Cauchy-Neumann Problem for Second-Order General Schrödinger Equations in Cylinders with Nonsmooth Bases

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Volume 2009, Article ID 231802,13pages doi:10.1155/2009/231802

Research Article

Cauchy-Neumann Problem for Second-Order

General Schr ¨odinger Equations in Cylinders with

Nonsmooth Bases

Nguyen Manh Hung,

1

_{Tran Xuan Tiep,}

2

and Nguyen Thi Kim Son

1

1_{Department of Mathematics, Hanoi University of Education, Hanoi, Vietnam}

2_{Faculty of Applied Mathematics and Informatics, Hanoi University of Technology, Hanoi, Vietnam}

Correspondence should be addressed to Nguyen Thi Kim Son,mt02 [email protected]

Received 26 February 2009; Accepted 18 June 2009

Recommended by Gary Lieberman

The main goal of this paper is to obtain the regularity of weak solutions of Cauchy-Neumann problems for the second-order general Schr ¨odinger equations in domains with conical points on the boundary of the bases.

Copyrightq2009 Nguyen Manh Hung et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Notations

Cauchy-Dirichlet problem for general Schr ¨odinger systems in domains containing conical points has been investigated in1,2. Cauchy-Neumann problems have been dealt with for hyperbolic systems in3and for parabolic equations in4–6. In this paper we consider the Cauchy-Neumann problem for the second-order general Schr ¨odinger equations in infinite cylinders with nonsmooth bases. The solvability of this problem has been considered in7. Our main purpose here is to study the regularity of weak solution of the mentioned problem. The paper consists of six sections. In Section 1, we introduce some notations and functional spaces used throughout the text. A weak solution of the problem is defined in

Section 2together with some results of its unique existence and smoothness with the time

variable. Our main result, the regularity with respect to both of time and spatial variables of the weak solution of the problem, is stated inSection 3. The proof of this result is given

inSection 4with some auxiliary lemmas. InSection 5we specify that result for the classical

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Let Ω be a bounded domain in Rn_{, n} _2; _Ω _and _∂_Ω _{denote the closure and the}

boundary of Ωin Rn_{. We suppose that}_Γ _∂_Ω_{\ {}₀_} _{is an infinitely di}_ﬀ_{erentiable surface}

everywhere except the coordinate origin andΩcoincides with the coneK{x: x/|x| ∈G}

in a neighborhood of the origin point 0,whereGis a smooth domain on the unit sphereSn−1

inRn.We begin by introducing some notations and functional spaces which are used fluently in the rest.

DenoteQ∞ Ω×0, ∞,Q∞is the closure ofQ∞, S∞ Γ×0, ∞, x x1, . . . , xn∈

Ω, ∂xj ∂/∂xj, uxj ∂xju, utk ∂ku/∂tk, r |x|

x2

1 · · · xn2. For each multi-indexα

α1, . . . , αn αi∈N, i1, . . . , n, set|α|α1 · · · αn,∂α∂αx∂αx11· · ·∂

αn

xn.

In this paper we will use usual functional spaces:Co∞Ω, L2Ω, HmΩ, wherem∈N

see1,2for the precise definitions.

DenoteH_βlΩis a space of all measurable complex functionsux, tthat satisfy

uHl βΩ

⎛ ⎝

|α|l

Ωr

2β |α|−l_|_∂α_u_|2_dx

⎞ ⎠

1/2

< ∞. 1.1

Hm,l_e−γt_{, Q}

∞ γ > 0—a space of all measurable complex functions ux, t that have generalized derivatives up to ordermwith respect toxand up to orderl with respect tot

with the norm

u_Hm,l_e−γt_,Q ∞ ⎛ ⎝ Q∞ ⎡ ⎣

|α|m

|∂αu|2

l

j1 |utj|2

⎤

⎦_e−2γt_{dx dt} ⎞ ⎠

1/2

< ∞. 1.2

H_βl,ke−γt_{, Q}

∞—a space of all measurable complex functionsux, twith the norm

u_Hl,k

β e−γt,Q∞

⎛ ⎝

Q∞

⎡ ⎣

|α|l

r2β |α|−l|∂αu|2

k

j1 |utj|2

⎤

⎦_e−2γt_{dx dt} ⎞ ⎠

1/2

< ∞. 1.3

H_βle−γt_{, Q}

∞—a weighted space with the norm

uHl

βe−γt,Q∞

⎛

⎝

|α| jl

Q∞

r2β |α| j−l|∂αutj|2e−2γtdx dt

⎞ ⎠

1/2

< ∞. 1.4

Let X be a Banach space. Denote by L∞0,∞;X a space of all measurable functions u :

