Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications

Nonlinear Analysis 42 (2000) 1161 – 1193

www.elsevier.nl/locate/na

Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications  David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Dedicated to Professor G. Strang, on the occasion of his 65th birthday

Keywords: Nonconvexity; Nonsmooth; Nonlinear dierential equations; Duality; Algebraic curves; Variational method; Nonconvex dynamical systems; Nonlinear functional analysis

1. Problems and motivations We are interested in the general analytic solution of the nonconvex variational problem: Z Z (1) (P) : P(u) = W ((u)) d x − fu d x → min∀u ∈ Ua ; I

I

where I ⊂ R is an open interval, f(x) is a given function,  is a nonlinear dierential operator, and W () ∈ L(I ) is a piecewise Gˆateaux dierentiable function of  = (u); Ua is a closed convex subspace of a re exive Banach space U. This general nonconvex, nonsmooth variational problem appears in many nonlinear systems. For example, in the nonlinear equilibrium problem of Ericksen’s bar subjected to axial extension [17], or the post-buckling analysis of extended nonlinear beam subjected to a compressed load [26], the nite strain  = (u) = 12 u;2x −  is a quadratic operator, E-mail address: [email protected] (D.Y. Gao) The main results of this paper were announced at the 45th SIAM Anniversary Meeting, Stanford, CA, July 12–19, 1997 and the 7th Conference on Nonlinear Vibrations, Stability and Dynamics of Structures, July 26 –30, 1998, Blacksburg, VA, USA. 

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 1 2 9 - 7

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and the stored energy W W ( 12 e2 − ) = 12 E( 12 e2 − )2 ; E ¿ 0

(2)

is a double-well function of e = u; x for any given parameter  ¿ 0. The corresponding Euler–Lagrange equation of the problem is then a nonlinear dierential equation [Eu; x ( 12 u;2x − )]; x + f = 0 ∀x ∈ I:

(3)

Even for this very simple problem, the analytic solutions for u(x) cannot be well determined by traditional analytic methods. If f = 0;  = 1=2, we should have u; x = {0; ±1} ∀x ∈ I: Hence, any zigzag function with slopes 0 and ±1 satis es this equation, but may not be a minimizer of the total potential energy P(u). Therefore, some numerical discretization approaches for this problem have been examined in [50]. The nonconvex variational problem with double-well structure was rst studied by Van der Waals in 1893 for a compressible uid whose free energy at constant temperature depends not only on the density, but also on the density gradient (see [51]). This problem also appears in hysteresis and phase transitions, super-conductivity, cosmology, mathematical economics, nonconvex dynamical systems, nonlinear bifurcation and post-buckling problems of large deformed structures (cf. e.g., [7,8,14,18,26,34,39]). The direct approaches and relaxation methods for solving nonlinear equilibrium equations have been discussed extensively for more than 30 years (see, for example, [5, 9 –11,13,15,19,21,38,40,41,43,46,47,55]). It is known that in nonlinear variational problems, traditional direct methods can provide only upper bound approaches to the solution. The so-called relaxation method can be used mainly for nding global minimizer of the nonconvex energy. However, in post-bifurcation analysis and phase transitions, local maximizers usually play more important roles. As was indicated in [40], the relaxation method for solving nonconvex variational problems with three or more phases (potential wells) is fundamentally more dicult. Duality theory for geometrically linear variational problem inf J (u; u), where  is a linear dierential operator (say, u = u; x ), has been well studied for both convex and nonconvex systems (see, for example, [1,3,4,12,16,27,31,44, 46 – 49,52,56– 61, 63–65]). The dual functional J ∗ (∗ e∗ ; e∗ ) in these classical principles is usually obtained by Fenchel–Legendre transformation, where e∗ is the dual variable of e = u. For nonconvex stored energy, to nd the Legendre dual function is very dicult, or even impossible. For example, if We =W ((e)) de ned by (2) is a double-well function of e, the dual variable e∗ of e de ned by
@We (e) = e 12 e2 −  e∗ = @e is nonlinearly dependent on the deformation rate e = u; x . The inverse form e(e∗ ) is very complicated, and therefore, the Legendre-conjugate function W c (e∗ ) de ned by Wec (e∗ ) = e∗ e(e∗ ) − We (e(e∗ )) does not have a simple algebraic expression (see [52]). The Fenchel sup-conjugate function, de ned by We∗ (e∗ ) = sup{e∗ e − We (e)} e

