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

1 Introduction

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

The unique strong solvability of this problem was proved in [38]. In [39] Softova studied the regularity of the solution in the Morrey spaces L^p,ƛ with p ∈ (1, ∞), ƛ ∈ (0, n + 2) and also its Hölder regularity. In [41] Softova extended these studies on generalized Morrey spaces L^p,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 L^p 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 M^p,ϕ with a weight ϕ satisfying (10) with w ≡ 1 (cf. [18, 19]). Precisely, a boundedness in M^p,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 M_w^p,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 M^p,ϕ and L^p,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 L_w^q if and only if the weight w satisfies the following integral condition called parabolic Muckenhoupt condition or parabolic A_q-condition. We say that the measurable, nonnegative function w : ℝⁿ → ℝ₊ satisfies the parabolic A_q-condition for q ∈ (1, ∞) if

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

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: x=(x',t)=(x",xn,t)∈ℝn+1,R+n+1=﹛x'∈ℝn,t>0﹜ and D+n+1=﹛x"∈Rn−1,xn>0,t>0﹜,Diu=∂u/∂xi,Diju=∂2u/∂xi∂xj,Dtu=ut=∂u/∂t stand for the corresponding derivatives while Du = (D₁u, … , D_nu) and D2u=﹛Diju﹜i,j=1n 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 C^1,1-domain, Q = Ω × (0, T) be a cylinder in ℝ+n+1, and S = ∂Ω × (0, T) stands for the lateral boundary of Q. We consider the problem

under the following conditions:

1. 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)﹛Λ−1|ξ|2≤aij(x)ξiξj≤Λ|ξ|2,∀ξ∈ℝn,aij(x)=aji(x),       i,j=1,…,n.

   The symmetry of the coefficient matrix a=﹛aij﹜i,j=1n implies essential boundedness of a^ij’s and we set ‖a‖∞,Q=∑i,j=1n‖aij‖∞,Q.

2. 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)〈l(x)⋅n(x)〉=li(x)ni(x)>0  on S,  li∈Lip(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

we use the equivalent one ρ(x)=(|x'|2+|x'|4+4t22)12 (see [9]). 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.

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.

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

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 A_p, 1 < p < ∞, if

where 1p+1p'=1. Note that, for any ball we have (see [11])

In case p = 1, we say that w ∈ A₁ if

and [w]A1 is the smallest A for which the above inequality holds. It is an immediate consequence of (7) that whenever w ∈ A_p than it satisfies the doubling property, precisely

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

3 Proof of the main result

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

We are going to show that f ∈ M^p,ϕ(Q, ω) implies uW˙2,1p,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 x0∈ℝ+n+1 consider the parabolic semi-cylinders 𝒞_r(x₀) = ℬ_r(x′₀) × (t₀ − r², t₀). Let ν𝒞₀^∞(𝒞_r) 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

where ν(ν₁, … , ν_n+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)/∂ζ_i ∂ζ_j.

Because of density arguments the representation formula (21) still holds for any ν ∈ W_{2, 1}^p(𝒞_r(x₀), ω). The properties of the fundamental solution (cf. [5, 25, 38]) imply Γ_ij are Calderon-Zygmund kernels in the sense of Definition 2.9. We denote by 𝒦_ij and C_ij the singular integrals defined in (12) with kernels 𝔎(x; x − y) = Γ_ij (x; x − y). Then we can write that

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, η_a(r), ║a║_∞,Q and ║DΓ║_∞,Q. Define a cut-off function ϕ(x) = ϕ₁(x′)ϕ₂(t), with ϕ₁C₀^∞(ℬ_r(x′₀)), ϕ₂C₀^∞(ℝ) such that

with θ(0, 1), θ′ = θ(3 − θ)/2 > θ and |D^sϕ| ≤ C ∈ [θ(1 − θ)r]^−s, s = 0, 1, 2, |ϕ_t| ∼ |D²ϕ|. For any solution u ∈ W_2,1^p(Q, ω) of (3) define ν(x) = ϕ(x)u(x)W_2,1^p(𝒞_r, 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

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 u_t 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

where C depends on n,p,[ω]Ap12Λ,T‖DΓ‖∞;Q,ηa(r),‖a‖∞,Q 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 ℬ_R⁺ = ﹛|x′| < R, x_n > 0﹜, S_R⁺ = ﹛|ʺ| < R, x_n = 0, t(0, R²)﹜ and consider the problem

Let u ∈ W_{2, 1}^p (C_R⁺, ω) with u = 0 for t ≤ 0 and x_n ≤ 0, then the following representation formula holds (see [26, 38])

where

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

Here I_ij 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 J_ij we apply the estimates obtained in Theorem 2.13 and Corollary 2.14 that give

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

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

Taking a parabolic cylinder ℐ_r(x) centered in some point x𝒞_R⁺ we have