0, ∞ → X, t→utwith the norm

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2. Formulation of the Problem and Obvious Results

In this paper we consider following problem:

iLu−utf inQ∞, 2.1

ux,0 0 onΩ, 2.2

Nu0 onS∞, 2.3

whereLis a formal self-adjoint diﬀerential operator of second-order defined inQ∞:

LuLx, t, ∂u

n

j,k1

∂ ∂xj

ajkx, t∂u

∂xk

ax, tu, 2.4

ajkx, t akjx, tfor allj, k1,2, . . . , n; ax, t ax, t, for all x, t∈Q∞,and

NuNx, t, ∂u

n

j,k1

ajkx, t∂u

∂xkcos

xj, ν

2.5

is the conormal derivative onS∞, νis the unit exterior normal toS∞, fis a given function. Set

Bt, u, v

Ω

⎛ ⎝n

j,k1

ajkx, t∂u

∂xk

∂v

∂xj −ax, tuv ⎞

⎠_dx. _2.6

Throughout this paper, we assume that the coeﬃcients ofLare infinitely diﬀerentiable and bounded in Q_∞ together with all their derivatives. Moreover, suppose that ajk are

continuous inx∈Ωuniformly with respect tot∈0, ∞for allj, k 1, . . . , n.In addition, assume thatBt,·,·isH1_Ω_{—coercive uniformly with respect to}_t_∈₀_,_∞_,_{that is,}

Bt, u, uμ0u2_H1_Ω ∀u∈H1Ω, t∈0, ∞, 2.7

whereμ0is a positive constant independent ofuandt.

The functionux, tis calleda weak solutionin the spaceH1,0_e−γt_{, Q}

∞of the problem

2.1–2.3ifux, t∈H1,0_e−γt_{, Q}

∞, satisfying for eachT ∈0, ∞

n

j,k1

Q∞ ajk ∂u

∂xk

∂η

∂xjdx dt−

Q∞

auη dx dt i

Q∞

uηtdx dti

Q∞

fη dx dt, 2.8

for all test functionsηx, t∈H1,1_e−γt_{, Q}

∞,ηx, t 0 for allt∈T, ∞.

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Lemma 2.1. The solvability of the problem, (see [7, Theorems 3.1, 3.2]). There exists a positive number γ0such that iff, ft∈L∞0,∞, L2Ωthen for everyγ > γ0,the Cauchy-Neumann problem2.1– 2.3has exactly one weak solutionux, tinH1,0_e−γt_{, Q}

∞, that satisfies

u2_H1,0_e−γt_,Q_∞C

f2_L∞₀_,_∞_,L₂_Ω ft2_L∞₀_,_∞_,L₂_Ω

, 2.9

where the constantCdoes not depend onu,f.

The constantγ0depends only on the operatorLand the dimension of the spacen.

Lemma 2.2. The regularity with respect to time variable of the weak solution (see [7, Theorem 4.1]).Let hbe a nonnegative integer. Suppose thatftk ∈L∞0,∞, L2Ωfor allkh 1, fx,0 0and if

h 2thenftkx,0 0for allk h−1,for allx∈ Ω.Then for everyγ > 2h 1γ₀, the weak

solutionux, tof the problem2.1–2.3has generalized derivatives with respect to time variable up to orderh, which belong toH1,0_e−γt_{, Q}

∞,moreover

uts2_H₁_,₀_e₋γt_,Q

∞C

h 1

k0

_f_tk2_L∞₀_,_∞_,L

2Ω, ∀s0,1, . . . , h, 2.10

whereCis a constant independent ofu,f.

3. Formulation of the Main Result

LetL0x, t, ∂be the principal homogenous part of Lx, t, ∂.We can writeL00, t, ∂in the form

L00, t, ∂ r−2Lω, t, ∂ω, r∂r, 3.1

wherer|x|, ω ω1, . . . , ωn−1is an arbitrary local coordinate system onSn−1,Lis a linear operator with smooth coeﬃcients.

