Journal of Mathematical Sciences, Vol. 175, No. 6, August, 2011
AN OPTIMIZATION PROBLEM FOR THE BIHARMONIC EQUATION WITH SOBOLEV CONDITIONS G. Buttazzo Dipartimento di Matematica, Universit` a di Pisa 5, Largo B. Pontecorvo, Pisa 56127, Italy
[email protected]
S. A. Nazarov∗ Institute of Problems in Mechanical Engineering RAS 61, Bolshoi pr. V.O., St. Petersburg 199178, Russia
[email protected],
[email protected]
UDC 517.9 Dedicated to Vasilii Vasil’evich Zhikov
We state and solve an optimization problem about distribution of several supporting points under a Kirchhoff plate clamped along the boundary: the biharmonic equation is supplied with the Dirichlet boundary conditions and pointwise Sobolev conditions. Some open questions are formulated. Bibliography: 23 titles.
1. Equation of plate deflection. Let Ω ∈ R2 be a bounded planar domain with smooth (of class C ∞ for the sake of simplicity) one-dimensional boundary ∂Ω. It is known (cf., for example, [1]) that the biharmonic equation Δ2 w(x) = f (x),
x ∈ Ω,
(1)
governs, with an appropriate accuracy, the deflection w(x) of a thin isotropic cylindrical plate under the transverse load D −1 f (x), where f (x) is the right-hand side of Equation (1) and D is the reduced cylindrical rigidity (it will not be required below, but we note that its expression can be found in Subject Index of the book [1]). The differential equation (1) bears the name of Marie-Sophie Germain since she was the first who suggested [2] a model of plate transverse deformation (deflection), but the complete theory of plates based on the so-called Kirchhoff hypothesis was constructed in [3]. A rigorous mathematical justification of the procedure for reducing dimension (a thin three-dimensional body is described by equations in a planar domain) has been discussed in a large number of publications (cf., in particular, the papers [4]–[8], the books [9]–[11], and the references therein). ∗
To whom the correspondence should be addressed.
Translated from Problems in Mathematical Analysis 58, June 2011, pp. 0–00. c 2011 Springer Science+Business Media, Inc. 1072-3374/11/1756-0000
1
The Dirichlet conditions w(x) = 0,
x ∈ Γ = ∂Ω,
∂n w(x) = 0,
(2)
mean that the plate boundary is rigidly clamped and, together with Equation (1), form an elliptic boundary value problem. Here, ∂n is the derivative along the outward normal. The variational statement of the problem (1), (2) includes the integral identity [12] o
v ∈ H 2 (Ω),
(Δw, Δv)Ω = (f, v)Ω ,
(3) o
where ( , )Ω is a natural product in the Lebesgue space L2 (Ω) and H 2 (Ω) is the subspace of functions in the Sobolev class H 2 (Ω) satisfying the conditions (2). o o For any linear continuous functional f ∈ H 2 (Ω)∗ on the space H 2 (Ω) the problem (3) has a o unique (weak) solution w ∈ H 2 (Ω) that satisfies the estimate o
w; H 2 (Ω) cf ; H 2 (Ω)∗ .
(4)
In the case f ∈ L2 (Ω), the weak solution becomes a classical one, belongs to the Sobolev space H 4 (Ω), and satisfies the inequality w; H 4 (Ω) cf ; L2 (Ω).
