PARABOLIC OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 3 LUBOMIRA G. SOFTOVA 1 0 2 n a Abstract. Westudytheregularityofthesolutionsoftheoblique J derivative problem for linear uniformly parabolic equations with 3 VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized Morrey space ] Mp,ϕ(Q)thanthestrongsolutionbelongstothegeneralizedSobolev- P ◦ A Morrey space Wp,ϕ(Q). 2,1 . h t 1. Introduction a m In the present work we consider the regular oblique derivative prob- [ lem for linear non-divergence form parabolic equations in a cylinder 1 v u aij(x)D u = f(x) a.e. in Q, t ij 8 − 8 u(x′,0) = 0 on Ω, 3 ∂u/∂ℓ = ℓi(x)D u = 0 on S. 0 i 1. The unique strong solvability of this problem was proved in [23]. In 0 [24] we study the regularity of the solution in the Morrey spaces Lp,λ 3 1 with p (1, ), λ (0,n+2) and also its Hölder regularity. In [26] we ∈ ∞ ∈ : extend these studies on generalized Morrey spaces Lp,ω with a weight v i ω satisfying the doubling and integral conditions introduced in [18, 20]. X The approach associated to the names of Calderoń and Zygmund and r a developed by Chiarenza, Frasca and Longo in [7, 8] consists of obtain- ing 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 [24] and [25] we study the regularity of the corresponding operators in the Morrey and generalized Morrey spaces while in [23] we dispose of the corresponding results obtained in Lp by [9] and [5]. In recent works we study the regularity of the solutions of elliptic andparabolic problems with Dirichlet 1991 Mathematics Subject Classification. 35K20,35D35,35B45, 35R05. Keywords andphrases. Uniformlyparabolicoperator,regularobliquederivative problem, VMO, generalized Morrey spaces, unique solvability. 1 2 L.G. SOFTOVA data on the boundary in Morrey-type spaces Mp,ϕ with a weight ϕ satisfying (2.4) (cf. [12, 13]). Precisely, we obtain boundedness in Mp,ϕ for sub-linear integrals generated by singular integrals as the Calderoń- Zygmund. More results concerning sub-linear integrals in generalized Morreyspacescanbefoundin[3,11,25], seealsothereferencestherein. Throughout this paper the following notations are to be used, x = (x′,t) = (x′′,x ,t) Rn+1, Rn+1 = x′ Rn,t > 0 and Dn+1 = x′′ Rn−1,xn > ∈0,t > 0 ,+D u ={∂u/∈∂x , D u =} ∂2u/∂+x ∂x , n i i ij i j { ∈ } D u = u = ∂u/∂t stand for the corresponding derivatives while Du = t t (D u,... ,D u) and D2u = D u n mean the spatial gradient and 1 n { ij }i,j=1 the Hessian matrix of u. For any measurable function f and A Rn+1 ⊂ we write 1/p 1 f = f(y) pdy , f = f(y)dy p,A A k k | | A (cid:18)ZA (cid:19) | | ZA 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 i⊂n Rn+1, ≥and S = ∂Ω (0,T) stands for the late×ral boundary + × of Q. We consider the problem Pu := u aij(x)D u = f(x) a.e. in Q, t ij − (2.1) Iu := u(x′,0) = 0 on Ω, Bu := ∂u/∂ℓ = ℓi(x)D u = 0 on S. i under the following conditions: (i) The operator P is supposed to beuniformly parabolic, i.e. there exists a constant Λ > 0 such that for almost all (a.a.) x Q ∈ Λ−1 ξ 2 aij(x)ξ ξ Λ ξ 2 ξ Rn, i j (2.2) | | ≤ ≤ | | ∀ ∈  aij(x) = aji(x), i,j = 1,... ,n.  