Denoteλtis an eigenvalue of Neumann problem for following equation:

Lω, t, λt, ∂ωvω 0, ω∈G. 3.2

It is well known in8that for eacht∈0, ∞,the spectrum of this problem is an enumerable set of eigenvalues.

Recall thatγ0is the positive real number inLemma 2.1. Now, let us give the main result

of the present paper.

Theorem 3.1. Letl be a nonnegative integer. Assume that ux, tis a weak solution in the space H1,0_e−γt_{, Q}

∞withγ >2l 5γ0of the problem2.1–2.3andftk ∈L∞0,∞, H₀lΩifk3,

ftkx,0 0ifkl 1. In addition, suppose that in the strip

1−ε−n

2 Imλl 2−

n

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whereε >0orε0according ton2orn >2,there is no point from the spectrum of the Neumann problem for the equation3.2for allt∈0, ∞. Then we haveu∈Hl 2

0 e−γt, Q∞and the following estimate holds

u2_Hl 2

0 e−γt,Q∞C

3

k0

_f_tk2

L∞0,∞,Hl

0Ω, 3.4

whereCis a constant independent ofu, f.

4. Proof of Theorem 3.1

By using the same arguments as in1,2and Lemmas2.1,2.2, we can prove following lemma.

Lemma 4.1. Let γ > 3γ0 arbitrary. Assume thatux, tis a weak solution of the problem 2.1– 2.3in the spaceH1,0e−γt, Q∞andf, ft, ftt ∈L∞0,∞, L2Ω, fx,0 0. Then for almost all

t∈0, ∞the equation

n

k,j1

Ωajk

∂u ∂xk

∂χ ∂xjdx−

Ωauχ dxi

Ω

ut f

χ dx 4.1

holds for all functionsχχx∈H1_Ω_.

Now we surround the origin by a neighborhoodU0with a suﬃciently small diameter

such that the intersection ofΩandU0coincides with the coneK.We begin by proving some

auxiliary lemmas.

Lemma 4.2. Letux, tbe a weak solution inH1,0_e−γt_{, Q}

∞ γ >3γ0of the problem2.1–2.3

such thatux, t 0outsideU0. Moreover, we assume thatf, ft, ftt∈L∞0,∞, L2Ω, fx,0 0.

Then for almost allt∈0, ∞,one has

iifn3thenu∈H2

1Ω,

iiifn2thenu∈H₁2 _εΩ,whereε >0arbitrary.

Proof. Because f, ft, ftt ∈ L∞0,∞, L2Ω, fx,0 0, from Lemma 2.2 we have ut ∈

H1,0_e−γt_{, Q}

∞ orut ∈ L2Ω for almost allt ∈ 0, ∞. FollowingLemma 4.1,ux, t is a

solution of the Neumann problem for elliptic equation

LuF, 4.2

whereF −iut f ∈L2Ωfor almost allt ∈0, ∞.DenoteΩk {x∈ Ω : 2−k |x|

2−k 1_}_,_k₁_,₂_{, . . . .}_Let_k

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boundary of domainsee9for reference, we haveu∈ H2_Ω

k0for almost allt ∈0, ∞ and the following inequality holds

ux, t|2_H2_Ω k0 C

Fx, t2_L₂_Ω

k0

ux, t2_L₂_Ω

k0

, 4.3

whereCis a positive constant independent ofu, F. It follows

Ωk0

|∂αux, t|2dxC

Ωk0−1∪Ωk0∪Ωk01

|Fx, t|2 |ux, t|2dx, ∀|α|2. 4.4

By choosingk1 > k0and settingx 2k0/2k1x, one has

Ωk0

_∂α_u_x_{, t}2

dx C

⎡

⎣_F_x_{, t}2

2k0 2k1

4

_u

x, t2

⎤

⎦_dx_, _|_α_|2.

4.5

Return to the variablex, we get

2k0 2k1

2|α|

Ωk₁|∂

α_u_{x, t}_|2_dx_C

Ωk₁−1∪Ωk1∪Ωk11

⎡

⎣_|_F_{x, t}_|2

2k0 2k1

4

|ux, t|2

⎤

⎦_dx, 4.6

where the positive constantCis independent ofu, f, k1.