(5)
A complete list of results about lifting the smoothness of solutions to the boundary value problems for the biharmonic equation can be found, for example, in the book [13]. We emphasize o that the bilinear form on the left-hand side of (3) is coercive on H 2 (Ω) in view of the second main inequality [12]. 2. Pointwise Sobolev conditions. Let P = {P 1 , . . . , P J } be a collection of interior points of Ω. By the Sobolev embedding theorem H 2 ⊂ C 0,α on a plane, the linear set o
o
H 2 (Ω; P) = {w ∈ H 2 (Ω) : w(P 1 ) = . . . = w(P J ) = 0}
(6)
o
is a closed subspace of H 2 (Ω). Consequently, the variational problem (Δu, Δv)Ω = (f, v)Ω ,
o
v ∈ H 2 (Ω; P),
(7)
is also uniquely solvable, but, in this case, in the space (6). Conventionally, this problem is called the Sobolev problem and the connections u(P 1 ) = . . . = u(P J ) = 0,
(8)
included into the definition (6), are referred to as the (pointwise) Sobolev conditions on the (supporting) set P. The variational problem (3) (or the boundary value problem (1), (2)) will be referred to as the Dirichlet problem. The following assertion is a direct consequence of the Riesz theorem on representation of a linear functional in a Hilbert space. o
Lemma 1. The Sobolev problem (7) has a unique solution u ∈ H 2 (Ω; P) for any right-hand o side f ∈ H 2 (Ω; P)∗ , and this solution satisfies the estimate o
u; H 2 (Ω) cf ; H 2 (Ω; P)∗ . 2
o
o
o
o
Since H 2 (Ω; P) ⊂ H 2 (Ω), we have the inclusion H 2 (Ω)∗ ⊂ H 2 (Ω; P)∗ . Consequently, for o o f ∈ H 2 (Ω)∗ both problems, (3) and (7), have solutions of class H 2 (Ω) that satisfy the estimate (4), however, they can coincide only by chance. The estimate (5) for the solution u to the Sobolev problem fails. Indeed, the differential statement of the variational problem (7) is reduced to the equation (9) Δ2 u(x) = f (x), x ∈ Ω• = Ω \ P which is supplied by the pointwise conditions (8) and boundary conditions (2). Moreover, the differential equation (9) is stated exactly in the punctured domain Ω• . Thus, a solution to the Sobolev problem with the right-hand side f ∈ L2 (Ω) is not necessarily of class H 4 (Ω) because of possible singularities at interior points P 1 , . . . , P J of the boundary ∂Ω ∪ P. However, the 4 (Ω \ P) and, together with its first order derivatives, is H¨ older function u belongs to class Hloc continuous, i.e., the pointwise conditions are well imposed. Furthermore, the solution u is contained in the Kondrat’ev space V04 (Ω; P). We recall that the space Vβl (Ω; P) was introduced [14] as the completion of the lineal Cc∞ (Ω \ P) relative to the weight norm v; Vβl (Ω; P)
=
l
β−l+q
ρ
q
2
2
∇ v; L (Ω)
1/2 ,
(10)
q=0
where ∇ = grad, l ∈ {0, 1, 2, . . . } and β ∈ R are the smoothness and weight exponents, ∇qx v denotes all the qth order derivatives of a function v, and ρ(x) = min{|x − P j | : j = 1, . . ., J} is l (Ω) the distance from a point x to the set P. The space Vβl (Ω; P) consists of functions v ∈ Hloc with finite weight norm (10). Isolated points P j of the boundary Ω• can be regarded as vertices of complete angles R2 \P j . According to the general theory of elliptic boundary value problems in domains with conical (angular) points (cf., for example, the monographs [15, 16]), we have the following assertion providing an asymptotic representation of solutions as x → P j . o
Lemma 2. Under the condition f ∈ L2 (Ω), the solution u ∈ H 2 (Ω; P) to the problem (7) takes the form u(x) =
J j=1
+
χj (x) aij (xi − Pij ) i=1,2
a0j 1 iq |x − P j |2 ln |x − P j | + aj (xi − Pij )(xq − Pqj ) + u (x), 8π 2
(11)
i,q=1,2
where χj is a smooth cut-off function such that χj = 1 in a neighborhood V j ⊂ Ω of the point ∈ H 4 (Ω) belongs to the P j and χj (x) = 0 for x ∈ V k in the case k = j. The remainder u 4 Kondrat’ev space Vβ (Ω; P) for any β > 0 and, together with the coefficients aij , a0j , and aiq j , satisfies the inequality u; Vβ4 (Ω; P)
+
J j=1
|a0j |
iq i + |aj | cβ f ; L2 (Ω), |aj | + i=1,2
(12)
q=1,2
where the factor cβ is independent of the right-hand side f , but cβ → ∞ as β → +0. 3
Proof. The decomposition (11) and estimate (12) are guaranteed by the inclusion f ∈ = V00 (Ω; P) and a theorem about asymptotics [14] (cf., for example, the monographs [15, Theorem 4.2.1 and Chapter 2] and [17, Section 5.2]). All necessary computations and arguments are contained in the book [17, Section 5.5.2] and the paper [18]. L2 (Ω)
Lemma 2 presents a result about the smoothness of a solution u to the Sobolev problem and determines coefficients of its representations in neighborhoods of the points P j . In particular, the equalities ∂u j (P ) =: u,i (P j ) (13) aij = ∂xi are not do not follow from the Sobolev embedding theorem H 2 ⊂ C 0,α, which is somewhat unexpected fact. 3. Minimization and optimization problems. The Dirichlet problem (3) and the Sobolev problem (7) can be stated as the minimization problem for the functional E(u; f ) =
1 (Δu, Δu)Ω − (f, u)Ω 2
o
(14)
o
on the corresponding function space (H 2 (Ω) or H 2 (Ω; P)), with loading f in the dual spaces o o (H 2 (Ω)∗ or H 2 (Ω; P)∗ ). The solutions to these boundary value problems are minimizers [1] of the functional (14) which is interpreted as the potential energy of plate deformation (the elastic o o energy plus the work of external forces). Thus, by the obvious inclusion H 2 (Ω) ⊂ H 2 (Ω; P), the solutions w and u to the Dirichlet and Sobolev problems are initially connected by the relation E(w; f ) E(u; f ).
