Cauchy-Neumann Problem for Second-Order

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,

Tran Xuan Tiep,

and Nguyen Thi Kim Son

1Department of Mathematics, Hanoi University of Education, Hanoi, Vietnam

2Faculty 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 Schr ¨odinger equations in quantum mechanics. Finally, some conclusions of our results are given inSection 6.

Let Ω be a bounded domain in Rⁿ, n 2; Ω and ∂Ω denote the closure and the boundary of Ωin Rⁿ. We suppose thatΓ ∂Ω\ {0} is an infinitely differentiable 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, ut^k ∂^ku/∂t^k, r |x|

x²₁ · · · xn². For each multi-indexα α1, . . . , αn αi∈N, i1, . . . , n, set|α|α1 · · · αn,∂^α∂^α_x∂^αx¹1· · ·∂^αxⁿn.

In this paper we will use usual functional spaces:C^o∞Ω, L2Ω, H^mΩ, wherem∈N see1,2for the precise definitions.

DenoteH_β^lΩis a space of all measurable complex functionsux, tthat satisfy

u_H^l

βΩ

⎛

⎝

|α|l

Ωr^{2β |α|−l}|∂^αu|²dx

⎞

⎠

1/2

< ∞. 1.1

H^m,le^−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,le^−γt,Q∞

⎛

⎝

Q∞

⎡

⎣

|α|m

|∂^αu|^{2 l}

j1

|u_t^j|²

⎤

⎦e^−2γtdx dt

⎞

⎠

1/2

< ∞. 1.2

H_β^l,ke^−γt, Q_∞—a space of all measurable complex functionsux, twith the norm

u_H^l,k

β e^−γt,Q∞

⎛

⎝

Q_∞

⎡

⎣

|α|l

r^{2β |α|−l}|∂^αu|^{2 k}

j1

|ut^j|²

⎤

⎦e^−2γtdx dt

⎞

⎠

1/2

< ∞. 1.3

H_β^le^−γt, Q∞—a weighted space with the norm

u_H^l

βe^−γt,Q∞

⎛

⎝

|α| jl

Q∞

r^{2β |α| j−l}|∂^αut^j|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

u_L∞0,∞;Xess sup0<t< ∞ut_X < ∞. 1.5

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,·,·isH¹Ω—coercive uniformly with respect tot∈0, ∞,that is,

Bt, u, uμ0u²_H1Ω ∀u∈H¹Ω, t∈0, ∞, 2.7 whereμ0is a positive constant independent ofuandt.

The functionux, tis called a weak solution in the spaceH^1,0e^−γt, Q_∞of the problem 2.1–2.3ifux, t∈H^1,0e^−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∈H^1,1e^−γt, Q_∞,ηx, t 0 for allt∈T, ∞.

Now we derive here some our obvious results of the unique existence and smoothness with respect to time variable of the weak solution of the problem2.1–2.3as lemmas of main results.

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, tinH^1,0e^−γt, Q∞, that satisfies

u²_H1,0e^−γt,Q∞Cf²

L^∞0,∞,L2Ω ft²

L^∞0,∞,L2Ω

, 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 thatft^k ∈L^∞0,∞, L2Ωfor allkh 1, fx,0 0 and if h 2 thenft^kx,0 0 for 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 toH^1,0e^−γt, Q_∞,moreover

ut^s2_H1,0e^−γt,Q_∞C

h 1

k0

ft^k2

L^∞0,∞,L2Ω, ∀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 H^1,0e^−γt, Q_∞withγ >2l 5γ0of the problem2.1–2.3andft^k ∈L^∞0,∞, H₀^lΩifk3, ft^kx,0 0 ifkl 1. In addition, suppose that in the strip

1−ε−n

2 Imλl 2− n

2, 3.3

whereε >0 orε0 according ton2 orn >2,there is no point from the spectrum of the Neumann problem for the equation3.2for allt∈0, ∞. Then we haveu∈H₀^{l 2}e^−γt, Q_∞and the following estimate holds

u²_Hl 2

0 e^−γt,Q_∞C 3 k0

ft^k2

L^∞0,∞,H0^lΩ, 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 spaceH^1,0e^−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∈H¹Ω.

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 inH^1,0e^−γt, Q∞ γ >3γ0of the problem2.1–2.3 such thatux, t 0 outsideU0. Moreover, we assume thatf, ft, ftt∈L^∞0,∞, L2Ω, fx,0 0.

Then for almost allt∈0, ∞,one has iifn3 thenu∈H₁²Ω,

iiifn2 thenu∈H_{1 ε}² Ω,whereε >0 arbitrary.

