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The geometry of an eigenvalue problem
Appl. Math. Lett. Vol. 11, No. 1, pp. 111-119, 1998
Pergamon
Copyright(~1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/98 $19.00 + 0.00
PII: $0893-9659(97)00143-2
T h e G e o m e t r y of an E i g e n v a l u e P r o b l e m M. J. KAISER Department of Industrial and Manufacturing Engineering Wichita State University, Wichita, KS 67260-0035, U.S.A. kaiser©ie, twsu. edu
(Received January 1996; revised and accepted November 1996) A b s t r a c t - - T h e geometry of an eigenvalue problem associated with the classical Dirichlet problem is illustrated using constructive geometric techniques. Based on a result of Guggenheimer, a geometric optimization problem defined over a convex domain is formulated and solved, and provides a measure of deviation of the domain from circular shape. An extension of the formulation is suggested and illustrated through example. K e y w o r d s - - C o n s t r u c t i v e geometry, Dirichlet problem, Geometric analysis, Partial differential equations. 1. I N T R O D U C T I O N A plane membrane of uniform density in the shape of a convex domain K can only vibrate at certain eigenfrequencies. Selecting all material constants equal to unity, the possible frequencies A of vibration are those for which
A(u) = f f ( u ~ , +u~2)dxldx2 f f u 2 dxl dx2
u(Og) = 0 ,
is stationary for the function u compared to other functions that vanish on the boundary of K , OK. This expression can be solved using variational calculus and leads to various bounds and inequalities related to characteristic values associated with K [1, Chapters 5, 6]. A particularly interesting eigenvalue problem examined in this note is related to the classical Dirichlet problem and based on the inclusion of a geometric functional, the support function H ( K , u), associated with K:
g(K,u) = gK(u) = sup{(u,z) I x e K}, where (u, x) is the inner product of u and x, and u E S 1, the unit circle. The support function HK of K t h a t contains the origin O is a nonnegative real-valued function in R '~ with the properties (i) HK(Ax) = AHK(x), A > O, (ii) HK(X + y) <_ g g ( x ) + HK(y), and conversely, a nonnegative real-valued function in R '~ satisfying (i) and (ii) is the support function of a unique convex figure t h a t contains the origin, i.e., the support function completely defines K . The following result is due to Guggenheimer [2, Proposition 3]. THEOREM. I f the first eigenfunction of
Au + Ap(x)u = O, u = O,
in K, on OK, Typeset by .AA4S-TEX
Iii
M. J. KAISER
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Figure 1. The support function H(P6, u) for P6 -- {(-7, -8), (-4, 3), (5, 7), (5.5, 3), (5, 1), (3, -2)}.
p(x) > O, is concave, then A 1 / / p ( x ) dXl dx2 > 2zr + ~r(K), where A1 = min A(u) is the lowest principal frequency, and
(
7r(K) = n ~ n / \ H(K, u) ] du, where r(K) is the deviation of K from circular shape.
(1)
Eigenvalue Problem
113
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Figure 2. The derivative of the support function
H(P6, u) for P6.
The minimum exists since the integral is nonnegative, and the integrand tends to cx~ as O approaches O K and is uniformly continuous if O E K °. By using basic properties of the support function, it is not difficult to show that (1) attains its minimum at a unique point within K . See also [3, p. 194]. The value of 7r(K) serves as a measure of deviation of K from circularity.
2. C O N S T R U C T I V E
SOLUTION
METHODOLOGY
The task of this note is to construct the solution to the minimization problem defined by (1) using geometric procedures. A convex polygon P m = { A 1 , . . . , A m } , where Ai = (x~,y~), i = 1 , . . . , m, will be used for the representation of K. The procedure we adopt to solve problem (1) is as follows, for a given point Q E p o .