(15)
We first formulate a problem about an optimal distribution of pointwise Sobolev conditions in an integral form. We fix a natural number J and try to place J points P1 , . . . , PJ inside the domain Ω in such a way that the flexibility (compliance) functional of a plate clamped along the boundary ∂Ω and supported at the above-mentioned points, i.e., the integral (16) C(P; f ) = u(x)f (x) dx Ω
takes the minimal value. On the right-hand side of (16) there is a given transverse load f and a o solution u ∈ H 2 (Ω; P) to the Sobolev problem minimizing the potential energy functional (14). The integration by parts formula shows that
o (17) C(P; f ) = −2 min E(v; f ) : v ∈ H 2 (Ω; P) = (Δu, Δu)Ω 0. Thus, the optimization problem is formulated as
min C(P; f ) : P ⊂ Ω, #P J .
(18)
Here, #P is the number of distinct points in the set P. As was established in [18], if f ∈ L2 (Ω), then the functional (17) is a continuous function of 2J variables P 1 , . . . , P J ∈ Ω. Hence the proof of Theorem 3 about the existence of a solution to the problem (18) becomes obvious. However, hypothetically, the following situation can happen: because of the singularity of the 4
o
right-hand side f ∈ H 2 (Ω; P)∗ \ L2 (Ω), the potential energy (14) computed on the solution uP to the Sobolev problem (9), (2), (8) is unboundedly increasing if one of the points P j of the set P approaches to some other point P k ∈ P. At the same time, how to construct such an example is still an open question. Theorem 3. For f ∈ L2 (Ω) the optimization problem (18) has a solution for any J. Proposition 4. If f ∈ L2 (Ω) and P opt is an optimal solution, then #Popt = J. o
Proof. As was mentioned at the beginning of this section, the Sobolev space H 2 (Ω; P) shrinks under extending the set P, i.e., the corresponding potential energy increases, which was required to verify by formulas (17) and (14), (15). Further we focus only on the problem (18), although the problem
min W(P; f ) : P ⊂ Ω, #P J
(19)
with the maximal flexibility functional
W(P; f ) = max uP (x) : x ∈ Ω
(20)
seems more reasonable from the physical point of view. The problem (19) is discussed in [18], and the definition of the function (20) is well posed in view of the Sobolev embedding theorem H 2 ⊂ C 0,α . The optimization problem (18) is more attractive because of simple algebraic formulas for the solution. For example, if the Sobolev conditions are imposed at a single point P 1 (in other words, J = 1 and P = {P 1 }), the optimal choice of its location (cf. Remark 6 below) corresponds to the extremum points of the function Ω x → G(x, x)−1/2 w(x),
(21)
where G is the Green function for the problem (1), (2) described in Section 4 and w is the solution to the Dirichlet problem (1), (2). In the case of one or several supporting points, it is easy to indicate an algorithm for numerical searching an optimal location of one or several points for setting the Sobolev conditions. If J is too large (for example, the foundation of a heavy building is supported by frequently located piles), the computational costs for determining the optimal location of points in the set PJ become unimaginably huge since the functionals (17) and (20) are not convex. In this case, it is reasonable to study the asymptotic density defined by the formula 1 μ = lim δ(x − P j ). (22) J→+∞ J opt P j ∈PJ