Thesymmetryofthecoefficient matrixa = aij n implieses- { }i,j=1 sentialboundednessofaij’sandweset a = n aij . k k∞,Q i,j=1k k∞,Q (ii) The boundary operator B is prescribed in terms of a direc- P tional derivative with respect to the unit vector field ℓ(x) = (ℓ1(x),...,ℓn(x)) x S. We suppose that Bis a regular oblique ∈ OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 3 derivative operator, i.e., the field ℓ is never tangential to S: (2.3) ℓ(x) n(x) = ℓi(x)n (x) > 0 on S, ℓi Lip(S¯). i h · i ∈ Here Lip(S¯) is the class of uniformly Lipschitz continuous func- ¯ tions on S and n(x) stands for the unit outward normal to ∂Ω. In the following, besides the parabolic metric ̺(x) = max( x′ , t 1/2) | | | | and the defined by it parabolic cylinders (x) = y Rn+1 : x′ y′ < r, t τ < r2 = Crn+2. r r I ∈ | − | | − | |I | n o |x′|2+√|x′|4+4t2 1/2 we use the equivalent one ρ(x) = (see [9]). The 2 (cid:18) (cid:19) balls with respect to this metric are ellipsoids x′ y′ 2 t τ 2 (x) = y Rn+1 : | − | + | − | < 1 = Crn+2. Er ∈ r2 r4 |Er| n o 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. ([14, 22]) Let a L1 (Rn+1), denote by ∈ loc 1 η (R) = sup f(y) f dy for every R > 0 a | − Er| Er,r≤R |Er| ZEr where ranges over all ellipsoids in Rn+1. The Banach space BMO r E (bounded mean oscillation) consists of functions for which the following norm is finite a = supη (R) < . ∗ a k k ∞ R>0 A function a belongs to VMO (vanishing mean oscillation) with VMO- modulus η (R) provided a lim η (R) = 0. a R→0 For any bounded cylinder Q we define BMO(Q) and VMO(Q) taking a L1(Q) and Q = Q instead of in the definition above. r r r ∈ ∩I E According to [1, 15] having a function a BMO/VMO(Q) it is possible to extend it in the whole Rn+1 prese∈rving its BMO-norm or VMO-modulus, respectively. In the following we use this property without explicit references. Definition 2.2. Let ϕ : Rn+1 R R be a measurable function + + and p (1, ). A function f ×Lp (→Rn+1) belongs to the generalized ∈ ∞ ∈ loc 4 L.G. SOFTOVA parabolic Morrey space Mp,ϕ(Rn+1) provided 1/p f = sup ϕ(x,r)−1 r−(n+2) f(y) pdy < . p,ϕ;Rn+1 k k (x,r)∈Rn+1×R+ ZEr(x)| | ! ∞ The space Mp,ϕ(Q) consists of Lp(Q) functions provided the following norm is finite 1/p f = sup ϕ(x,r)−1 r−(n+2) f(y) pdy . p,ϕ;Q k k (x,r)∈Q×R+ ZQr(x)| | ! The generalized Sobolev-Morrey