114
M.J. KAISER
X= -4.75888 Y=-3,33333
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0.07956
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Figure 3. The functional H'(P6,u)/H(P6,u) for P6. Step 1. Construct H(Pm,u). Step 2. Using H(Pm, u), compute H'(Pm, u), H'(Pm, u)/H(Prn, u), (H'(Pm, u)/g(Pm, u)) 2. Step 3. Using (g'(Pm, u)/H(Pm, u)) 2, calculate f (H'(Pm, u)/H(Pm, u)) 2 du. The support function H(Pm, u) is constructed directly from the input data and is completely determined by the extreme (vertex) points of Pro. As Q varies throughout p o , the support functional will change in the following manner [4, p. 27]. If the convex body K with support function H(K, u) is translated by the vector e, then the support function of the displaced body K ~ is H(KI,u) = H(K, u) + (e, u). The geometric form of the functional is obtained by computing f(H'(Pm, u)/H(Pm, u)) 2 du for points (defined, say, over a grid) throughout the domain of
Eigenvalue Problem
115
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Figure 4. The functional (H'(P6,
°
o
u)/H(P6, u)) 2 for P6.
definition Pro. Optimization by exhaustive search is a computationally intensive method (normally a method of last resort), and a more sophisticated approach such as gradient search would decrease the amount of computation required. A gradient search procedure would also yield a more accurate solution. Nonetheless, in the present discussion considering the geometric nature of the computations involved and our desire to illustrate the functional form of (1), an enumera-
116
M . J . KAISER
Figure 5. The graph of the functional
f(H'(P6, u)/H(P6, u)) 2 du over
P6.
0.00212..
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Figure 6. The functional (H/(P6, 0.6333).
u)/H(P6, u)) 2 at
the minimum point Q* = (0.0167,
Eigenvalue Problem
117
X= - 4 . 7 5 0 L ~ ) Y = - 4 . 3 3 3 3 3
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Figure 7. The radial function r(P6,u) for P6 = {(-7, -8), (-4, 3), (5, 7), (5.5,3), (~, 1), (3, -2)}. tive (naive) approach works quite satisfactorily with reasonable time requirements. An example illustrates the procedure. EXAMPLE. P6 ----{(-7, - 8 ) , ( - 4 , 3), (5, 7), (5.5, 3), (5, 1), (3, -2)}.
118
M. 3. KAISER
Figure 8. The graph of the functional f(r,(p6, u)/r(p6, u)) 2 du over P6.
o. oo~za.
o.oo1~. 0.00137. 4"
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functional (rJ(P6, U)/r(p6, u)) 2 at the minimum point Q*
Eigenvalue Problem
119
The values of the functional H(P6, u), H'(P6, u), H'(P6, u)/H(P6, u), (H'(P6, u)/H(P6, u)) 2 are illustrated in Figures 1-4 for variable points within P6. In Figure 5, the integral f(H'(P6, u)/ H(P6, u)) 2 du is computed at 3800 points within P6, and in Figure 6, the graph of f(H'(P6, u)/ H(P6, u)) 2 du is shown at the minimum point Q* = (0.0167, 0.6333). Total computation time on a 486 PC to graph the functional (Figure 5) is on the order of a few seconds.