Proof. Because f, ft, ftt ∈ L^∞0,∞, L2Ω, fx,0 0, from Lemma 2.2 we have ut ∈ H^1,0e^−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, ∞.DenoteΩ^k {x∈ Ω : 2^−k |x|

2^{−k 1}},k1,2, . . . .Letk0be large enough such thatΩk0−1⊂U0. By choosing a smooth domain Ω_k0such thatΩk0 ⊂Ω_k0⊂Ωk0−1∪Ωk0∪Ωk0 1, from the theory of the regular of solutions of the boundary value problem for elliptic systems in smooth domains and near the piece smooth

boundary of domainsee9for reference, we haveu∈ H²Ω_k0for almost allt ∈0, ∞ and the following inequality holds

ux, t|²_H2Ω_k0 C

Fx, t²_L2Ω

k0

ux, t²_L2Ω

k0

, 4.3

whereCis a positive constant independent ofu, F. It follows

Ωk0

|∂^αux, t|²dxC

Ωk0−1∪Ωk0∪Ωk0 1

|Fx, t|² |ux, t|²

dx, ∀|α|2. 4.4

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

Ωk0

∂^αux, t²dx C

Ωk0−1∪Ωk0∪Ωk0 1

⎡

⎣F

x, t² 2^k0 2^k1

4

u

x, t²⎤

⎦dx, |α|2.

4.5

Return to the variablex, we get 2^k0

2^k1 _2|α|

Ωk1

|∂^αux, t|²dxC

Ωk1−1∪Ωk1∪Ωk1 1

⎡

⎣|Fx, t|² 2^k0

2^k1 ₄

|ux, t|²

⎤

⎦dx, 4.6

where the positive constantCis independent ofu, f, k1. Case 1n3. Then

Ωr⁻²|u|²dxC

Ωr⁻ⁿ|u|²dx < ∞. 4.7 It follows from4.6that

Ωk1

r^2|α|−1|∂^αu|²dxC

Ωk1−1∪Ωk1∪Ωk1 1

|F|²r² r⁻²|u|²

dx, 4.8

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

k1>k0

Ωk1

r^2|α|−1|∂^αu|²dxC

k1k0

Ωk1

|F|²r² r⁻²|u|²

dx. 4.9

This implies

k>k0Ωk

r^2|α|−1|∂^αu|²dxC

kk0Ωk

|F|²r² r⁻²|u|²

dx. 4.10

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

Ωr^2|α|−1|∂^αu|²dxC

Ω

|F|²r² r⁻²|u|²

dx 4.11

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

Case 2n 2. Sinceu∈H^1,0e^−γt, Q_∞so for almost allt∈0, ∞one has

Ωr⁰|∂^βu|²dx <

∞,|β| 1.This implies

Kr^2ε|∂^βu|²dx C

K|∂^βu|²dx < ∞,whereε > 0 arbitrary,Cis a positive constant. Becauseu≡0 outsideU0, so we have

Ωr^2ε∂^βu²dxC

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|²dxC

|β|1

Ωr^2ε∂^βu²dxC

|β|1

Ω

∂^βu²dx < ∞. 4.13

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

Ωk1

r^{2|α|−1 ε}|∂^αu|²dxC

Ωk1−1∪Ωk1∪Ωk1 1

|F|²r^{2ε 1} r^2ε−1|u|²

dx, 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}|∂^αu|²dxC

Ω

⎡

⎣|F|²

|β|1

∂^βu²

⎤

⎦dx < ∞, 4.15

for all|α|2,almost allt∈0, ∞.That isu∈H_{1 ε}² Ω. The lemma is proved.

Lemma 4.3. Letft^k ∈L^∞0,∞, L2Ω, k 3, andfx,0 ftx,0 0 forx∈Ω. Assume that ux, tis a weak solution inH^1,0e^−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

whereε0 orε >0 according ton3 orn2, does not contain any point of the spectrum of the Neumann problem for the equation3.2for allt∈0, ∞. Thenu∈H₀²e^−γt, Q_∞.

Proof. We can rewrite2.1in the form L00, t, ∂uFx, t −i

ut f

L00, t, ∂−Lx, t, ∂u. 4.17 Ifn 3 then by applyingLemma 4.2we haveu ∈ H₁²Ω. 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

ut²_L2Ω f²

L2Ω u²_H1Ω

, 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

ft^k2

L^∞0,∞,L2Ω< ∞. 4.20

Thenuis a function in the spaceH₀^2,0e^−γt, Q∞.

Ifn 2 then followingLemma 4.2we haveu ∈ H_{1 ε}² Ωfor almost allt ∈ 0, ∞.

This and the property of the functionsajk continuous in x ∈ Ω uniformly with respect to t∈0, ∞followsF∈H₁⁰Ω. 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

H⁰₁Ω u²_H2 1 ε

. 4.21

Repeating the proof in the casen3 we achieveu∈H₀^2,0e^−γt, Q∞, too.

Now differentiating2.1with respect tot, we have LvF1−i

vt ft

Ltu, 4.22

wherevut, Lt ⁿ_j,k1∂/∂xjajk_t∂/∂xk a_t. From the hypotheses of the operatorL andLemma 2.2we haveF1∈L2Ωfor almost allt∈0, ∞. Repeating arguments used for functionuwe receivev∈H₀^2,0e^−γt, Q_∞orut∈H₀^2,0e^−γt, Q_∞.