space Wp,ϕ(Q), p (1, ) consist 2,1 ∈ ∞ of all Sobolev functions u Wp (Q) with distributional derivatives ∈ 2,1 DlDsu Mp,ϕ(Q), 0 2l+ s 2, endowed by the norm t x ∈ ≤ | | ≤ u Wp,ϕ(Q) = ut p,ϕ;Q + Dsu p,ϕ;Q. k k 2,1 k k k k |sX|≤2 ◦ Wp,ϕ(Q) = u Wp,ϕ(Q) : u(x) = 0 x ∂Q , 2,1 ∈ 2,1 ∈ kukW◦p,ϕ(Q) =nkukW2p,,1ϕ(Q) o 2,1 where ∂Q means the parabolic boundary Ω ∂Ω (0,T) . ∪{ × } Theorem 2.3. (Main result) Let (i) and (ii) hold, a VMO(Q) and ∈ ◦ u Wp (Q), p (1, ) be a strong solution of (2.1). If f Mp,ϕ(Q) ∈ 2,1 ∈ ∞ ∈ with ϕ(x,r) being measurable positive function satisfying n+2 essinf ϕ(x,ζ)ζ p ∞ s s<ζ<∞ (2.4) 1+ln ds C Zr (cid:18) r(cid:19) sn+p2+1 ≤ ◦ for each (x,r) Q R then u Wp,ϕ(Q) and ∈ × + ∈ 2,1 (2.5) kukW◦p,ϕ(Q) ≤ Ckfkp,ϕ;Q 2,1 with C = C(n,p,Λ,∂Ω,T,kak∞;Q,ηa) and ηa = ni,j=1ηaij. Ifϕ(x,r) = r(λ−n−2)/p thenMp,ϕ Lp,λ andthPecondition(2.4)holds ≡ withaconstant depending onn,pandλ.Ifϕ(x,r) = ω(x,r)1/pr−(n+2)/p with ω : Rn+1 R R satisfying the conditions + + × → ω(x ,s) k 0 k x Rn+1, r s 2r 1 2 0 ≤ ω(x ,r) ≤ ∀ ∈ ≤ ≤ 0 ∞ ω(x ,s) 0 ds k ω(x ,r) k > 0, i = 1,2,3 3 0 i s ≤ Zr OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 5 thanwe obtainthe spaces Lp,ω studied in[18, 20]. The following results areobtained in [13] and treat continuity in Mp,ϕ of certain singular and nonsingular integrals. Definition 2.4. A measurable function K(x;ξ) : Rn+1 Rn+1 0 R × \{ } → is called variable parabolic Calderón-Zygmund kernel (PCZK) if: i) K(x; ) is a PCZK for a.a. x Rn+1 : a) K· (x; ) C∞(Rn+1 0 )∈, · ∈ \{ } b) K(x;µξ) = µ−(n+2)K(x;ξ) µ > 0, ∀ c) K(x;ξ)dσ = 0, K(x;ξ) dσ < + . ξ ξ Sn Sn | | ∞ Z Z ii) DβK M(β) < for each multi-index β. ξ ∞;Rn+1×Sn ≤ ∞ (cid:13) (cid:13) Consi(cid:13)der th(cid:13)e singular integrals (cid:13) (cid:13) f(x) = P.V. K(x;x y)f(y)dy K Rn+1 − Z (2.6) C[a,f](x) = P.V. K(x;x y)[a(y) a(x)]f(y)dy. Rn+1 − − Z Theorem 2.5. For any f Mp,ϕ(Rn+1) with (p,ϕ) as in Theorem 2.3 ∈ and a BMO there exist constants depending on n,p and the kernel ∈ such that f C f , p,ϕ;Rn+1 p,ϕ;Rn+1 kK k ≤ k k (2.7) C[a,f] C a f . p,ϕ;Rn+1 ∗ p,ϕ;Rn+1 k k ≤ k k k k Corollary 2.6. Let Q be a cylinder in Rn+1, f Mp,ϕ(Q), a BMO(Q) and K(x,ξ) : Q Rn+1 0 + R. Th∈en the operator∈s × + \ { } → (2.6) are bounded in Mp,ϕ(Q) and f C f , p,ϕ;Q p,ϕ;Q kK k ≤ k k (2.8) C[a,f] C a f p,ϕ;Q ∗ p,ϕ;Q k k ≤ k k k k with C independent of a and f. Corollary 2.7. Let a VMO and (p,ϕ) be as in Theorem 2.3. Then ∈ for any ε > 0 there exists a positive number r0 = r0(ε,ηa) such that for any (x ) with a radius r (0,r ) and all f Mp,ϕ( (x )) r 0 0 r 0 E ∈ ∈ E (2.9) C[a,f] Cε f k kp,ϕ;Er(x0) ≤ k kp,ϕ;Er(x0) where C is independent of ε, f, r and x . 