The neumann problem for second order hyperbolic equations in nonsmooth domains

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2012, Vol. 57, No. 3, pp. 53-59 THE NEUMANN PROBLEM FOR SECOND ORDER HYPERBOLIC EQUATIONS IN NONSMOOTH DOMAINS Nguyen Van Trung Hong Duc University, Thanh Hoa City E-mail: trungnv2000@gmail.com Abstract. In this paper, we study the Neumann problem for second order hyperbolic equations without initial data in nonsmooth domains. Our inten- sion was to prove the existence of a generalized solution to this problem by applying the results of this problem to the initial data. Keywords: Neumann problem, nonsmooth domains, hyperbolic equations, generalized solution. 1. Introduction We are concerned with boundary value problems and value for hyperbolic equations in nonsmooth domains. The problems with the Dirichlet boundary condi- tions and initial data have been investigated in [2,3,4]. The boundary value problem without initial condition for parabolic equation has been investigated in [5,6,7,8]. The main goal of this work is to prove the existence of a generalized solution of the Neumann problem without having initial data. Let Ω be a nonsmooth domain in Rn (n ≥ 2). For h ∈ R, set Qh = Ω × (h,∞), Sh = ∂Ω× (h,∞), S = ∂Ω× R, Q = Ω× R. Let L(x, t; ∂) = − n∑ i,j=1 ∂i(aij(x, t)∂j) + n∑ i=1 bi(x, t)∂i + c(x, t), be a second order partial differential operator, where ∂i = ∂xi , and aij , bi, c are bounded functions from C∞(Q). We study the present paper hyperbolic equation utt + L(x, t;D)u = f in Q, (1.1) with Neumann boundary conditions ∂u(x, t) ∂νL = 0, (x, t) ∈ S (1.2) where ∂u(x,t) ∂νL = ∑n j=1 ∂ju(x, t)aij(x, t)νi(x, t). 53 Nguyen Van Trung Let m, k be non-negative integers. Denote by Hm(Ω) the Sobolev spaces in [1]. By the notation (., .) we mean the inner product in L2(Ω). Denote by H−1(Ω) the dual space of H1(Ω). The pairing between H1(Ω) and H−1(Ω) is denoted by 〈., .〉. Let X be a Banach space, γ = γ(t) be a real functions. We denote by L2(a,+∞, γ;X), the space of functions f : (a,+∞)→ X with the norm ‖f‖Lp(a,+∞,γ;X) = (∫ +∞ a ‖f(t)‖2Xe −γ(t)tdt ) 1 2 <∞. Finally, we introduce the Sobolev spaceH1,1∗ (Qa, γ) which consists all functions u defined on Qa such that u ∈ L2(a,+∞, γ;H1(G)), ut ∈ L2(a,+∞, γ;L2(Ω)) and utt ∈ L2(a,+∞, γ;H −1(Ω)) with the norm ‖u‖2 H 1,1 ∗ (Qa,γ) = ‖u‖2L2(a,+∞,γ;H1(Ω)) + ‖ut‖ 2 L2(a,+∞,γ;L2(Ω)) + ‖utt‖ 2 L2(a,+∞,γ;H−1(Ω)) . To simplify notation, we set L2(Q, γ0) = L2(R, γ0;L2(Ω)). We assume that the operator L is uniformly strong elliptic, that is, there exists a constant µ0 > 0 such that n∑ i,j=1 aij(x, t)ξξ ≥ µ0|ξ| 2, ξ ∈ R, (x, t) ∈ Q. (1.3) Let us introduce the following bilinear form B(u, v; t) = ∫ Ω ( n∑ i,j=1 (aij(x, t)∂ju∂iv + n∑ i=1 bi(x, t)∂iuv + c(x, t)uv ) dx. Then we have from the classical Green formula B(u, v; t)− (L(x, t;D)u, v) = ∫ ∂Ω ∂u ∂νL vds, u, v ∈ C∞(Ω). Definition 1.1. Let f ∈ L2(Q, γ0), then a function u ∈ H1,1∗ (Q, γ) is called a generalized solution of problem (1.1) - (1.2) if and only if the equality 〈utt, v〉+B(u, v; t) = (f, v), a.e. t ∈ R, (1.4) holds for all v ∈ H1(Ω). 