Here, δ(x − P ) is the Dirack function supported at the point P ∈ Ω. Optimization problems similar to (22) were intensively studied for other differential equations (cf., for example, [19]– [21]). However, to the knowledge of the authors, the biharmonic equations was not considered from this point of view. 4. The Green function. The Green function G(x, y) is a solution to the problem Δ2 G(x, y) = δ(x − y),
x ∈ Ω,
G(x, y) = ∂n(x) G(x, y) = 0,
x ∈ Γ,
(23)
5
Since the fundamental solution to the biharmonic equation Φ(x) = (8π)−1 |x|2 ln |x|
(24)
2 (R2 ), the solution G to the problem (23) is simultaneously a belongs to the class H 2 (Ω) ⊂ Hloc o weak solution in the Sobolev space H 2 (Ω), i.e., it satisfies the integral identity
(ΔG, Δv)Ω = v(y),
o
v ∈ H 2 (Ω),
(25)
o
where H 2 (Ω) v → v(y) is a continuous functional in view of the Sobolev embedding theorem. Moreover, the regular part of the Green function G(x, y) = G(x, y) − Φ(x − y)
(26)
is infinitely differentiable in Ω x. Finally, it is symmetric, i.e., G(x, y) = G(y, x).
(27)
Fixing the second variable, we see that the function Ω \ P j x → Gj (x) = G(x, P j ) (called the Green function, as above) admits the decomposition 0 i + Gjk (xi − Pik ) Gj (x) = Φ(x − P j )δj,k + Gjk i=1,2
1 iq Gjk (xi − Pik )(xq − Pqk ) + O(|x − P k |3 ), + 2
x → P k,
(28)
i,q=1,2
0 , G i and G iq = G qi from the Taylor formula for the regular part (26). with the coefficients Gjk jk jk jk Taking Gk for a test function in the integral identity (25) for Gj , we see that the (J × J)matrix G 0 with entries 0 = Gj (P k ) = G(P k , P j ) = (ΔGj , ΔGk )Ω Gjk
(29)
is a Gram matrix, symmetric and positive definite, because it is constructed from linearly indeo pendent functions G1 , . . . , GJ by using the inner product (Δu, Δv)Ω in the space H 2 (Ω). Since Φ(0, 0) = 0 by the definition (24), formula (29) shows, in particular, that the diagonal G(x, x) of the Green function appearing in (21), is positive inside the domain Ω. Further, we need derivatives of the Green function with respect to the second variable: ∂G (i) (x, y) . (30) Gj (x) = − ∂yi y=P j Within the framework of the theory of distributions, the function (30) turns out to be a solution to the problem ∂δ (i) (x − P j ), x ∈ Ω, Δ2 Gj (x) = ∂xi (i)
(i)
Gj (x, y) = ∂n(x) Gj (x, y) = 0,
x ∈ Γ.
Furthermore, (i)
i + δj,k Gj (x) = −Gkj
6
iq ∂Φ (x − P k ) − Gjk (xq − Pqk ) + O(|x − P k |2 ), ∂xi q=1,2
x → P k,
(31)
iq where Gjk are coefficients and
xi Φ,i (x) = 4π
1 ln |x| + 2
.
i coincide with those appearing in the decomposition (28), which is confirmed The coefficients Gkj by the formula (i) (i) (i) Gj (P k ) = Gj (x)δ(x − P k ) dx = Gj (x)Δ2 Gk (x) dx Ω
Ω
(i)
Gk (x)Δ2 Gj (x) dx =
= Ω
Gk (x) Ω
∂δ ∂Gk j i (x − P j ) dx = − (P ) = −Gkj . ∂xi ∂xi
(32)
(i)
Since the derivative Gj does not belong to the space C 1 (Ω), a simple calculation (32) cannot help to clarify connections between other coefficients in the relations (28) and (31). Nevertheless, the same notation, used already in the above relations, can be justified by the method in [22] which is required more informal computations; this was done in a more general situation in [18]. 5. Connection between the solutions to the Dirichlet and Sobolev problems. We set u(x) = w(x) + GP (x)A, where A = (A1 , . . . , AJ ) is a column in RJ , is the transposition symbol, and (33) GP (x) = (G1 (x), . . . , GJ (x)); moreover, GP (P 1 ), . . . , GP (P J ) are rows of the matrix G 0 . Then the pointwise conditions (8) are expressed as the equation RJ 0 = uP = wP + G 0 A, where uP = (u(P 1 ), . . . , u(P J )) . Now, we find
and, consequently,
A = −(G 0 )−1 wP = −(G 0 )−1 (w(P 1 ), . . . , w(P J ))
(34)
u(x) = w(x) − GP (x)(G 0 )−1 wP .