Case 1n3. Then

Ωr

−2_|_u_|2_dx_C

Ωr

−n_|_u_|2_{dx <}_∞_. _4.7

It follows from4.6that

Ωk1

r2|α|−1|∂αu|2dxC

|F|2r2 r−2|u|2dx, 4.8

whereCdoes not depend onk1.Taking sum with respect tok1> k0, one has

k1>k0

Ωk₁r

2|α|−1_|_∂α_u_|2

dxC

k1k0

Ωk₁

|F|2r2 r−2|u|2dx. 4.9

This implies

k>k0Ωk

r2|α|−1_|_∂α_u_|2_dx_C

kk0Ωk

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Because in out of a neighborhood of conical pointΩis a smooth domain, so we have

Ωr

2|α|−1_|_∂α_u_|2

dxC

Ω

|F|2r2 r−2|u|2dx 4.11

for all|α|2,almost allt∈0, ∞.From4.7,4.11andF∈L2Ωwe receiveu∈H₁2Ω

for almost allt∈0, ∞.

Case 2n 2. Sinceu∈H1,0_e−γt_{, Q}

∞so for almost allt∈0, ∞one has_Ωr0|∂βu|2dx <

∞,|β| 1.This implies_Kr2ε_|_∂β_u_|2_dx _C K|∂βu|

2

dx < ∞,whereε > 0 arbitrary,Cis a positive constant. Becauseu≡0 outsideU0, so we have

Ωr

2ε_∂β_u2_dx_C

Ω

∂βu2dx < ∞. 4.12

For allε >0 we have 2ε >01−n/2, so it follows from8, Lemma 7.1.1, page 268that

Ωr

2ε−1_|_u_|2_dx_C

|β|1

Ωr

2ε_∂β_u2_dx_C

|β|1

Ω

∂βu2dx < ∞. 4.13

From the inequality4.6, for all|α|2 one gets

Ωk₁r

2|α|−1 ε_|_∂α_u_|2

dxC

Ωk₁−1∪Ωk1∪Ωk11

|F|2r2ε 1 r2ε−1|u|2dx, 4.14

whereCdoes not depend on u, f, k1.By using analogous arguments used in Case1, from 4.13,4.14we have

Ωr

21 ε |α|−2_|_∂α_u_|2

dxC

Ω

⎡

⎣_|_F_|2

|β|1

∂βu2

⎤

⎦_{dx <} ∞, 4.15

for all|α|2,almost allt∈0, ∞.That isu∈H2

1 εΩ. The lemma is proved.

Lemma 4.3. Letftk ∈L∞0,∞, L2Ω, k 3, andfx,0 f_tx,0 0forx∈Ω. Assume that

ux, tis a weak solution inH1,0_e−γt_{, Q}

∞ γ > 5γ0of the problem2.1–2.3such thatu ≡ 0

outsideU0. In addition, suppose that the strip

1−ε− n

2 Imλ2−

n

2, 4.16

whereε0orε >0according ton3orn2, does not contain any point of the spectrum of the Neumann problem for the equation3.2for allt∈0, ∞. Thenu∈H2

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Proof. We can rewrite2.1in the form

L00, t, ∂uFx, t −iut f

L00, t, ∂−Lx, t, ∂u. 4.17

Ifn 3 then by applyingLemma 4.2we haveu ∈ H2

1Ω. In another way, because

ajkare continuous inx∈ Ωuniformly with respect tot ∈0, ∞for allj, k 1, . . . , nthen

|ajkx, t−ajk0, t|C|x|, for allt∈0, ∞andCis a constant independent oft. Therefore,

from the hypotheses of this lemma one getsF∈L2Ωfor almost allt∈0, ∞. Since in the strip 1−n/2 Imλ 2−n/2 there is no spectral point of the Neumann problem for the equation3.2for allt∈0, ∞, then following results of the work9, one getsu∈H₀2Ω

and satisfies

u2_H2 0ΩC

F2

L2Ω

u2_H2 1Ω

, 4.18

for almost allt ∈ 0, ∞, whereCis a positive constant. Using the same arguments in the proof ofLemma 4.2, we have

u2_H2 0ΩC

ut2L2Ω f

2

L2Ω u

2

H1_Ω

, 4.19

for almost allt∈0, ∞. Multiplying this inequality withe−2γt_{, then integrating with respect}

totfrom 0 to ∞, fromLemma 2.2one gets

u2_H2,0

0 e−γt,Q∞C

3

k0

_f_tk2

L∞0,∞,L2Ω< ∞. 4.20

Thenuis a function in the spaceH₀2,0e−γt_{, Q}

∞.