In another way, it follows fromLemma 2.2that

Q∞

|ut²|²e^−2γtdx dt < ∞. 4.23

From4.23and the assertion that bothuandut are in the spaceH₀^2,0e^−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 inH^1,0e^−γt, Q∞of the problem2.1–2.3such thatu ≡0 outsideU0. Assume thatft^k ∈ L^∞0,∞, H₀^lΩ, k 3, andft^kx,0 0 fork 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, ∞, whereε0 orε >0 according ton3 orn2. Thenu∈H₀^{l 2}e^−γt, Q∞, satisfying

u²_Hl 2

0 e^−γt,Q∞C 3 k0

ft^k2

L^∞0,∞,H0^lΩ, 4.25

where the constantCis independent ofu, f.

Proof. We use the induction byl. Forl0 then we hadLemma 4.3with noting thatH₀⁰Ω≡ 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:

u_t^j2

H^{l 2−j}₀ e^−γt,Q∞C 3 k0

ft^k2

L^∞0,∞,H₀^lΩ, 4.26

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

Sinceft^k ∈L^∞0,∞, H₀^lΩfork3, soft^k ∈L^∞0,∞, L2Ωforkl 3. In another way,ft^kx,0 0 fork l 1. Then fromLemma 2.2we have ut^{l 1} ∈ H^1,0e^−γt, Q∞, ut^s ∈ H^1,0e^−γt, Q_∞for allsl.Hence, by using similar arguments in the proof ofLemma 4.3we getut^l ∈H₀²e^−γt, Q∞. This means that4.26holds forj l.

Assume that4.26holds forj l, l−1, . . . , s 1. By puttingvut^s ∈H₀^{l−s 1}e^−γt, Q∞ by inductive hypothesisand differentiating2.1s-times with respect tot, we have

Lv−i vt ft^s

^s

p1

C^psLt^put^s−p, 4.27

whereLt^{p n}_j,k1∂/∂xjajk_tp∂/∂xk a_tp.Following the assumptions of the induction ofsand the hypotheses of the functionfone hasvt∈H₀^{l−s 1}e^−γt, Q∞, ft^s ∈H₀^l−se^−γt, Q∞. It follows Fs −ivt ft^{s s}_p1C^psLt^put^s−p ∈ H₀^l−se^−γt, Q_∞. In another way since

H₀^l−se^−γt, Q∞ ⊆ H₋₁^l−s−1,0e^−γt, Q∞, so we have Fs ∈ H₋₁^l−s−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₋₁^{l 1−s}Ω. This implies v ∈ H₋₁^{l 1−s,0}e^−γt, Q∞. Note thatFs ∈H₀^l−s,0e^−γt, Q_∞then by applying8, Theorem 7.3.2one getsv∈H₀^{l 2−s,0}e^−γt, Q_∞ satisfying

v²_Hl 2−s,0

0 e^−γt,Q∞C

Fs²_Hl−s,0

0 e^−γt,Q∞ v²_Hl 1−s,0

−1 e^−γt,Q∞

, 4.28

whereCis a positive constant. In another way, it is easy to see that ut^s2_Hl 2−s

0 e^−γt,Q_∞ut^{s 12}_Hl 2−s−1

0 e^−γt,Q_∞ ut^s2_Hl 2−s,0

0 e^−γt,Q_∞. 4.29

Hence from the inductive assumptions we receive

ut^s2_Hl 2−s

0 e^−γt,Q_∞C 3 k0

ft^k2

L^∞0,∞,H₀^lΩ, 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. Denote u0 ϕ0u, where ϕ0 ∈ C^o∞U0 andϕ0 ≡ 1 in a neighborhood of coordinate origin. The functionu0satisfies

iLu0−u0_tϕ0f 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 ∈ H₀^{l 2}Ωfor almost all t ∈ 0, ∞.By applying Lemma 2.2we receiveu1∈H₀^{l 2}e^−γt, Q_∞and

u1²_Hl 2

0 e^−γt,Q∞C 3 k0

ft^k2

L^∞0,∞,H^l₀Ω, 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 ∈ H₀^{l 1}e^−γt, Q_∞.By using analogous arguments in the proof ofLemma 4.4, with note thatft^s∈ H₀^l−se^−γt, Q_∞ from the hypothesis of induction, we can prove thatu0 ∈H₀^{l 2}e^−γt, Q_∞. So

u∈H₀^{l 2}e^−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, ωinRⁿcan 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−nλ

v0, 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 spaceH^1,0e^−γt, Q∞ γ >5γ0of the Cauchy- Neumann problem5.1–5.3andft^k ∈ L^∞0,∞, L2Ωifk 3,fx,0 ftx,0 0. Then u∈H₀²e^−γt, Q_∞.

Proof. Notekbe nonnegative eigenvalues of the Neumann problem for equation

Δωv kv0, ω∈G. 5.6

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∈H₀²e^−γt, Q_∞. The theorem is proved.

6. Conclusions

The Schr ¨odinger 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 ¨odinger 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 ¨odinger 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 ¨odinger 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 in St. 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.

8 V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, vol. 52 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1997.

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

10 J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. 1-2, Springer, New York, NY, USA, 1972.