2. The main results Theorem 2.1. Suppose that the coefficients of the operator L satisfy sup{|aij|, |aijt|, |bi|, |c| : i, j = 1, . . . , n; (x, t) ∈ Q} ≤ µ, µ = const. 54 The Neumann problem for second order hyperbolic equations in nonsmooth domains Then for each γ(t)t > γ0(t)t, t ∈ R, problem (1.1)-(1.2) has a generalized solution u in the space H1,1∗ (Q, γ) and the following estimate holds ‖u‖2 H 1,1 ∗ (Q,γ) ≤ C‖f‖2L2(Q,γ0) (2.1) here C is a constant independent of u and f . Proof. To prove the theorem, we construct a family approximate solution uh of solution u of the problem (1.1)-(1.2). It is known that there is a smooth function χ(t) which is equal to 1 on [1,+∞), is equal to 0 on (−∞, 0] and assumes value in [0, 1] on [0; 1] (see [4, Th. 5.5] for more details). Moreover, we can suppose that all derivatives of χ(t) are bounded. Let h ∈ (−∞, 0] be an integer. Setting fh(x, t) = χ(t−h)f(x, t) then fh = { f if t ≥ h + 1 0 if t ≤ h. Moreover, if f ∈ L2(Q, γ0), fh ∈ L2(h,∞, γ;L2(Ω)), fh ∈ L2(Q; γ0) and ‖fh‖2L2(Qh;γ0) = ‖f h‖2L2(Q;γ0) ≤ ‖f‖ 2 L2(Q;γ0). (2.1) Fixed f ∈ L2(Q; γ0), we consider the following problem in the cylinder Qh: utt + L(x, t,D)u = f h(x, t) in Qh, (2.2) ∂u(x, t) ∂νL = 0, (x, t) ∈ Sh, (2.3) u |t=h= 0, ut |t=h= 0 on Ω. (2.4) This is the initial boundary value problem for hyperbolic equations in cylinders Qh. A function uh ∈ H1,1∗ (Qh, γ) is called a generalized solution of the problem (2.2)-(2.4) iff uh(., h) = 0, uht (., h) = 0 and the equality 〈uhtt, v〉+B(u h, v; t) = (fh, v), holds for a.e. t ∈ (h,∞) and all v ∈ H1(Ω). Lemma 2.1. For any h fixed, there exists a solution of the problem (2.2)-(2.4). Firstly, we will prove the existence by Galerkin’s approximating method. Let {ωk(x)} ∞ k=1 be an orthogonal basis of H1(Ω) which is orthonormal in L2(Ω). Put uN(x, t) = N∑ k=1 CNk (t)ωk(x) where CNk (t), k = 1, . . . , N, is the solution of the following ordinary differential system: (uNtt , ωk) +B(u N , ωk; t) = (f h, ωk), k = 1, . . . , N, (2.2) 55 Nguyen Van Trung with the initial conditions CNk (h) = 0, C N kt(h) = 0, k = 1, . . . , N. (2.3) Let us multiply (2.2) by CNkt(t), then take the sum with respect to k from 1 to N to arrive at (uNtt , u N t ) +B(u N , uNt ; t) = (f h, uNt ). Since (uNtt , uNt ) = ddt ( ‖uNt ‖ 2 L2(Ω) ) , we get d dt ( ‖uNt ‖ 2 L2(Ω) ) + 2B(uN , uNt ; t) = 2(f h, uNt ). (2.4) By the Cauchy-Schwarz inequality, we have 2|(fh, uNt )| ≤ ‖f h‖2L2(Ω) + ‖u N t ‖ 2 L2(Ω). (2.5) Furthermore, we can write B(uN , uNt ; t) = ∫ Ω n∑ i,j=1 Aij(x, t)Dju NDiu N t dx+ ∫ Ω n∑ i=1 Bi(x, t)Diu NuNt +C(x, t)u NuNt dx =: B1 +B2. (2.6) It is easy to see B1 = d dt (1 2 A[uN , uN , t] ) − 1 2 ∫ Ω n∑ i,j=1 AijtDju NDiu Ndx, (2.7) for the symmetric bilinear form A[uN , uN , t] = ∫ Ω n∑ i,j=1 AijDju NDiu Ndx. The equality (2.7) implies B1 ≥ d dt (1 2 A[uN , uN , t] ) − µ‖uN‖2H1(Ω), (2.8) and we note also |B2| ≤ µ ( ‖uN‖2H1(Ω) + ‖u N t ‖ 2 L2(Ω) ) , (2.9) Combining estimates (2.4)-(2.9), we obtain d dt ( ‖uNt ‖ 2 L2(Ω) + A[uN , uN , t] ) ≤ 2µ ( ‖uNt ‖ 2 L2(Ω) + ‖uN‖2H1(Ω) ) + ‖fh‖2L2(Ω) ≤ µ1 ( ‖uNt ‖ 2 L2(Ω) + A[uN , uN , t] ) + ‖fh‖2L2(Ω) (2.10) 56 The Neumann problem for second order hyperbolic