We consider the regular oblique derivative problem in generalized weighted Sobolev- Morrey space for linear non-divergence form parabolic equations in a cylinder
The unique strong solvability of this problem was proved in . In  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  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  and  the regularity of the corresponding operators in the Morrey and generalized Morrey spaces is studied, while in  we can find the corresponding results obtained in Lp by  and . 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  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  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  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  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) for all parabolic cylinders ℐ in ℝn+1. Then w(ℐ) means the weighted measure of ℐ, that is
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) where c1 and τ1 depend on n and q but not on ℐ and 𝒜.
Throughout this paper the following notations are to be used: and stand for the corresponding derivatives while Du = (D1u, … , Dnu) and mean the spatial gradient and the Hessian matrix of u. For any measurable function f and A ⊂ ℝn+1 we write 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 , and S = ∂Ω × (0, T) stands for the lateral boundary of Q. We consider the problem (3) under the following conditions:
The operator 𝔅 is supposed to be uniformly parabolic, i.e. there exists a constant Λ > 0 such that for almost all x ∈ Q (4)
The symmetry of the coefficient matrix implies essential boundedness of aij’s and we set
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)), x ∈ S. We suppose that 𝔅 is a regular oblique derivative operator, i.e., the field l is never tangential to S: (5) 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 we use the equivalent one (see ). The balls with respect to this metric are ellipsoids
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.
A function a belongs to VMO (vanishing mean oscillation) with VMO- modulus ηa(R) provided
For any bounded cylinder Q we define BMO(Q) and VMO(Q) taking a ∈ L1(Q) and Qr = Q ⋂ ℐr instead of ℰr in the definition above.
According to [1, 21] having a function a ∈ BMO/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.
(John-Nirenberg lemma, ). Let a ∈ BMO and p ∈ (1, ∞). Then for any ball ℬ there holds
As an immediate consequence of Lemma 2.2 we get the following property.
Let a ∈ BMO then for all 0 < 2r < t holds (6) 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 the w-measure of 𝕊. A weight w belongs to the Muckenhoupt class Ap, 1 < p < ∞, if (7) where Note that, for any ball we have (see ) (8)
In case p = 1, we say that w ∈ A1 if and is the smallest A for which the above inequality holds. It is an immediate consequence of (7) that whenever w ∈ Ap than it satisfies the doubling property, precisely (9)
The following lemma collects some of the most important properties of the Muckenhoupt weights.
(). We have the following:
If w ∈ Ap for some 1 ≤ p < ∞, then for all ƛ > 1 we have
The following equality is valid: A∞ = ⋃1≤p<∞ Ap.
If w ∈ A∞, then for all ƛ > 1 we have
If w ∈ Ap for some 1 ≤ p ≤ ∞, then there exist C > 0 and δ > 0 such that for any ball ℬ and a measurable set 𝕊 ⊂ ℬ,
([30, Theorem 5]). Let w ∈ A∞. Then the norm of BMO(w) is equivalent to the norm of BMO(ℝn), where and
(The John-Nirenberg inequality). Let a ∈ BMO,
there exist constants C1, C2 > 0, such that for all β > 0
for all p ∈ (1, ∞)
for all p ∈ [1, ∞) and w ∈ A∞
Let ϕ(x, r) be weight in ϕ: ℝn × ℝ+ → ℝ+ and ω ∈ Ap, p ∈ [1, ∞). The generalized weighted Morrey space Mp,ϕ(ℝn, ω) or Mp,ϕ(ω) consists of all functions such that
The space Mp,ϕ(Q, ω) consists of functions provided the following norm is finite
The generalized Sobolev-Morrey space consist of all Sobolev functions with distributional derivatives endowed by the norm and where ∂Q means the parabolic boundary Ω ∪ ﹛∂Ω × (0, T)﹜.
(Main result) Let (i) and (ii) hold, aVMO(Q, ω) and u ∈ , ω), p(1, ∞), ω ∈ Ap be a strong solution of (3). If f ∈ Mp,ϕ (Q, ω) with ϕ(x, r) being measurable positive function satisfying (10) for each (x, r) ∈ Q × ℝ+, then u ∈ and (11) with and
If ϕ(x, r) = r(ƛ−n−2)/p, then Mp,ϕ ≡ Lp,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 than we obtain the spaces Lp,ω studied in [27, 31]. The following results are obtained in  and treat continuity in Mp,ϕ(ℝn+1, ω) of certain singular and nonsingular integrals.
A measurable function 𝔎(x; ζ) : ℝn+1 × ℝn+1 \ ﹛0﹜ → ℝ is called variable parabolic Calderon-Zygmund kernel (PCZK) if:
𝔎(x; ·) is a PCZK for a.a. xℝn+1:
𝔎(x; ·)C∞(ℝn+1 \ ﹛0﹜),
𝔎(x; μζ) = μ−(n+2) 𝔎(x; ζ) ∀μ > 0,
for each multi-index β.