(35)
By the estimate (5), the Sobolev embedding theorem H 4 ⊂ C 2,α , and the Taylor formula Taylor, we have w(x) =
J j=1
1 iq j j 0 i j χj (x) bj + bj (xi − Pi ) + bj (xi − Pi )(xq − Pq ) + w j (x); 2 i=1,2
i,q=1,2
moreover, w ∈ Vβ4 (Ω; P) with any weight exponent and b0j = w(P j ),
bij = w,i (P j ),
j biq j = w,iq (P ).
Furthermore, using the identity (3) with v = Gj and (25) with v = w, we find 1 1 1 E(u, f ) = − (f, u)Ω = − (f, w)Ω + (f, GP )Ω (G 0 )−1 wP 2 2 2 J 1 1 −1 = E(w; f ) + (wP ) (G 0 )−1 wP = E(w; f ) + w(P j )w(P k ). G j,k 2 2
(36)
j,k=1
7
Here, (G 0 )−1 j,k is an entry of the inverse of the matrix (G 0 )−1 , which is symmetric and positive definite. Consequently, the energy increment E(w; f ) − E(u; f ) caused by supporting the plate at the points P 1 , . . . , P J is negative. This fact was already mentioned in Section 3 on the basis of other arguments. It is clear that the quantity (36) continuously depends on the 2N coordinates of points P 1 , . . . , P J if P j = P k for j = k and P j ∈ ∂Ω (the matrix G 0 is not singular). One can check that both excluded cases (the coincidence of two points and displacement of points on the boundary) are a priori nonoptimal (we refer to [18] for details). 6. Necessary condition for optimal location of the pointwise Sobolev conditions. We perturb a point in the set P and put Pεj = P j + εe(i) , where j = 1, . . . , J, i = 1, 2 and e(i) is the unit vector of the x1 -axis. Let uε be a solution to the Sobolev problem with supporting points P ε = {P 1 , . . . , P j−1 , P jε , P j+1 , . . . , P J }. Denote w,i = ∂w/∂xi (cf. (13)). Assume that f ∈ L2 (Ω), which means that w ∈ H 4 (Ω) in view of the inequality (5). Since w ∈ C 2 (Ω) because of the Sobolev embedding theorem, we have w(P jε ) = w(P j ) + εw,i (P j ) + O(ε2 ). Consequently, ε
wP = wP + εe(j) w,i (P j ) + O(ε2 ),
(37)
where e(j) = (δj,1 , . . . , δj,J ) is the unit vector in the Euclidean space RJ . For k = 1, . . . , J and k = j the Green function G(x, y) satisfies the relations G(P jε , P k ) = G(P j , P k ) + ε
∂ G(P j , P k ) + O(ε2 | ln ε|), ∂xi
G(P k , P jε ) = G(P k , P j ) + ε
∂ G(P k , P j ) + O(ε2 | ln ε|), ∂yi
G(P jε , P jε ) = G(P j , P j ) + ε
∂ ∂ G(P j , P j ) + ε G(P j , P j ) + O(ε2 | ln ε|). ∂xi ∂yi
By (27), we have ∂G k j ∂G j k (P , P ) = (P , P ), ∂xi ∂yj i.e., the matrix G 0ε corresponding to the perturbed location of points in the set P ε takes the form P 2 ) + G e G 0 + ε e(j) (GP j,i j,i (j) + O(ε | ln ε|), where, by (28), 1 J i i GP j,i = (Gj,i (P ), . . . , Gj,i (P )) = (Gj1 , . . . , GjJ ) .