3. A R E L A T E D
GEOMETRIC
OPTIMIZATION
PROBLEM
Motivated by Guggenheimer's formulation of the Dirichlet problem, it is natural to consider the analogous optimization problem
a(K) = n~n
r(g, u)
an
(2)
for the radial function r(K, u) defined by
r(K,u) = rg(u) ----max{A > 0 I Ax E K}. The quantity a ( K ) also has the property that a(K) =- 0 only for the circular disk, although much less is known about the structure and properties of this problem. The radial functional r(K, u) is what we normally think of as tracing out the boundary of K. The radial functional r(K, u) depends on the angle u E S 1 made by a ray emanating from Q c K ° with the x-axis. The quantity r(K, u) is equal to the length of the line segment from Q to the contour point in which the ray intersects the boundary OK. The radial functional completely characterizes K , and is closely related to the geometry of K. The radial functional does not exhibit properties similar to the support function, however, and thus the proof of the uniqueness of the solution point to problem (2) appears to be open. This is due in large part to the fact that no simple relation exists between radial functions defined at different reference points within K. Using K = P6 from the previous example, we can numerically solve (2) by applying the constructive techniques described in Steps 1-3 previously outlined (with H(K, u) replaced by
r(K,u)). EXAMPLE. P6 = ( ( - 7 , - 8 ) , ( - 4 , 3), (5, 7), (5.5, 3), (5, 1), (3, -2)}. In Figure 7, the form of the radial function r(P6, u) is illustrated. In Figure 8, the functional f(r'(P6, u)/r(P6, u)) 2 du is computed over the domain P6, and in Figure 9, the graph of J(r'(P6, u)/r(P6, u)) 2 du is shown at the minimum point Q* -- (0.317, 0.933). Computation time to graph the functional and obtain the solution point for this example is again on the order of a few seconds.
REFERENCES 1. G. P61ya and G. Szeg6, Isoperimetric inequalities in mathematical physics, Ann. of Math. Studies, No. 27, Princeton University Press, Princeton, NJ, (1951). 2. H.M. Guggenheimer, Concave solutions of a Dirichlet problem, Proc. Amer. Math. Soe. 40 (2), 501-506 (1973). 3. H.M. Guggenheimer, Applicable Geometry, Global and Local Convexity, R.E. Krieger, Huntington, NY, (1977). 4. T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS Associates, Moscow, ID, (1987).
Pergamon
Copyright(~1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/98 $19.00 + 0.00
PII: $0893-9659(97)00143-2
T h e G e o m e t r y of an E i g e n v a l u e P r o b l e m M. J. KAISER Department of Industrial and Manufacturing Engineering Wichita State University, Wichita, KS 67260-0035, U.S.A. kaiser©ie, twsu. edu
(Received January 1996; revised and accepted November 1996) A b s t r a c t - - T h e geometry of an eigenvalue problem associated with the classical Dirichlet problem is illustrated using constructive geometric techniques. Based on a result of Guggenheimer, a geometric optimization problem defined over a convex domain is formulated and solved, and provides a measure of deviation of the domain from circular shape. An extension of the formulation is suggested and illustrated through example. K e y w o r d s - - C o n s t r u c t i v e geometry, Dirichlet problem, Geometric analysis, Partial differential equations. 1. I N T R O D U C T I O N A plane membrane of uniform density in the shape of a convex domain K can only vibrate at certain eigenfrequencies. Selecting all material constants equal to unity, the possible frequencies A of vibration are those for which
A(u) = f f ( u ~ , +u~2)dxldx2 f f u 2 dxl dx2
u(Og) = 0 ,
is stationary for the function u compared to other functions that vanish on the boundary of K , OK. This expression can be solved using variational calculus and leads to various bounds and inequalities related to characteristic values associated with K [1, Chapters 5, 6]. A particularly interesting eigenvalue problem examined in this note is related to the classical Dirichlet problem and based on the inclusion of a geometric functional, the support function H ( K , u), associated with K:
g(K,u) = gK(u) = sup{(u,z) I x e K}, where (u, x) is the inner product of u and x, and u E S 1, the unit circle. The support function HK of K t h a t contains the origin O is a nonnegative real-valued function in R '~ with the properties (i) HK(Ax) = AHK(x), A > O, (ii) HK(X + y) <_ g g ( x ) + HK(y), and conversely, a nonnegative real-valued function in R '~ satisfying (i) and (ii) is the support function of a unique convex figure t h a t contains the origin, i.e., the support function completely defines K . The following result is due to Guggenheimer [2, Proposition 3]. THEOREM. I f the first eigenfunction of
Au + Ap(x)u = O, u = O,
in K, on OK, Typeset by .AA4S-TEX
Iii
M. J. KAISER
112
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Figure 1. The support function H(P6, u) for P6 -- {(-7, -8), (-4, 3), (5, 7), (5.5, 3), (5, 1), (3, -2)}.
p(x) > O, is concave, then A 1 / / p ( x ) dXl dx2 > 2zr + ~r(K), where A1 = min A(u) is the lowest principal frequency, and
(
7r(K) = n ~ n / \ H(K, u) ] du, where r(K) is the deviation of K from circular shape.