0 For any x′ Rn and any fixed t > 0 define the generalized reflection ∈ + an(x′,t) (2.10) (x) = ( ′(x),t), ′(x) = x′ 2x n T T T − ann(x′,t) 6 L.G. SOFTOVA where an(x) is the last row of the coefficients matrix a(x) of (2.1). The function ′(x) maps Rn into Rn and the kernel K(x; (x) y) = K(x; ′(x) yT′,t τ) is a n+onsingula−r one for any x,y DTn+1. T−aking T − − ∈ + x = (x′′, x ,t) there exist positive constants κ and κ such that n 1 2 − (2.11) κ ρ(x y) ρ( (x) y) κ ρ(x y). 1 2 e − ≤ T − ≤ − For any f Mp,ϕ(Dn+1) with a norm ∈ e+ e 1/p f = sup ϕ(x,r)−1 r−(n+2) f(y) pdy k kp,ϕ;Dn++1 (x,r)∈Dn++1×R+ ZEr(x)| | ! and a BMO(Dn+1) define the nonsingular integral operators ∈ + f(x) = K(x; (x) y)f(y)dy K Dn+1 T − Z + (2.12) e C[a,f](x) = K(x; (x) y)[a(y) a(x)]f(y)dy. Dn+1 T − − Z + Theoreme 2.8. Let a BMO(Dn+1) and f Mp,ϕ(Dn+1) with (p,ϕ) ∈ + ∈ + as in Theorem 2.3. Then the operators f and C[a,f] are continuous in Mp,ϕ(Dn+1) and K + e e f C f , kK kp,ϕ;Dn++1 ≤ k kp,ϕ;Dn++1 (2.13) kCe[a,f]kp,ϕ;Dn++1 ≤ Ckak∗kfkp,ϕ;Dn++1 with a constant indeependend of a and f. Corollary 2.9. Let a VMO, then for any ε > 0 there exists a positive number r0 = r0(ε∈,ηa) such that for any Er+(x0) = Er(x0)∩Dn++1 with a radius r (0,r ) and center x0 = (x′′,0,0) and for all f 0 ∈ ∈ Mp,ϕ( +(x0)) holds Er (2.14) C[a,f] Cε f , k kp,ϕ;Er+(x0) ≤ k kp,ϕ;Er+(x0) where C is independent of ε, f, r and x0. e 3. Proof of the main result ◦ Asitfollowsby[24],theproblem(2.1)isuniquelysolvableinWp (Q). 2,1 ◦ We are going to show that f Mp,ϕ(Q) implies u Wp,ϕ(Q). For this ∈ ∈ 2,1 goal we obtain an a priori estimate of u. The proof is divided in two steps. Interior estimate. For any x Rn+1 consider the parabolic semi- 0 ∈ + cylinders (x ) = (x′) (t r2,t ). Let v C∞( ) and suppose Cr 0 Br 0 × 0 − 0 ∈ 0 Cr OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 7 that v(x,t) = 0 for t 0. According to [5, Theorem 1.4] for any x ≤ ∈ suppv the following representation formula for the second derivatives of v holds true D v(x) = P.V. Γ (x;x y)[ahk(y) ahk(x)]D v(y)dy ij ij hk Rn+1 − − Z (3.15) +P.V. Γ (x;x y)Pv(y)dy+Pv(x) Γ (x;y)ν dσ , ij j i y Rn+1 − Sn Z Z where ν(ν ,... ,ν ) is the outward normal to Sn. Here Γ(x;ξ) is the 1 n+1 fundamentalsolutionoftheoperatorPandΓ (x;ξ) = ∂2Γ(x;ξ)/∂ξ ∂ξ . ij i j Because of density arguments the representation formula (3.15) still holds for any v Wp ( (x )). The properties of the fundamental ∈ 2,1 Cr 0 solution (cf. [5, 16, 23]) imply Γ are Calderoń-Zygmund kernels in ij the sense of Definition 2.4. We denote by and C the singular ij ij K integrals defined in (2.6) with kernels K(x;x y) = Γ (x;x y). Then ij − − we can write that D v(x) =C [ahk,D v](x) ij ij hk (3.16) + (Pv)(x)+Pv(x) Γ (x;y)ν dσ . ij j i y K Sn Z Because of Corollaries 2.6 and 2.7 