Consider the singular integrals (12)
For any f ∈ Mp,ϕ(ℝn+1, ω) with (p, ϕ) as in Theorem 2.8 and a ∈ BMO there exist constants depending on n, p, ϕ, ω and the kernel such that (13)
Let Q be a cylinder in and Then the operators (12) are bounded in Mp,ϕ (Q, ω) with p, ϕ, and ω as in Theorem 2.10. Then (14) with
Let a ∈ VMO 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 any ℰr(x0) with a radius r(0, r0) and all f ∈ Mp,ϕ(ℰr(x0), ω) (15) where C is independent of ϵ, f, r and x0.
For any and any fixed t > 0 define the generalized reflection (16) where an(x) is the last row of the coefficients matrix a(x) of (3). The function 𝒯′(x) maps into and the kernel 𝔎(x; 𝒯(x) − y) = 𝒦(x; 𝒯′(x) − y′, t − τ) is a nonsingular one for any Taking x = (ʺ, −xn, t) there exist positive constants k1 and k2 such that (17)
For any with a norm and defines the nonsingular integral operators (18)
Let and with (p, ϕ) as in Theorem 2.8. Then the operators and are continuous in and (19) with a constant independend of a and f.
Let a∈2 ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for any with a radius r(0, r0) and center x0 = (ʺ, 0, 0) and for all holds (20) where C is independent of ϵ, f, r and x0.
3 Proof of the main result
As it follows by , the problem (3) is uniquely solvable in .
We are going to show that f ∈ Mp,ϕ(Q, ω) implies u. For this goal we obtain an a priori estimate of u. Following the method used by Chiarenza, Frasca and Longo in  and , we prove the results considering two steps.
Interior estimate. For any consider the parabolic semi-cylinders 𝒞r(x0) = ℬr(x′0) × (t0 − r2, t0). Let ν𝒞0∞(𝒞r) and suppose that ν(x, t) = 0 for t ≤ 0. According to [, Theorem 1.4] for any xsupp ν the following representation formula for the second derivatives of ν holds true(21)
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)/∂ζi ∂ζj.
Because of density arguments the representation formula (21) still holds for any ν ∈ W2, 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)
Because of Corollaries 2.11 and 2.12 and the equivalence of the metrics we get
for some r small enough. Moving the norm of D2ν on the left-hand side we get
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
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 u ∈ W2,1p(Q, ω) of (3) define ν(x) = ϕ(x)u(x)W2,1p(𝒞r, ω). Then we get
By the choice of θ′ it holds θ(1 − θ) ≤ 2θ′(1 − θ′) which leads to
Introducing the semi-norms
and taking the supremo with respect to θ and θ′ we get(23)
The interpolation inequality [26, Lemma 4.2] gives that there exists a positive constant C independent of r such that
Thus (23) becomes
for each θ(0, 1). Taking θ = 1/2 we get the Caccioppoli-type estimate
To estimate ut we exploit the parabolic structure of the equation and the boundedness of the coefficients
Consider cylinders Q′ = Ω′ × (0, T) and Qʺ = Ωʺ × (0, T) with Ω′ ⋐ Ωʺ ⋐ Ω, by standard covering procedure and partition of the unity we get(24)
where C depends on and dist(Ω′, ∂Ωʺ).
Boundary estimates. For any fixed R > 0 and x0 = (ʺ, 0, 0) define the semi-cylinders
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)
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
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)
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)
Taking a parabolic cylinder ℐr(x) centered in some point x𝒞R+ we have
Moving on the left-hand side and taking the supremo with respect to (x, r)𝒞R+ × ℝ+ we get
An immediate consequence of (27) is the estimate
The convolution Γk * f is a Riesz potential. On the other hand
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 [, Theorem 3.7] we get
Further, the Rademacher theorem asserts existence almost everywhere of the derivatives Dhlk ∈ L∞, thus
The Mp,ϕ(ω) norm of the last term is estimated as above and(29)
Finally unifying (26), (28) and (29) we get
Taking we get
Choosing R small enough and moving the terms containing the norm of D2u on the left-hand side we get
Because of the parabolic structure of the equation analogous estimate holds also for ut. Further the Jensen inequality applied to gives
Choosing R smaller, if necessary, we get and therefore(30)
Making a covering ﹛Cα+﹜, α𝒜 such that , considering a partition of unity subordinated to that covering and applying (30) for each 𝒞α+ we get(31)
with a constant depending on , and ║Dl║∞,𝒮. The estimate (11) follows from (24) and (31).
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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Published Online: 2016-02-13
Published in Print: 2016-01-01