(1)
Eigenvalue Problem
113
X= 8.Z58~ ¥"-'0.666673
0.14~$
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lO -
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0.07~.
.
0, O000O
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\ 0.06773.
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Figure 2. The derivative of the support function
H(P6, u) for P6.
The minimum exists since the integral is nonnegative, and the integrand tends to cx~ as O approaches O K and is uniformly continuous if O E K °. By using basic properties of the support function, it is not difficult to show that (1) attains its minimum at a unique point within K . See also [3, p. 194]. The value of 7r(K) serves as a measure of deviation of K from circularity.
2. C O N S T R U C T I V E
SOLUTION
METHODOLOGY
The task of this note is to construct the solution to the minimization problem defined by (1) using geometric procedures. A convex polygon P m = { A 1 , . . . , A m } , where Ai = (x~,y~), i = 1 , . . . , m, will be used for the representation of K. The procedure we adopt to solve problem (1) is as follows, for a given point Q E p o .
114
M.J. KAISER
X= -4.75888 Y=-3,33333
\
0.07956
",.,
0. 03978.
\
--0. 0 0 0 0 0
-0.03978.
-0.
07956.
X= E I . 2 5 g l ~
\
0.03425J
0.017
Y=0.66~,6?33
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,
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Figure 3. The functional H'(P6,u)/H(P6,u) for P6. Step 1. Construct H(Pm,u). Step 2. Using H(Pm, u), compute H'(Pm, u), H'(Pm, u)/H(Prn, u), (H'(Pm, u)/g(Pm, u)) 2. Step 3. Using (g'(Pm, u)/H(Pm, u)) 2, calculate f (H'(Pm, u)/H(Pm, u)) 2 du. The support function H(Pm, u) is constructed directly from the input data and is completely determined by the extreme (vertex) points of Pro. As Q varies throughout p o , the support functional will change in the following manner [4, p. 27]. If the convex body K with support function H(K, u) is translated by the vector e, then the support function of the displaced body K ~ is H(KI,u) = H(K, u) + (e, u). The geometric form of the functional is obtained by computing f(H'(Pm, u)/H(Pm, u)) 2 du for points (defined, say, over a grid) throughout the domain of
Eigenvalue Problem
115
9.2~.~88 't=B.55567
0.0023~.
0.00186
f
0,00139,
I
O .00093.
O. O[I~. j O. O~D~
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•
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X= -~. 7 5 ~
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'lHI
Y=-S. =3.3333
0 . 0 JL6:~O.
0.0.11.3~-0.
0.00990.
l
0.00660.
I
i 0,00330.
O.O00OO
°
°
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=
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Figure 4. The functional (H'(P6,
°
o
u)/H(P6, u)) 2 for P6.
definition Pro. Optimization by exhaustive search is a computationally intensive method (normally a method of last resort), and a more sophisticated approach such as gradient search would decrease the amount of computation required. A gradient search procedure would also yield a more accurate solution. Nonetheless, in the present discussion considering the geometric nature of the computations involved and our desire to illustrate the functional form of (1), an enumera-
116
M . J . KAISER
Figure 5. The graph of the functional
f(H'(P6, u)/H(P6, u)) 2 du over
P6.
0.00212..
i
0 • 00169.
0. O0.1~v'i ' •
'
l
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Figure 6. The functional (H/(P6, 0.6333).
u)/H(P6, u)) 2 at
the minimum point Q* = (0.0167,
Eigenvalue Problem
117
X= - 4 . 7 5 0 L ~ ) Y = - 4 . 3 3 3 3 3
14.80.
