International Conference «Inverse and Ill-Posed Problems of Mathematical Physics»,
dedicated to Professor M.M. Lavrent’ev in occasion of his 75-th birthday, August 20-25, 2007, Novosibirsk,
Russia
ON ONE OPTIMIZATION PROBLEM FOR THE EIGENVALUES
Yusif S. Gasimov
Institute of Applied Mathematics Baku State University
Z. Khalilov 23, AZ1148, Baku, Azerbaijan
e-mail: [email protected]
We investigate one shape optimization problem involving the first eigenvalue of the Laplace
operator with Dirichlet condition. Taking the first eigenvalue of this problem as a functional of the
domain and using obtained in [3, 4] formulas for the first eigenvalue and its variation we obtain formula
for the first variation in the optimal domain. Some particular cases also are considered.
Consider the problem
 u  u, x  D ,
(1)
u ( x)  0 , x  S D ,
(2)
where D  R n is bounded convex domain with smooth boundary S D  C 2 . The set of such domains
denote by K .
In a wide class of works [1] the existence of the optimal domain is investigated for the various
functionals of the eigenvalues of the problem (1), (2).
Here we consider the functional
J ( D)  1 ( D)   f ( x)dx  min ,
(3)
D
where 1 ( D) is the first eigenvalue of the problem (1), (2) corresponding to D  R n , f(x)- given
continuously differentiable in R n function.
We suppose that the problem (1)-(3) has a solution on K .
Theorem. If the domain D  K is a solution of the problem (1)-(3), then
1 ( D) 
1
f ( x) PD (n( x)) ds .
2 S
(4)
D
Here PD ( x)  sup (l , x), x  R is a support function of the domain D  R n .
n
lD
Proof. It is known that for fixed D the first eigenvalue of the problem (1), (2) is calculated by
formula [1]
(5)
1 ( D)  inf I (u , D),
u
where
I (u, D) 
 u
2
( x)dx
,
D
u
2
( x)dx
D
 u 
u ( x)   

i 1 x 
i
2
n
2
and inf is taken over all functions, u  С 2 ( D)  С 1 ( D ), being equal to zero at
As we see, formula (5) defines
1 as a functional of D .
SD .
In the works [2, 3] the differentiability of the functional 1 ( D) with respect to D on K is
proved and the expressions are obtained for its first variation under different boundary conditions. It is
known that the first eigenvalue of the problem (1), (2) is simple. In this case for this problem the
expression is true
2
(6)
 1 ( D)    u1 ( x) PD (n ( x))  PD (n ( x))ds,
SD
where D , D  K , n(x)  outward normal to
As one may obtain from (6)
SD
at the point
x.
1 (t  t )  1 (t )  1 ( D(t  t ))  1 ( D(t )) 

 u ( x) P
2
D ( t  t )
1

(n( x))  PD (t ) (n( x)) ds  o(t ) .
S (t )
From this dividing by t we get
 u1 ( x) PD (t ) (n ( x)) ds ,
1' (t )  
2
(7)
S D(t )
where
PD ( t ) ( x) 
d
PD ( t ) ( x).
dt
Now let’s show that for the first eigenvalue of the problem (1), (2) in the domain D is true the formula
1
2
u1 ( x) PD (n( x)) ds .

2 SD
1 ( D) 
(8)
D0  K , D(t )  tD0 , t  0. The first eigenfunction of the problem
(1), (2)
corresponding to the domain D0 denote by u1 ( x ). Then it is clear that ~
 x  , x  D(t ),
u1 ( x)  u1  
Take
D  D(t ). Really, considering
1
 x
u~1 ( x)  2  u1  
t
t
t
would be an eigenfunction of the problem (1),(2) by
we see that
u~1 ( x ) is an eigenfunction of
(1), (2) corresponding for the eigenvalue
1 (t ) 
1 ( D0 ) .
t2
Considering this in (7) we have
2
Taking
1 ( D0 )
t  1 we obtain (8).
t3
1
 2
t
2

SD
 x
u1   PD0 (n ( x))ds .
t
The functional
F ( D)   f ( x)dx  min
D
from (3) is also differentiable under the given condition and for its first variation the formula
 F ( D) 
 f ( x)P
D
(n ( x))  PD (n ( x)) ds
SD
is true [4].
Now, let D  K be a solution of the problem (1)-(3). Then according to the optimality condition
 u1 ( x)  f ( x)  0 , x  S D .
2
Multiplying (9) by PD (n( x)) and integrating over

SD
one may get
1
1
2
u1 ( x) PD (n( x)) ds   f ( x) PD (n( x)) ds  0 .

2 SD
2 SD
(9)
Considering here (8) we get (4).
Theorem is proved.
Consider some particular cases. Suppose that f ( x)  1, x  R n . In this case the functional (3)
takes a form
(10)
J ( D)  1 ( D)  mesD  min .
From (4) we obtain
1 ( D) 
1
PD (n( x)) ds .
2 S
D
In two dimensional case ([3])
1
PD (n( x)) ds  mesD .
2 S
D
Thus
1 ( D)  mesD .
In one dimensional case the problem (1)-(3) has the form
u ''   u , x  [ a , b ]
u (a )  u (b)  0,
b
1 (a, b)   f ( x)dx  min .
a
In this case finding of domain is equivalent to the finding of a and b . For this problem
1
2
1 (a, b)  [ f (b)b  f (a)a] .
For considered problem the eigenvalue may be determined, so it is not difficult to check the obtained
formulae ([1]).
Note, that when f ( x)  0 the problem (1)-(3) has no solution. In this case the “increasing” of
domain leads to decreasing of the eigenvalue. From (8) also follows the condition 1  0 .
Other particular cases also may be considered.
Note that using the obtained formula (6) for the first variation of the eigenvalue of the problem
(1), (2) with respect to domain and corresponding optimality condition the numerical algorithm is
proposed for the finding of the optimal shape. This algorithm is similar to conditional gradient method.
The numerical experiments have been carried out for some functionals of the first eigenvalue of the
problem (1), (2), in two dimensional case.
References :
1. G. Buttazo, G. Dal Mazo. An existence rezult for a class of shape optimization problem. Arch.
Rational Mech. Anal., 122, ( 1993), pp.183-195.
2. Vladimirov V. S. Equations of mathematical physics. M.: Nauka 1988, 512 p.
3. Gasimov Y. S., Niftiev A.A. On a minimization of the eigenvalues of Shrodinger operator over
domains.Doclady RAS, 2001, v.380, №3, p.305-307.
4. A.A.Niftiyev, Y.S.Gasimov. Control by boundaries and eigenvalue problems with variable
domains. Publishing House of Baku State University, 2004,185 p.
5. Vasilyev F. P.Numerical methods of solution of the exstremal problems. M.: Nauka, 1980, 518
p.