Expanding the matrix into the Neumann series, we find
8
G 0ε
−1
P 0 −1 = (G 0 )−1 − ε(G 0 )−1 e(j) (GP ) + G e + O(ε2 | ln ε|). j,i j,i (j) (G )
(38)
Now, using (36) and (37), (38), we find 1 ε ε E(uε ; f ) = E(w; f ) + (wP ) (G 0ε )−1 wP 2 1 1 = E(w; f ) + (wP ) (G 0 )−1 wP + ε (wP ) (G 0 )−1 e(j) w,i (P j ) 2 2
−1 P j 0 −1 P P 0 −1 P ) + G e w + O(ε2 | ln ε|) +(e(j) w,i (P )) (G ) w − (w ) (G ) e(j) (GP j,i j,i (j) G 0 −1 P ) (G ) w + O(ε2 | ln ε|). = E(u; f ) + ε((G 0 )−1 wP ) e(j) w,i (P j ) − (GP j,i Thus, the necessary condition for the optimal location of points P 1 , . . ., P J is written as 2J equations 0 −1 P ) (G ) w = 0, j = 1, . . . , J, i = 1, 2. (39) ((G 0 )−1 wP ) e(j) w,i (P j ) − (GP j,i Moreover, in the case of nondegenerate scalars ((G 0 )−1 wP ) e(j) ,
j = 1, . . . , J,
(40)
the representation (35) for the solution uP to the Sobolev problem (9), (8), (2) shows that the conditions (39) imply the equalities ∇x uP (P j ) = 0,
j = 1, . . . , J.
We emphasize that, by the symmetry and positive definiteness of the matrix (G 0 )−1 , all the scalars (40) may vanish only if wP = 0 ∈ RJ . Theorem 5. If f ∈ L2 (Ω) and P opt is a solution to the optimization problem (18), then the relations (39) hold, which are equivalent to the following: ((G 0 )−1 wP ) e(j) uP ,i
opt
(P j ) = 0,
j = 1, . . . , J, i = 1, 2.
(41)
Remark 6. If J = 1, i.e., the set P consists of a single point P , then 1 E(u; f ) = E(w; f ) + G(P, P )−1 |w(P )|2 2 and, consequently, P should be an extremum point for the function (21). This conclusion agrees with the condition (39) which, in the case P = {P }, takes the form ∂w ∂G −1 −1 (P ) − G(P, P ) w(P ) (P, P ) = 0, i = 1, 2. (42) G(P, P ) w(P ) ∂xi ∂xi Indeed, the left-hand side of (42) is equal to −3/2
G(P, P )
and formula (21) is unchallenged.
∂ −1/2 w(P ) w(x) , G(x, x) ∂xi x=P 9
7. Sufficient condition for local extremum. In spite of the fact that the second order derivatives of the Green function G have logarithmic singularities, the matrix G 0 defined by formula (33) smoothly depends on the coordinates of points P 1 , . . . , P J since this matrix is 0 = G(P k , P j ) = G(P k , P j ) of the regular part (26) which is infinitely composed of the values Gjk differentiable in the Cartesian product Ω × Ω. Furthermore, w ∈ C 2,α (Ω). Consequently, the function ΩJ P → M (P) = w(P) (G 0 )−1 w(P) and its derivative of order up to and including 2 are H¨older continuous. We perturb the set P and construct P ε from the points P jε = P j + εX j , where j = 1, . . ., J, X j ∈ R2 and ε > 0 is a small positive parameter. We have M (P ε ) = M (P) + ε
J 2 ∂M j=1 i=1
j j (P)Xi +
∂Xi
moreover, Miq jk (P) =
2 J ε2 j k 2+α Miq ); jk (P)Xi Xq + O(ε 2 j,k=1 i,q=1
∂2M ∂Xij ∂Xqk
(P).
(43)
Note that the derivative ∂M/∂Xij was computed in Section 6. Thus, we arrive at the following sufficient condition for local minimum of the flexibility (compliance) functional (16). Theorem 7. Assume that (39)–(41) hold for the set P = {P 1 , . . . , P J } and the matrix M with entries (43) is negative definite. Then the functional (16) attains a local minimum over the set P.
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