Ifn 2 then followingLemma 4.2we haveu ∈ H2

1 εΩfor almost allt ∈ 0, ∞.

This and the property of the functionsajk continuous in x ∈ Ω uniformly with respect to

t∈0, ∞followsF∈H0

1Ω. Because the strip 1−ε−n/2Imλ1−n/2 does not contain

any spectral point of the Neumann problem for3.2, so from results of the work9we have

u∈H₁2Ωsatisfying

u2_H2 1ΩC

F2

H0 1Ω

u2_H2 1 ε

. 4.21

Repeating the proof in the casen3 we achieveu∈H₀2,0e−γt_{, Q}

∞, too. Now diﬀerentiating2.1with respect tot, we have

LvF1−i

vt ft

Ltu, 4.22

wherevut, Lt nj,k1∂/∂xjajkt∂/∂xk at. From the hypotheses of the operatorL

andLemma 2.2we haveF1∈L2Ωfor almost allt∈0, ∞. Repeating arguments used for

functionuwe receivev∈H₀2,0e−γt_{, Q}

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In another way, it follows fromLemma 2.2that

Q∞

|ut2|2e−2γtdx dt < ∞. 4.23

From4.23and the assertion that bothuandut are in the spaceH02,0e−γt, Q∞we

haveu∈H₀2e−γt_{, Q}

∞. This lemma is proved.

Lemma 4.4. Letlbe a nonnegative integer number, γ be a real number satisfyingγ > 2l 5γ0,

ux, tbe a weak solution inH1,0_e−γt_{, Q}

∞of the problem2.1–2.3such thatu ≡0outsideU0. Assume thatftk ∈ L∞0,∞, H₀lΩ, k 3, and f_tkx,0 0fork l 1, x ∈ Ω. Moreover,

suppose that the strip

1−ε−n

2 Imλl 2−

n

2 4.24

does not contain any point of the spectrum of the Neumann problem for the equation 3.2 for all t∈0, ∞, whereε0orε >0according ton3orn2. Thenu∈H₀l 2e−γt, Q∞, satisfying

u2_Hl 2

0 e−γt,Q∞C

3

k0

_f_tk2

L∞0,∞,Hl

0Ω, 4.25

where the constantCis independent ofu, f.

Proof. We use the induction byl. Forl0 then we hadLemma 4.3with noting thatH0

0Ω≡

L2Ω. Assume that lemma’s assertion holds up tol−1, we need to prove this holds up tol. It means that we have to prove following inequality:

utj2

Hl ₀2−je−γt_,Q_∞C

3

k0

_f_tk2_L_∞₀_,_∞_,Hl

0Ω, 4.26

forjl, l−1, . . . ,0, whereCis a positive constant.

Sinceftk ∈L∞0,∞, H₀lΩfork3, sof_tk ∈L∞0,∞, L2Ωforkl 3. In another way,ftkx,0 0 fork l 1. Then fromLemma 2.2we have u_tl 1 ∈ H1,0e−γt, Q_∞, u_ts ∈

H1,0_e−γt_{, Q}

∞for allsl.Hence, by using similar arguments in the proof ofLemma 4.3we getutl ∈H₀2e−γt, Q_∞. This means that4.26holds forj l.

