The regularity in Lp-sobolev spaces for Cauchy-Dirichlet problem for second-order parabolic equations in a domain with conical points

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2012, Vol. 57, No. 3, pp. 48-52 THE REGULARITY IN LP -SOBOLEV SPACES FOR CAUCHY-DIRICHLET PROBLEM FOR SECOND-ORDER PARABOLIC EQUATIONS IN A DOMAIN WITH CONICAL POINTS Nguyen Thanh Anh and Tran Manh Cuong Hanoi National University of Education Abstract. In this paper, we study the regularity of the solution for Cauchy- Dirichlet problem for the second order parabolic equations in domains with conical points. The regularity of the solution in Lp-weighted Sobolev spaces will be established. Keywords: Parabolic equation, Cauchy-Dirichlet problem, nonsmooth do- mains, conical points, regularity. 1. Introduction The Lp-theory of second order parabolic equations have been studied widely under various regularity assumptions on the coefficients and the domains. Let us mention some works related to this topic. For the case of continuous leading coeffi- cients and smooth domains, theW 2,1p -solvability has been known for a long time, see, for example, [5]. Bramanti and Cerutti established the W 2,1p -solvability the Cauchy- Dirichlet problem for second order parabolic equations with VMO coefficients in domains being of the C1,1 class in [2]. Byun obtained W 1p -solvability for second or- der divergence parabolic equations with small BMO coefficients in Lipschitz domains with small Lipschitz constants in [3]. In domains of a more general class, analogous estimates were received by Alkhutov and Gordeev in [1] for parabolic equations with continuous coefficients. The present paper is concerned with Lp-estimates for of the Dirichlet-Cauchy problem for parabolic equations of second order in domains with conical points. The unique existence of weak solutions of such problems is reduced in [1]. Our goal is to study the regularity of the weak solutions in Lp-weighted Sobolev spaces. 2. Preliminaries and the statement of the main result Suppose that G is a bounded domain in Rn(n > 2) with the boundary ∂G infinitely smooth everywhere except at the origin 0 of coordinates, and that in a neighborhood of 0 the domain G coincides with the cone K = {x : x/|x| ∈ Ω}, where Ω is a smooth domain on the unit sphere Sn−1 in Rn. Let T be a positive 48 The regularity in Lp-Sobolev spaces for Cauchy-Dirichlet problem... number. Set QT = G× (0, T ), KT = K × (0, T ) and ST = (∂G\{0})× [0, T ]. In this paper, the letter p stands for some real number, 1 < p <∞. Let l ∈ N. We denote by W lp(G) the usual Sobolev space of functions defined in G with the norm ‖u‖W lp(G) = ( ∫ G ∑ |α|6l |∂αxu| pdx ) 1 p . By W˚ 1p (G) we denote the closure of C∞0 (G) in W 1p (G). We define the weighted Sobolev space V lp,β(G) (β ∈ R) as the closure of C∞0 (G \ {0}) with respect to the norm ‖u‖V l p,β (G) = (∑ |α|6l ∫ G rp(β+|α|−l)|∂αxu| pdx ) 1 p , where r = |x| = (∑n k=1 x 2 k ) 1 2 . The weighted Sobolev space V lp,β(K) is defined simi- larly with G replaced by K. Let X, Y be Banach spaces. We denote by Lp((0, T );X) the space of all func- tions f : (0, T )→ X with ‖f‖Lp(0,T ;X) = (∫ T 0 ‖f(t)‖pXdt ) 1 p <∞. By W 1p ((0, T );X, Y ) the space of all functions u ∈ Lp((0, T );X) such that ut ∈ Lp((0, T ); Y ) with the norm ‖u‖W 1p ((0,T );X,Y ) = ‖u‖Lp((0,T );X) + ‖ut‖Lp((0,T );Y ). For shortness, we set W˚ 1,0p (QT ) = Lp((0, T ); W˚ 1 p (G)), W 2,1p (QT ) = Lp((0, T );W 2 p (G), Lp(G)), V 2,1p,β (QT ) = Lp((0, T );V 2 p,β(G), V 0 p,β(G)), V 2,1p,β (KT ) = Lp((0, T );V 2 p,β(K), V 0 p,β(K)). Let L be a linear parabolic operator of the form Lu = ut − n∑ i,j=1 ∂i(aij∂ju), where aji = aij are real valued functions defined on QT . We assume that the func- tions aij are infinitely smooth on QT and there exists positive constant µ0 such that n∑ i,j=1 aij(x, t)ξiξj > µ0|ξ| 2 (2.1) 49 Nguyen Thanh Anh and Tran Manh Cuong for all ξ ∈ Rn and for a.e. (x, t) ∈ QT . In this paper, we consider the following problem Lu = f in QT , (2.2) u = 0 on ST , (2.3) u|t=0 = 0 on G. (2.4) Let f ∈ Lp(QT ). A function u ∈ W˚ 1,0p (QT ) is called a weak solution of the problem (2.2)-(2.4) if the equality∫ QT ( − uvt + n∑ i,j=1 aij∂ju∂iv ) dxdt = ∫ QT fvdxdt (2.5) holds for all smooth test functions v in QT vanishing in a neighborhood of the lateral surface and the upper base of the cylinder QT . The following assertion on the unique existence of weak solutions is deduced in [1]. Proposition 2.1. If f ∈ Lp(QT ), then there exists a unique weak solution u ∈ W˚ 1,0p (QT ) of the problem (2.2)− (2.4) which satisfies ‖u‖W˚ 1,0p (QT ) 6 C‖f‖Lp(QT ), (2.6) where C is a constant independent of f and u. Let us state the main theorem of the present paper: Theorem 2.1. Let f ∈ Lp(QT ). Then the weak solution of the problem (2.2)- (2.4) in fact belongs to V 2,1p,1 (QT ). Moreover, ‖u‖V 2,1p,1 (QT ) 6 C‖f‖Lp(QT ), (2.7) where C is a constant independent u and f . 3. The proof of Theorem 2.1 Let ζk be infinitely smooth functions on K depending only on r = |x| such that supp ζk ⊂ {x : 2 k−1 < r < 2k+1}, ∞∑ k=−∞ ζk = 1, |∂ α x ζk| 6 cα2 −k|α| (3.1) for all α, where cα is a constant independent of k and x. It is well-known that (see, for example, [6, Le. 2.1.4]) the norm in the space V lp,β(K) is equivalent to the norm 9u9V l p,β (K) = ( ∞∑ k=−∞ ‖ζku‖ p V l p,β (K) ) 1 p . 50 The regularity in Lp-Sobolev spaces for Cauchy-Dirichlet problem... From this it follows that the norm in V 2,1p,β (QT ) is equivalent to the norm 9u9V 2,1 p,β (KT ) = ( ∞∑ k=−∞ ‖ζku‖ p V 2,1 p,β (KT ) ) 1 p . Proof of Theorem 2.1. According to results for the Cauchy-Dirichlet problem for parabolic equations in smooth domains (see, for example, [5, Th. 9.2] or [2, Th. 4.1, 4.2]), the weak solution u belongs to W 2,1p,loc(QT\{0}) and u satisfies (2.2) a.e. in QT . Without loss of generality we assume that the domain G coincides with the cone K in the unit ball. Let ϕ be a infinitely smooth function in G which equals one for |x| < 1 2 and vanishes for |x| > 1. Since (1 − ϕ)u ∈ V 2,1p,1 (QT ), it suffices to prove that ϕu ∈ V 2,1p,1 (QT ). Thus, without loss of generality, we can assume that the solution u vanishes outside of the unit ball. This implies that f vanishes for |x| > 1. By extension by zero, u can be sometime considered as a function defined in KT , so does f . Let ζk be the infinitely smooth functions introduced in (3.1). We have from (2.2) L(ζku) = fk ≡ ζkf − n∑ i,j=1 aij(2∂iζk∂ju+ ∂ 2 ijζku) + n∑ i,j=1 ∂iaij(∂jζku+ ζk∂ju). (3.2) For each integer k, we introduce the function ηk = ζk−1 + ζk + ζk+1 which is equal to one on the support of ζk. Also from interior and boundary estimates for the case of smooth domains (see [2, Th. 4.1, 4.2]), we get from (3.2) that ‖ζku‖W 2,1p (KT ) 6 C(‖fk‖Lp(KT ) + ‖ζku‖Lp(KT )) 6 C ( ‖ζkf‖Lp(KT ) + 2 −k‖ηk∇u‖Lp(KT ) + 2 −2k‖ηku‖Lp(KT ) ) , (3.3) for k 6 −1, where and throughout this proof the letter C stands for a positive constant independent of k, u and f . Since u = 0 for |x| > 1, (3.3) also holds for all k > 0. Especially, we have ∑ |α|=2 ‖∂αx (ζku)‖Lp(KT ) + ‖∂t(ζku)‖Lp(KT ) 6 C ( ‖ζkf‖Lp(KT ) + 2 −k‖ηk∇u‖Lp(KT ) + 2 −2k‖ηku‖Lp(KT ) ) . Now multiplying both sides of this equality by 2k and noting that 2k−1 6 r 6 2k+1 51 Nguyen Thanh Anh and Tran Manh Cuong on the support of ζk, we obtain∑ |α|=2 ‖r∂αx (ζku)‖Lp(KT ) + ‖r∂t(ζku)‖Lp(KT ) 6 C ( ‖rζkf‖Lp(KT ) + ‖ηk∇u‖Lp(KT ) + ‖r −1ηku‖Lp(KT ) ) . Summing up over all k and adding to both sides of the obtained equality with ‖r−1u‖Lp(QT ), we receive∑ |α|=2 ‖r∂αxu‖Lp(QT ) + ‖r∂tu‖Lp(QT ) + ‖r −1u‖Lp(QT ) 6 C ( ‖rf‖Lp(QT ) + ‖∇u‖Lp(QT ) + ‖r −1u‖Lp(QT ) ) . Noting that the left-hand side of this equality is equivalent to the norm in V 2,1p,1 (QT ) (see [6, Le. 2.1.6]), we have ‖u‖ V 2,1 p,1 (QT ) 6 C ( ‖f‖Lp(QT ) + ‖∇u‖Lp(QT ) + ‖r −1u‖Lp(QT ) ) . (3.4) It follows directly from the Hardy’s inequality (see, for example, [4, Th. 85]) that there exists a constant C such that ‖r−1u‖Lp(G) 6 C‖u‖W 1p (G) for all u ∈ W˚ 1p (G). Thus, we have from (??) that ‖u‖V 2,1p,1 (QT ) 6 C ( ‖f‖Lp(QT ) + ‖u‖W˚ 1,0p (QT ) ) . (3.5) Now using (2.6) we obtain (2.7). The theorem is completely proved. REFERENCES [1] Yu. A. Alkhutov and A. N. Gordeev, Lp-solvability of the Dirichlet problem for second order parabolic equations, J. of Math. Sci. 172 (2011), no. 4, 423–445. [2] M. Bramanti and M. 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