equations in nonsmooth domains where we used (1.3), µ1 = max{2µ, 2µµ0 }. Now write η(t) := ‖uNt (., t)‖ 2 L2(Ω) + A[uN , uN , t]; ξ(t) := ‖fh(., t)‖2L2(Ω), t ∈ [h,∞). Then (2.10) implies η′(t) ≤ µ1η(t) + ξ(t), for a.e. t ∈ [h,∞). Thus the differential form of Gronwall-Belmann’s inequality yields the esti- mate η(t) ≤ C t∫ h eµ1(t−s)ξ(s)ds, t ∈ [h,∞). (2.11) We obtain from (2.11) and (1.3) the following estimate ‖uN(., t)‖2L2(Ω) + ‖u N‖2H1(Ω) ≤ C t∫ h eµ1(t−s)‖fh‖2L2(Ω)ds ≤ Ce µ1t‖fh‖2L2(Q,γ0), where γ0(t) ∈ [µ1,+∞) for t 0. Now multiplying both sides of this inequality by e−γ(t)t, then integrating them with respect to t from h to ∞, we obtain ‖uN(., t)‖2L2(h,∞,γ,L2(Ω)) + ‖u N‖2L2(h,∞,γ,H1(Ω)) ≤ C‖f h‖2L2(Q,γ0), (2.12) where γ(t) ∈ [µ1,+∞) for t > 0 and γ(t) ∈ [0, µ1) for t < 0. Fix any v ∈ H1(Ω), with ‖v‖2H1(Ω) ≤ 1 and write v = v 1 + v2 where v1 ∈ span{ωk}Nk=1 and (v2, ωk) = 0, k = 1, . . . , N, (v2 ∈ span{ωk}Nk=1 ⊥ ). We have ‖v1‖H1(Ω) ≤ ‖v‖H1(Ω) ≤ 1. Utilizing (2.2), we get (uNtt , v 1) +B(uN , v1; t) = (fh, v1) for a.e. t ∈ [h,+∞). From uN(x, t) = N∑ k=1 CNk (t)ωk, we can see that (uNtt , v) = (u N tt , v 1) = (fh, v1)− B(uN , v1; t). Consequently, |(uNtt , v)| ≤ C ( ‖fh‖2L2(Ω) + ‖u N‖2H1(Ω) ) . Since this inequality holds for all v ∈ H˚1(Ω), ‖v‖H1(Ω) ≤ 1, the following inequality will be inferred ‖uNtt ‖ 2 H−1(Ω) ≤ C ( ‖fh‖2L2(Ω) + ‖u N‖2H1(Ω) ) . (2.13) 57 Nguyen Van Trung Multiplying (2.13) by e−γ(t)t, then integrating them with respect to t from h to ∞, and by using (2.12), we obtain ‖uNtt ‖ 2 L2(h,∞,γ,H−1(Ω)) ≤ C‖fh‖2L2(Q,γ0). (2.14) Combining (2.12) and (2.14), we arrive at ‖uN‖2 H 1,1 ∗ (Qh,γ) ≤ C‖fh‖2L2(Q,γ0) (2.15) where C is a absolute constant. From the inequality (2.15), by standard weakly convergent arguments, we can conclude that the sequence {uN}∞N=1 possesses a subsequence convergent to a function uh ∈ H1,1∗ (Qh, γ), which is a generalized solution of problem (2.2)-(2.4). Let k be a integer less than h, denote uk a generalized solution of the problem (2.2)-(2.4) when we replaced h by k. We define uh in the cylinder Qk by setting uh(x, t) = 0 for k ≤ t ≤ h. Putting uhk = uh − uk, fhk = fh − fk, so uhk is the generalized solution of the following problem uhktt + L(x, t,D)u hk = fhk(x, t) in Qk, (2.12) uhk = 0 on Sk, j = 1, ..., m, (2.13) uhk |t=k= 0, u hk t |t=k= 0 on Ω. (2.14) We have ‖uhk‖2 H 1,1 ∗ (Qk,γ) ≤ C‖fh − fk‖2L2(Q,γ0), and ‖fh − fk‖2L2(Q,γ0) = h+1∫ k e−γ0(t)t‖fh − fk‖2L2(Ω)dt. ≤ 2 h+1∫ k e−γ0(t)t‖f‖2L2(Ω)dt. Thus ‖uh − uk‖2 H 1,1 ∗ (Q(k,∞),γ) ≤ 2C h+1∫ k e−γ0(t)t‖f‖2L2(Ω)dt. (2.15) Since f ∈ L2(Q, γ0), h∫ k e−γ0(t)t‖f‖2L2(Ω)dt → 0 when h, k → −∞. It follows that {uh}−∞h=0 is a Cauchy sequence. So {uh} is convergent to u in H1,1∗ (Qk, γ) (Consider uh in the cylinder Q by setting uh(x, t) = 0 for all t < h). Because ‖fh − f‖2L2(Q,γ0) = h+1∫ h e−γ0(t)t|χ(t− h)− 1|.‖f‖2L2(Ω)dt+ h∫ −∞ e−γ0(t)t‖f‖2L2(Ω)dt, 58 The Neumann problem for second order hyperbolic equations in nonsmooth domains so ‖fh − f‖2L2(Q,γ0) ≤ 2 h+1∫ h e−γ0(t)t‖f‖2L2(Ω)dt+ h∫ −∞ e−γ0(t)t‖f‖2L2(Ω)dt, {fh} is convergent to f in L2(Q, γ0). 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