Assume that4.26holds forj l, l−1, . . . , s 1. By puttingvuts ∈Hl−s 1

0 e−γt, Q∞ by inductive hypothesisand diﬀerentiating2.1s-times with respect tot, we have

Lv−ivt fts

s

p1

CpsLtpu_ts−p, 4.27

whereLtp n_j,k₁∂/∂x_ja_jk_tp∂/∂xk atp.Following the assumptions of the induction ofsand the hypotheses of the functionfone hasvt∈H₀l−s 1e−γt, Q∞, fts ∈H₀l−se−γt, Q_∞. It follows Fs −ivt fts s_p₁Cp_sL_tpu_ts−p ∈ Hl−s

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H₀l−se−γt_{, Q}

∞ ⊆ H₋l−1s−1,0e−γt, Q∞, so we have Fs ∈ H−l−1s−1e−γt, Q∞ for almost all t ∈ 0, ∞. Because the stripl 1 −s−n/2 Imλ l 2−s−n/2 does not contain any point of the spectrum of the Neumann problem for3.2for allt ∈ 0, ∞, then following results of the work 9, one getsv ∈ H₋l₁1−sΩ. This implies v ∈ H₋l₁1−s,0e−γt_{, Q}

∞. Note thatFs ∈H₀l−s,0e−γt, Q∞then by applying8, Theorem 7.3.2one getsv∈H0l 2−s,0e−γt, Q∞

satisfying

v2_Hl 2−s,0

0 e−γt,Q∞C

Fs2_Hl−s,0

0 e−γt,Q∞ v

2

Hl ₋₁1−s,0e−γt_,Q_∞

, 4.28

whereCis a positive constant. In another way, it is easy to see that

uts2

Hl 2−s

0 e−γt,Q∞ut s 12

Hl 2−s−1

0 e−γt,Q∞ ut s2

Hl ₀2−s,0e−γt_,Q

∞. 4.29

Hence from the inductive assumptions we receive

uts2

Hl 2−s

0 e−γt,Q∞C

3

k0

_f_tk2

L∞0,∞,Hl

0Ω, 4.30

whereCis a constant independent ofu, f. It means that4.26is proved. Finally we only need to fixj 0 in4.26to complete the proof of this lemma.

Now let us proveTheorem 3.1.

Proof. Denoteu0 ϕ0u, where ϕ0 ∈

o

C∞U0 andϕ0 ≡ 1 in a neighborhood of coordinate

origin. The functionu0satisfies

iLu0−u0tϕ0f L1u, 4.31

whereL1uis a linear diﬀerential operator order 1. Coeﬃcients of this operator depend on the

choice of the functionϕ0and equal to 0 outsideU0.Denoteu1ϕ1u 1−ϕ0u. It is easy to see

thatu1is equal to 0 in a neighborhood of conical point. Therefore we can apply the theorem

on the smoothness of a solution of elliptic problem in a smooth domain to this function to conclude that u1 ϕ1u ∈ H₀l 2Ωfor almost all t ∈ 0, ∞.By applying Lemma 2.2we

receiveu1∈H0l 2e−γt, Q∞and

u12_Hl 2

0 e−γt,Q∞C

3

k0

_f_tk2_L_∞₀_,_∞_,Hl

0Ω, Cconst>0. 4.32

Now, let us proveTheorem 3.1by induction byl.Whenl 0 then functionsu0,f

ϕ0f L1u satisfy the hypotheses ofLemma 4.3. So u0 ∈ H02e−γt, Q∞.It follows thatu

u0 u1 is in H₀2e−γt, Q∞. Assume that the theorem holds up to l−1 then we have u ∈

Hl 1

0 e−γt, Q∞.By using analogous arguments in the proof ofLemma 4.4, with note thatfts∈

Hl−s

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u∈H₀l 2e−γt_{, Q}

∞.The inequality inTheorem 3.1can derive from inequality4.25 foru0 and inequality4.32. The theorem is proved completely.

5. Cauchy-Neumann Problem For Classical Schr ¨odinger Equation In

Quantum Mechanics

In this section we apply the previous result to the Cauchy-Neumann problem for classical Schr ¨odinger equations in quantum mechanics. It is shown that the smoothness of the weak solution of this problem depends on the structure of the boundary of the domain, the right hand side and the dimensionnof the spaceRn_.

The classical Schr ¨odinger equation in quantum mechanics has the form

iΔux, t−utx, t fx, t, 5.1

whereΔis the Laplace operator. Now we consider the Cauchy-Neumann problem for5.1 in infinite cylinderQ∞with the initial condition

ux,0 0 onΩ, 5.2

and the boundary condition

∂u ∂ν

n

k1

∂u

∂xkcosxk, ν 0 on S∞, 5.3

whereνis the unit exterior normal toS∞.