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is always convex, lower semicontinuous. The traditional Fenchel–Rockafellar dual problem for this one-dimensional nonconvex variational problem (P) can be simply given by (cf., e.g., [16]) Z (4) (P∗ ) : P ∗ (e∗ ) = F ∗ (∗ e∗ ) − We∗ (e∗ ) d x → max ∀e∗ ; I R where F ∗ is the Fenchel inf-conjugate of F(u) = I uf d x, ∗ is the adjoint operator of . Since We is not a convex function of e, there exists a duality gap between (P) and (P∗ ): inf P(u) ≥ sup P ∗ (e∗ ):

(5)

The inequality shows that the solution of the Fenchel–Rockafellar’s dual problem is not equivalent to the solution of the primal problem. Therefore, the classical, well-developed Fenchel–Rockafellar’s dual variational principles are mainly applicable to convex variational problems. In nonconvex dynamics, the geometric operator  = {(t) ; (s) } is a time-space differential operator. The stored energy function W is the so-called action density: W ((u)) = K((t) (u)) − U ((s) (u)):

(6)

In Newtonian systems, the kinetic energy K(v) is usually a quadratic function of the velocity v = u; t . However, the stored potential energy U ((s) (u)) is often a nonconvex function of u. For example, if K(v) = 12 v2 , and U (u) = 12 E( 12 u2 − )2 is a double-well potential, then on the given time interval I = (0; T ), the critical condition of P(u) leads to the well-known forced Dung equation: u; tt = Eu( − 12 u2 ) − f(t):

(7)

It is known that this system is very sensitive to the initial conditions. For some given parameter  ¿ 0, the direct methods may produce the so-called chaotic numerical “solutions”. Hence, the dual solution will be very important in understanding the behavior of this nonconvex dynamical system. Duality theory for general fully nonlinear, nonsmooth variational problems was originally studied by Gao and Strang in [35,36] for n-dimensional nite deformation systems, where the geometrical equation  = (u) is nonlinear and the constitutive relation  ∈ @W is nonsmooth. In order to recover the duality gap between the primal and dual variational problems, a so-called complementary gap function was discovered. They proved that the convexity of the primal problem depends directly on the sign of this gap function. A general complementary variational principle for fully nonlinear, nonsmooth systems was proposed. An open problem left by Hellinger–Reissner was partially solved for convex problems. Applications of this general theory have been given in a series publications on nite deformation mechanics (see Gao et al. [22–37]). Recently, the remained open problem has been solved for nonconvex systems, where in the gap function is usually negative. An interesting triality theory was proposed and a so-called nonlinear dual transformation method has been developed for solving general nonconvex variational=boundary value problems [26,29]. In the present paper, this nonlinear dual transformation method is generalized for obtaining the general analytic solutions of the nonconvex, nonsmooth variational problem

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(P). The main results for mixed boundary value problems with quadratic operator  are given in Section 2. The canonic dual transformation method and the complementary variational principle for quadratic operator  are discussed in Section 3. A triality theory for general nonlinear operator  and nonconvex stored energy W is presented in Section 4. The generalization and applications in nonsmooth bifurcation problem, phase transitions with multi-well energy and nonconvex Hamilton systems are discussed in the last two sections.

