume it is zero, then since |Dv| ≥ 0 has an integral of zero, this implies that |Dv| = 0, but this is only true if the gradient of v vanishes at every point in the parabolic cylinder, and this would then imply that the coefficients aαβ ij (z) are constant for every z, which is the case discussed already. By squaring both sides of the last inequality and applying Young’s inequality, we obtain the following estimate

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¨

¨

¨

|Dv| dz ≤ const · ε

|Du| dz + const

2

QR (z0 )

9

|f |2 dz.

2

QR (z0 )

QR (z0 )

Therefore, we obtain the following 

¨ |Du|2 dz ≤ Qρ (z0 )

const

R

 ¨ |Du|2 dz + const

+ε

ϕ2 (R)Rλ ϕ2 (R)Rλ

QR (z0 )

 ≤

 ρ n+2

const

 ρ n+2 R

¨ |f |2 dz QR (z0 )

 ¨

|Du|2 dz + constϕ2 (R)Rλ f 2L2,λ dz.

+ε

ϕ

QR (z0 )

Then, the desired result follows immediately from Lemma 2.2. Thus, the theorem has been proved. 2 We only have to show that the Morrey estimate is valid for continuous functions on the closure of the parabolic domain, which can be considered a finite cylinder. Before the desired estimate can be proved, we need two additional lemmas. The first lemma is the “reverse” Hölder inequality given by [6]. We use this to prove the second lemma, which we need before the final estimate is obtained. Lemma 4.2. Let u ∈ V2 (QT ) be a weak solution to the following system   j uit − Dα aαβ =0 ij (z)Dβ u in QT with i = 1, . . . , N . Assuming that aαβ ij satisfy the uniform ellipticity condition (1.2), then some s > 2 exists such that Du ∈ Lsloc (QT ) and for every QR ⊂ Q4R ⊂ QT , the following inequality holds ⎞ 1s ⎛ ⎞ 12 ⎛ ˆˆ ˆˆ ⎝ −− |Du|s dz ⎠ ≤ C ⎝ −− |Du|2 dz ⎠ . QR

Q4R

The proof of this lemma is omitted because it was provided by [6]. We now consider the second lemma needed for the last Morrey estimate that we require. Lemma 4.3. Let u ∈ V2 (QT ) be a weak solution to the following system   j uit − Dα aαβ (z)D u =0 β ij ∞ in QT with i = 1, . . . , N . Assume that aαβ ij ∈ L (QT ) ∩ V M O(QT ) and they satisfy the uniform ellipticity condition. Then, for any 0 < μ < n + 2, R0 and C exist, where they depend only on n + 2, N , μ, Λ, and   such that for ρ ≤ R ≤ 12 min(R0 , dist(z0 , ∂p QT )), the following inequality holds aαβ ij BM O(QT ;σ)

¨ |Du| dz ≤ C 2

 ρ μ ¨ R

Qρ (z0 )

|Du|2 dz.

QR (z0 )

Proof. First, we define the following 

aαβ ij

 z0 R

ˆˆ := −− aαβ ij (x, t) dxdt. QR (z0 )

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Doctopic: Partial Differential Equations

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M. McBride / J. Math. Anal. Appl. ••• (••••) •••–•••

10

As in Theorem 4.1, let w and v = u − w satisfy the following systems of partial differential equations, respectively:   ⎧ j ⎨ wti − Dα aαβ =0 ij (z0 )Dβ w ⎩

in QR (z0 )

(4.3)

w = u on ∂p QR (z0 )

and  ⎧  ⎪ ⎨ vti − Dα aαβ ij ⎪ ⎩

 z0 R

Dβ v j

= Dα

   αβ aαβ (z) − a ij ij

 z0 R

 Dβ uj

in QR (z0 )

v=0

.

(4.4)

on ∂p QR (z0 )

Similar to the beginning of the proof for Theorem 4.1, we can obtain the following estimate ¨ |Du| dz ≤ const 2

 ρ n+2 ¨ R

Qρ (z0 )

¨ |Du| dz + const

|Dv|2 dz.

2

QR (z0 )

QR (z0 )

Multiplying v by Equation (4.4), applying integration by parts, and reasoning in a similar manner to Theorem 4.1, we obtain the following inequality    2  αβ a (z) − aαβ  |Du|2 dz. ij  ij  z0 R

¨

¨ |Dv| dz ≤ const 2

QR (z0 )

QR (z0 )

Using Hölder’s inequality, we obtain the following inequality ¨

⎛ ⎜ |Dv|2 dz ≤ const ⎝

QR (z0 )

¨

⎞ p1 ⎛  2p ¨    αβ  ⎜ a (z) − aαβ  dz ⎟ ⎠ ⎝ ij  ij  z0 R

QR (z0 )

⎞ q1 ⎟ |Du|2q dz ⎠

QR (z0 )

= const · m (QR (z0 )) ⎛ ⎞ s−2 ⎞ 2s ⎛ s 2s  s−2 ˆˆ ˆˆ      ⎜ ⎟ ⎟ ⎜ αβ s  dz ⎠ × ⎝ −− aαβ ⎝ −− |Du| dz ⎠ . ij (z) − aij  z0 R QR (z0 )

QR (z0 )

Using the fact that aαβ ij ∈ V M O, we deduce that ⎛

¨ |Dv|2 dz ≤ const · ε

s−2 s

⎞ 2s ˆˆ ⎜ ⎟ · m (QR (z0 )) ⎝ −− |Du|s dz ⎠ .