and the equivalence of the metrics we get D2v C(ε D2v + Pu ) k kp,ϕ;Cr(x0) ≤ k kp,ϕ;Cr(x0) k kp,ϕ;Cr(x0) for some r small enough. Moving the norm of D2v on the left-hand side we get D2v C Pv k kp,ϕ;Cr(x0) ≤ k kp,ϕ;Cr(x0) with a constant depending on n,p,ηa(r), a ∞,Q and DΓ ∞,Q. Define k k k k a cut-off function φ(x) = φ (x′)φ (t), with φ C∞( (x′)), φ C∞(R) such that 1 2 1 ∈ 0 Br 0 2 ∈ 0 1 x′ (x′) 1 t (t (θr)2,t ] φ (x′) = ∈ Bθr 0 , φ (t) = ∈ 0 − 0 1 0 x′ 6∈ Bθ′r(x′0) 2 0 t < t0 −(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, φ D2φ . Fo−r any solution u | Wp| (≤Q) of (2.−1) define | t| ∼ | | ∈ 2,1 v(x) = φ(x)u(x) Wp ( ). Then we get ∈ 2,1 Cr D2u D2v C Pv k kp,ϕ;Cθr(x0) ≤ k kp,ϕ;Cθ′r(x0) ≤ k kp,ϕ;Cθ′r(x0) Du u C f + k kp,ϕ;Cθ′r(x0) + k kp,ϕ;Cθ′r(x0) . ≤ k kp,ϕ;Cθ′r(x0) θ(1 θ)r [θ(1 θ)r]2 ! − − 8 L.G. SOFTOVA By the choice of θ′ it holds θ(1 θ) 2θ′(1 θ′) which leads to − ≤ − 2 θ(1 θ)r D2u − k kp,ϕ;Cθr(x0) h i C r2 f +θ′(1 θ′)r Du + u . ≤ k kp,ϕ;Q − k kp,ϕ;Cθ′r(x0) k kp,ϕ;Cθ′r(x0) (cid:16) (cid:17) Introducing the semi-norms s Θ = sup θ(1 θ)r Dsu s = 0,1,2 s − k kp,ϕ;Cθr(x0) 0<θ<1 h i and taking the supremo with respect to θ and θ′ we get (3.17) Θ C r2 f +Θ +Θ . 2 p,ϕ;Q 1 0 ≤ k k (cid:16) (cid:17) The interpolation inequality [26, Lemma 4.2] gives that there exists a positive constant C independent of r such that C Θ εΘ + Θ for any ε (0,2). 1 2 0 ≤ ε ∈ Thus (3.17) becomes [θ(1 θ)r]2 D2u Θ C r2 f +Θ − k kp,ϕ;Cθr(x0) ≤ 2 ≤ k kp,ϕ;Q 0 (cid:16) (cid:17) foreachθ (0,1).Takingθ = 1/2wegettheCaccioppoli-typeestimate ∈ 1 D2u C f + u . k kp,ϕ;Cr/2(x0) ≤ k kp,ϕ;Q r2k kp,ϕ;Cr(x0) (cid:18) (cid:19) To estimate u we exploit the parabolic structure of the equation and t the boundedness of the coefficients u a D2u + f k tkp,ϕ;Cr/2(x0) ≤ k k∞;Qk kp,ϕ;Cr/2(x0) k kp,ϕ;Cr/2(x0) 1 C f + u . ≤ k kp,ϕ;Q r2k kp,ϕ;Cr(x0) (cid:16) (cid:17) Consider cylinders Q′ = Ω′ (0,T) and Q′′ = Ω′′ (0,T) with Ω′ ⋐ × × Ω′′ ⋐ Ω, by standard covering procedure and partition of the unity we get (3.18) u Wp,ϕ(Q′) C f p,ϕ;Q+ u p,ϕ;Q′′ k k 2,1 ≤ k k k k (cid:16) (cid:17) whereC dependsonn,p,Λ,T, DΓ ∞;Q,ηa(r), a ∞,Q anddist(Ω′,∂Ω′′). k k k k Boundary estimates. For any fixed R > 0 and x0 = (x′′,0,0) define the semi-cylinders +(x0) = (x0) Dn+1. CR CR ∩ + Without lost of generality we can take x0 = (0,0,0). Define + = x′ < R,x > 0 , S+ = x′′ < R,x = 0,t (0,R2) and BR {| | n } R {| | n ∈ } OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 9 consider the problem Pu := u aij(x)D u = f(x) a.e. in +, t − ij CR (3.19) Iu := u(x′,0) = 0 on +, Bu := ℓi(x)D u = 0 on SBR+. i R Let u Wp(C+) with u = 0 for t 0 and x 0, then the following ∈ 2,1 R ≤ n ≤ representation formula holds (see [17, 23]) D u(x) = I (x) J (x)+H (x) ij ij ij ij − where I (x) =P.V. Γ (x;x y)F(x;y)dy ij ij ZCR+ − +f(x) Γ (x;y)ν dσ , i,j = 1,...,n; j i y Sn Z J (x) = Γ (x; (x) y)F(x;y)dy; ij ij ZCR+ T − l ∂ (x) J (x) =J (x) = Γ (x; (x) y) T F(x;y)dy, in ni il ZCR+ T − ∂xn ! i,j = 1,... ,n 1 − l s ∂ (x) ∂ (x) J (x) = Γ (x; (x) y) T T F(x;y)dy; nn ls ZCR+ T − ∂xn ! ∂xn ! F(x;y) =f(y)+[ahk(y) ahk(x)]D u(y) hk − Hij(x) =(Gij 2 g)(x)+g(x′′,t) Gj(x;y′′,xn,τ)nidσ(y′′,τ), ∗ Sn Z i,j = 1,... ,n, ∂T(x) an1(x) ann−1(x) = 2 ,... , 2 , 1 . ∂xn − ann(x) − ann(x) − ! Here the kernel G = Γ , is a byproduct of the fundamental solution Q and a bounded regular function . Hence its derivatives G behave as ij Q Γ and the convolution that appears in H is defined as follows ij ij (G g)(x) = P.V. G (x;x′′ y′′,x ,t τ)g(y′′,0,τ)dy′′dτ, ij 2 ij n ∗ ZSR+ − − g(x′′,0,t) = ℓk(0) ℓk(x′′,0,t) D u ℓk(0)(Γ F) (x′′,0,t), k k − − ∗ h(cid:16) (cid:17) i(cid:12)(cid:12)xn=0 (cid:12) (Γk F)(x) = Γk(x;x y)F(x;y)dy. (cid:12) ∗ ZCR+ − 10 L.G. SOFTOVA Here I are a sum of singular integrals and bounded surface integrals ij hence the estimates obtained in Corollaries 2.6 and 2.7 hold true. On the nonsingular integrals J we apply the estimates obtained in Theo- ij rem 2.8 and Corollary 2.9 that give (3.20) kIijkp,ϕ;CR+ +kJijkp,ϕ;CR+ ≤ C kfkp,ϕ;CR+ +ηa(R)kD2ukp,ϕ;CR+ (cid:16) (cid:17) for all i,j = 1,... ,n. To estimate the norm of H we suppose that the ij vector field ℓ is extended in + preserving its Lipschitz regularity and CR the norm. This automatically leads to extension of the function g in + that is CR (3.21) g(x) = ℓk(0) ℓk(x) D u(x) ℓk(0)(Γ F)(x). k k − − ∗ (cid:16) (cid:17) Applying the estimates for the heat potentials [16, Chapter 4] and the trace theorems in Lp [2, Theorems 7.48, 7.53] (see also [23, Theorem 1]) we get (G g)(y) pdy C g(y) pdy + Dg(y) pdy . ij 2 ZCR+| ∗ | ≤ ZCR+ | | ZCR+ | | ! Taking a parabolic cylinder (x) centered in some point x + we Ir ∈ CR have rn+2 ϕ(x,r)−p (G g)(y) pdy C g(y) pdy ZCR+∩Ir(x)| ij ∗2 | ≤ ϕ(x,r)−p(cid:18) rn+2 ZCR+∩Ir(x)| | ϕ(x,r)−p + Dg(y) pdy rn+2 ZCR+∩Ir(x)| | (cid:19) ϕ(x,r)−p C g p + Dg p . ≤ rn+2 k kp,ϕ;C+ k kp,ϕ;C+ (cid:18) R R(cid:19) Moving ϕ(x,r)−p on the left-hand side and taking the supremo with rn+2 respect to (x,r) + R we get ∈ CR × + G g p C g p + Dg p . k ij ∗2 kp,ϕ;C+ ≤ k kp,ϕ;C+ k kp,ϕ;C+ R (cid:18) R R(cid:19) An immediate consequence of (3.21) is the estimate g [ℓk(0) ℓk( )]D u + ℓk(0)(Γ F) k kp,ϕ;CR+ ≤ k − · k kp,ϕ;CR+ k k ∗ kp,ϕ;CR+ CR ℓ Du + Γ f ≤ k kLip(S¯)k kp,ϕ;CR+ k k ∗ kp,ϕ;CR+ + Γ [ahk( ) ahk(x)]D u . k k ∗ · − hk kp,ϕ;CR+

Parabolic oblique derivative problem in generalized Morrey spaces

Similar Parabolic oblique derivative problem in generalized Morrey spaces