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11.90.
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Figure 7. The radial function r(P6,u) for P6 = {(-7, -8), (-4, 3), (5, 7), (5.5,3), (~, 1), (3, -2)}. tive (naive) approach works quite satisfactorily with reasonable time requirements. An example illustrates the procedure. EXAMPLE. P6 ----{(-7, - 8 ) , ( - 4 , 3), (5, 7), (5.5, 3), (5, 1), (3, -2)}.
118
M. 3. KAISER
Figure 8. The graph of the functional f(r,(p6, u)/r(p6, u)) 2 du over P6.
o. oo~za.
o.oo1~. 0.00137. 4"
o.
,1:/
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O.O004G.
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functional (rJ(P6, U)/r(p6, u)) 2 at the minimum point Q*
Eigenvalue Problem
119
The values of the functional H(P6, u), H'(P6, u), H'(P6, u)/H(P6, u), (H'(P6, u)/H(P6, u)) 2 are illustrated in Figures 1-4 for variable points within P6. In Figure 5, the integral f(H'(P6, u)/ H(P6, u)) 2 du is computed at 3800 points within P6, and in Figure 6, the graph of f(H'(P6, u)/ H(P6, u)) 2 du is shown at the minimum point Q* = (0.0167, 0.6333). Total computation time on a 486 PC to graph the functional (Figure 5) is on the order of a few seconds.
3. A R E L A T E D
GEOMETRIC
OPTIMIZATION
PROBLEM
Motivated by Guggenheimer's formulation of the Dirichlet problem, it is natural to consider the analogous optimization problem
a(K) = n~n
r(g, u)
an
(2)
for the radial function r(K, u) defined by
r(K,u) = rg(u) ----max{A > 0 I Ax E K}. The quantity a ( K ) also has the property that a(K) =- 0 only for the circular disk, although much less is known about the structure and properties of this problem. The radial functional r(K, u) is what we normally think of as tracing out the boundary of K. The radial functional r(K, u) depends on the angle u E S 1 made by a ray emanating from Q c K ° with the x-axis. The quantity r(K, u) is equal to the length of the line segment from Q to the contour point in which the ray intersects the boundary OK. The radial functional completely characterizes K , and is closely related to the geometry of K. The radial functional does not exhibit properties similar to the support function, however, and thus the proof of the uniqueness of the solution point to problem (2) appears to be open. This is due in large part to the fact that no simple relation exists between radial functions defined at different reference points within K. Using K = P6 from the previous example, we can numerically solve (2) by applying the constructive techniques described in Steps 1-3 previously outlined (with H(K, u) replaced by
r(K,u)). EXAMPLE. P6 = ( ( - 7 , - 8 ) , ( - 4 , 3), (5, 7), (5.5, 3), (5, 1), (3, -2)}. In Figure 7, the form of the radial function r(P6, u) is illustrated. In Figure 8, the functional f(r'(P6, u)/r(P6, u)) 2 du is computed over the domain P6, and in Figure 9, the graph of J(r'(P6, u)/r(P6, u)) 2 du is shown at the minimum point Q* -- (0.317, 0.933). Computation time to graph the functional and obtain the solution point for this example is again on the order of a few seconds.
REFERENCES 1. G. P61ya and G. Szeg6, Isoperimetric inequalities in mathematical physics, Ann. of Math. Studies, No. 27, Princeton University Press, Princeton, NJ, (1951). 2. H.M. Guggenheimer, Concave solutions of a Dirichlet problem, Proc. Amer. Math. Soe. 40 (2), 501-506 (1973). 3. H.M. Guggenheimer, Applicable Geometry, Global and Local Convexity, R.E. Krieger, Huntington, NY, (1977). 4. T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS Associates, Moscow, ID, (1987).