The Laplace operator in polar coordinater, ωinRn_{can be written in the form}

Δur, ω 1

rn−1

∂ ∂r

rn−1 ∂ ∂r

ur, ω 1

r2Δωur, ω, 5.4

where Δω is the Laplace-Beltrami operator on the unit sphere Sn−1. Therefore, the

corresponding spectral problem for3.2is the Neumann problem for following equation:

Δωv

iλ2 i2−nλv0, ω∈G. 5.5

The regularity of the weak solution of the problem5.1–5.3can be stayed as follows.

Theorem 5.1. Letn >4, ube a weak solution in the spaceH1,0_e−γt_{, Q}

∞ γ >5γ0of the

Cauchy-Neumann problem5.1–5.3andftk ∈ L∞0,∞, L2Ωifk 3,fx,0 f_tx,0 0. Then

u∈H2

0e−γt, Q∞.

Proof. Notekbe nonnegative eigenvalues of the Neumann problem for equation

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Thenλi2−n/2±

n−2/22 kare eigenvalues of the Neumann problem for5.5. It is easy to see that whenn >4 the strip

1−n

2 Imλ2−

n

2 5.7

does not contain any eigenvalue of the Neumann problem for5.5. By applying Theorem3.1

we haveu∈H2

0e−γt, Q∞. The theorem is proved.

6. Conclusions

The Schr ödinger equation has received a great deal of attention from mathematicians, in particular because of its application to quantum mechanics and optics. It is therefore important to research boundary value problems for it. Such problems have been previously proposed and analyzed for Schr ödinger equations whose coefficients are independent of the time variable and in finite cylindersQT T < ∞ see, e.g.,10. In infinite cylinderQ∞, the first initial boundary value problem for this kind of equation with coefficients depend on both of time and spatial variables has been consideredsee1,2. In this paper, for a general Schr ödinger equation in infinite cylinder Q∞ with conical points in the boundary of base, we proved regularity property of solution of second initial boundary value problem. As a special application of these new results, we received the regularity of solution of a classical Schr ödinger equation in quantum mechanics when the dimension of spacen >4. The similar questions for the casen4 can be answered after researching the asymptotic of solution in the case the strip 1−n/2Imλ2 l−n/2 contains eigenvalues of the associated spectral problem. This is also the aim of our future research.

References

1 N. M. Hung, “The first initial boundary value problem for Schr ¨odinger systems in non-smooth domains,”Diﬀ. Urav., vol. 34, pp. 1546–1556, 1998Russian.

2 N. M. Hung and C. T. Anh, “On the smoothness of solutions of the first initial boundary value problem for Schr ¨odinger systems in domains with conical points,”Vietnam Journal of Mathematics, vol. 33, no. 2, pp. 135–147, 2005.

3 A. Yu. Kokotov and B. A. Plamenevski˘ı, “On the asymptotic behavior of solutions of the Neumann problem for hyperbolic systems in domains with conical points,”Algebra i Analiz, vol. 16, no. 3, pp. 56–98, 2004Russian, English translation inSt. Petersburg Mathematical Journal, vol. 16, no. 3, pp. 477–506, 2005.

4 E. V. Frolova, “An initial-boundary value problem with a noncoercive boundary condition in a domain with edges,”Zapiski Nauchnykh Seminarov (POMI), vol. 213, no. 25, pp. 206–223, 1994.

5 N. M. Hung and N. T. Anh, “Regularity of solutions of initial-boundary value problems for parabolic equations in domains with conical points,”Journal of Diﬀerential Equations, vol. 245, no. 7, pp. 1801– 1818, 2008.

6 V. A. Solonnikov, “On the solvability of classical initial-boundary value problem for the heat equation in a dihedral angle,”Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, vol. 127, pp. 7–48, 1983.

7 N. M. Hung and N. T. K. Son, “Existence and smoothness of solutions to second initial boundary value problems for Schr ¨odinger systems in cylinders with non-smooth bases,”Electronic Journal of Diﬀerential Equations, vol. 2008, no. 35, pp. 1–11, 2008.

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9 L. Nirenberg, “Remarks on strongly elliptic partial diﬀerential equations,”Communications on Pure and Applied Mathematics, vol. 8, pp. 648–674, 1955.