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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 CauchyDirichlet 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 domains, 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 coefficients and smooth domains, the Wp2,1 -solvability has been known for a long time, see, for example, [5]. Bramanti and Cerutti established the Wp2,1 -solvability the CauchyDirichlet problem for second order parabolic equations with VMO coefficients in domains being of the C 1,1 class in [2]. Byun obtained Wp1 -solvability for second order 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 Wpl (G) the usual Sobolev space of functions defined in G with the norm Z X
1 kukWpl (G) = |∂xα u|p dx p . G |α|6l By W̊p1 (G) we denote the closure of C0∞ (G) in Wp1 (G). We define the weighted l Sobolev space Vp,β (G) (β ∈ R) as the closure of C0∞ (G \ {0}) with respect to the norm XZ
1 r p(β+|α|−l)|∂xα u|p dx p , kukVp,β l (G) = G |α|6l
1 Pn 2 2 l where r = |x| = x . The weighted Sobolev space Vp,β (K) is defined simik k=1 larly with G replaced by K. Let X, Y be Banach spaces. We denote by Lp ((0, T ); X) the space of all functions f : (0, T ) → X with kf kLp (0,T ;X) = Z T 0  p1 kf (t)kpX dt < ∞. By Wp1 ((0, T ); X, Y ) the space of all functions u ∈ Lp ((0, T ); X) such that ut ∈ Lp ((0, T ); Y ) with the norm kukWp1((0,T );X,Y ) = kukLp ((0,T );X) + kut kLp ((0,T );Y ) . For shortness, we set W̊p1,0 (QT ) = Lp ((0, T ); W̊p1(G)), Wp2,1 (QT ) = Lp ((0, T ); Wp2(G), Lp (G)), 2,1 2 0 Vp,β (QT ) = Lp ((0, T ); Vp,β (G), Vp,β (G)), 2,1 2 0 Vp,β (KT ) = Lp ((0, T ); Vp,β (K), Vp,β (K)). Let L be a linear parabolic operator of the form Lu = ut − n X ∂i (aij ∂j u), i,j=1 where aji = aij are real valued functions defined on QT . We assume that the functions aij are infinitely smooth on QT and there exists positive constant µ0 such that n X aij (x, t)ξi ξj > µ0 |ξ|2 (2.1) i,j=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 , u = 0 on ST , u|t=0 = 0 on G. (2.2) (2.3) (2.4) Let f ∈ Lp (QT ). A function u ∈ W̊p1,0 (QT ) is called a weak solution of the problem (2.2)-(2.4) if the equality Z  Z n  X − uvt + aij ∂j u∂i v dxdt = f vdxdt (2.5) QT QT i,j=1 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 ∈ W̊p1,0 (QT ) of the problem (2.2) − (2.4) which satisfies (2.6) kukW̊p1,0(QT ) 6 Ckf kLp (QT ) , 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) 2,1 in fact belongs to Vp,1 (QT ). Moreover, (2.7) kukV 2,1 (QT ) 6 Ckf kLp (QT ) , p,1 where C is a constant independent u and f . 3. that The proof of Theorem 2.1 Let ζk be infinitely smooth functions on K depending only on r = |x| such supp ζk ⊂ {x : 2 k−1 1. Since (1 − ϕ)u ∈ Vp,1 (QT ), it suffices to 2 2,1 prove that ϕu ∈ Vp,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(ζk u) = fk ≡ ζk f − n X aij (2∂i ζk ∂j u + ∂ij2 ζk u) i,j=1 + n X ∂i aij (∂j ζk u + ζk ∂j u). (3.2) i,j=1 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 kζk ukWp2,1(KT ) 6 C(kfk kLp (KT ) + kζk ukLp (KT ) )
6 C kζk f kLp (KT ) + 2−k kηk ∇ukLp (KT ) + 2−2k kηk ukLp (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 X k∂xα (ζk u)kLp (KT ) + k∂t (ζk u)kLp (KT ) |α|=2
6 C kζk f kLp (KT ) + 2−k kηk ∇ukLp (KT ) + 2−2k kηk ukLp (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 X kr∂xα (ζk u)kLp (KT ) + kr∂t (ζk u)kLp (KT ) |α|=2
6 C krζk f kLp (KT ) + kηk ∇ukLp (KT ) + kr −1 ηk ukLp (KT ) . Summing up over all k and adding to both sides of the obtained equality with kr −1 ukLp (QT ) , we receive X kr∂xα ukLp (QT ) + kr∂t ukLp (QT ) + kr −1ukLp (QT ) |α|=2
6 C krf kLp (QT ) + k∇ukLp (QT ) + kr −1 ukLp (QT ) . 2,1 Noting that the left-hand side of this equality is equivalent to the norm in Vp,1 (QT ) (see [6, Le. 2.1.6]), we have
kukV 2,1 (QT ) 6 C kf kLp (QT ) + k∇ukLp (QT ) + kr −1 ukLp (QT ) . (3.4) p,1 It follows directly from the Hardy’s inequality (see, for example, [4, Th. 85]) that there exists a constant C such that kr −1 ukLp (G) 6 CkukWp1(G) for all u ∈ W̊p1 (G). Thus, we have from (??) that   kukV 2,1 (QT ) 6 C kf kLp (QT ) + kukW̊p1,0(QT ) . p,1 (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. Cerutti, Wp2,1 -solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coeffcients, Comm. Partial Diff. Eq. 18 (1993), no. 9-10, 1735–1763. [3] S. Byun, Parabolic equations with BMO coefficients in Lipschitz domains, J. Diff. Eq. 209 (2005), 229–265. [4] Y. Egorov and V. A. Kondrat’ev, On spectral theory of elliptic operators, Birkhauser Verlag, 1996. [5] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Nauka, Moscow, 1967. [6] V.G. Maz’ya and J. Rossman, Elliptic equations in polyhedral domains, Mathematical Surveys and Monographs 162, Amer. Math. Soc., Providence, Rhode Island, 2010. 52