QR (z0 )

QR (z0 )

Then, from Lemma 4.2, it follows that ¨

¨ |Dv| dz ≤ const · ε 2

QR (z0 )

|Du|2 dz.

Q4R (z0 )

Therefore, by combining this inequality and the first inequality from the beginning of this proof, we obtain the following inequality

JID:YJMAA

AID:21218 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.209; Prn:23/03/2017; 17:48] P.11 (1-12)

M. McBride / J. Math. Anal. Appl. ••• (••••) •••–•••

¨ |Du| dz ≤ const 2

Qρ (z0 )

≤ const

¨

 ρ n+2 ¨

|Du| dz + const · ε

QR (z0 )

ρ n+2 R

|Du|2 dz

2

R 

11

Q4R (z0 )

 ¨ +ε

|Du|2 dz.

QR (z0 )

Then, by using Lemma 2.2, we can obtain the desired result. Thus, this completes the proof. 2 We are now in the position to state and prove the final theorem in this study. We have all that we require to obtain the final Morrey estimate, which extends the result for the system of linear elliptic partial differential equations. Theorem 4.4. Let u ∈ V2 (QT ) be a weak solution to the following system of parabolic partial differential equations   j uit − Dα aαβ = −div f i ij (z)Dβ u in QT with i = 1, . . . , N , and let aαβ ij satisfy the uniform ellipticity condition. Suppose that λ and γ exist γ−λ

∞ such that λ < γ < n + 2, and that the function ϕr 2 (r) is almost increasing. If aαβ ij ∈ L (QT ) ∩ V M O(QT ) 2,λ   and f i ∈ L2,λ ϕ (QT ), then Du ∈ Lϕ (Q ) for any Q ⊂⊂ QT and for QR ⊂ QT and ρ ≤ R. Moreover, the following interior integral estimate holds

¨ |Du|2 dz ≤ C

ρλ ϕ2 (ρ) Rλ ϕ2 (R)

Qρ

¨ |Du|2 dz + Cϕ2 (ρ)ρλ f 2L2,λ . ϕ

QR

Proof. Again, as in Theorem 4.1, let w and v = u − w satisfy the following systems of partial differential equations, respectively:   ⎧ j ⎨ wti − Dα aαβ (z)D w =0 β ij ⎩

in QR (z0 )

(4.5)

w = u on ∂p QR (z0 )

and   ⎧ j ⎨ vti − Dα aαβ = −div f i ij (z)Dβ v ⎩

in QR (z0 )

.

(4.6)

|Dv|2 dz.

(4.7)

on ∂p QR (z0 )

v=0

Applying Lemma 4.3 to w yields the following inequality ¨ |Du| dz ≤ const 2

 ρ μ ¨ R

Qρ (z0 )

¨ |Du| dz + const 2

QR (z0 )

QR (z0 )

Then, after multiplying v by Equation (4.6), applying integration by parts, and reasoning in a similar manner to the proof of Theorem 4.1, we obtain the following ¨

¨ |Dv| dz ≤ const 2

QR (z0 )

QR (z0 )

|f ||Dv|dz.

JID:YJMAA

AID:21218 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.209; Prn:23/03/2017; 17:48] P.12 (1-12)

M. McBride / J. Math. Anal. Appl. ••• (••••) •••–•••

12

By using the Cauchy–Schwarz inequality in a similar manner to the proof of Theorem 4.1, we deduce that ¨

¨ |Dv|2 dz ≤ const QR (z0 )

|f |2 dz.

(4.8)

QR (z0 )

We have f i ∈ L2,λ ϕ (QT ), so after combining Inequalities (4.7) and (4.8), we obtain the following inequality ¨ |Du| dz ≤ const 2

Qρ (z0 )

 ρ μ ¨ R

|Du|2 dz + const · ϕ2 (ρ)ρλ f 2L2,λ . ϕ

QR (z0 )

Therefore, the result follows again by applying Lemma 2.2. Thus, this completes the proof. 2 References [1] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. [2] Q. Huang, Estimates on the generalized Morrey spaces L2,λ ϕ and BM Oψ for linear elliptic systems, Indiana Univ. Math. J. 45 (1996) 397–439. [3] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. [4] G. Lieberman, A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with V M O coefficients, J. Funct. Anal. 201 (2003) 457–479. [5] W. Schlag, Schauder and Lp estimates for parabolic systems via Campanato spaces, Comm. Partial Differential Equations 21 (1996) 1141–1175. [6] M. Struwe, M. Giaquinta, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z. 179 (1982) 437–451.