2. Analytic solutions for quadratic operators Let us rst consider a mixed boundary value problem such that the feasible space Ua can be simply de ned by Ua = {u ∈ W;1 (0; ‘) | u(0) = u0 };

(8)

where W;1 is a standard notation for Sobolev space with  ∈ (1; +∞). Let Lÿ (I ) be a Lebesgue integrable space with 1 ¡ ÿ ¡ ∞, I = (0; ‘), and let the geometrical mapping  : Ua → E ⊂ Lÿ (I ) be a quadratic operator (u) = 12 a(x)u;2x + b(x)u; x + c(x);

with a(x) ¿ 0 ∀x ∈ I:

(9)

The coecients a; b; c are real-valued functions on I . We suppose that there exists a Gˆateaux dierentiable function W () such that the constitutive relation  = DW () =

@W () @

(10) ∗

is an one-to-one mapping from E to its dual space E∗ ⊂ Lÿ (I ), where ÿ∗ ∈ (1; +∞) is the dual number of ÿ, i.e. 1=ÿ + 1=ÿ∗ = 1. Let g‘ be a given load at x = ‘, the nonconvex variational problems (1) for mixed boundary value problem can be written as Z Z (11) (P): P(u) = W ((u)) d x − fu d x − g‘ u(‘) → min ∀u ∈ Ua : I

I

∗

Let  be a dual number of  ∈ (1; +∞). The critical condition of P(u) on Ua leads to the following mixed boundary value problem (BVP for short) : Problem 1 (BVP). For the given boundary data u0 , g‘ and the external load ∗ f(x) ∈ L (I ); nd u ∈ Ua such that [(b + au; x )D W ((u))]; x + f(x) = 0 (b + au; x )D W ((u)) = g‘

at x = ‘:

∀x ∈ I;

(12) (13)

where D W ((u)) = @W=@((u)) is the Gˆateaux derivative of W with respect to  = (u).

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This is a fully nonlinear (i.e., both geometrically and constitutively nonlinear) problem. Since P : Ua → R is not convex, the problems (P) and (BVP) are not equivalent.The traditional direct methods for solving this problem are dicult. However, by the nonlinear dual transformation method, this problem can be solved completely. Let S ⊂ E∗ be the range of the constitutive mapping DW () ∀ ∈ E, and Sa ⊂ S a subset of S: Sa = { ∈ S | (x) 6= 0 ∀x ∈ I }:

(14)

Since  = DW () is one-to-one, invertible on E, using the Legendre transformation, the conjugate function of W can be easily obtained as W c () = () − W (()):

(15)

The main result can be proposed in the following theorem. ∗

Theorem 1. For any given boundary data u0 ; g‘ ; and the external load f ∈ L (I ); let Z ‘ f(t)dt + g‘ : (16) g(x) = x

Then for every solution of the dual Euler–Lagrange equation; 1 2 b2 g (x); with  = − c; 2a 2a there is a unique u(x) deÿned by   Z x g(t) 1 − b(t) dt + u0 ; u(x) = 0 a(t) (t) 2 [DW c () + ] =

(17)

(18)

and this u solves the mixed boundary value problem (BVP). Conversely; every solution u of the (BVP) can be written in the form (18) for some solution  of (17). Moreover; if W () is convex; for every positive solution (x) ¿ 0 ∀x ∈ I of (17); the solution u(x) deÿned by (18) is a global minimizer of P. If W () is strictly convex and all the algebraic solutions of the dual Euler–Lagrange equation (17) are positive on I; then the solution of the (BVP) is unique. Proof. We rst prove that the function u(x) de ned by (18) solves the (BVP). For every solution (x) of (17), the derivative of (18) leads to a(x)u; x + b(x) =

g(x) : (x)

(19)

Substituting this into (17) gives DW c () = 12 au;2x + bu; x + c = (u):

(20)

Since the constitutive relation  = DW () is one-to-one, invertible, this inverse constitutive equation is equivalent to the geometric-constitutive equation  = D W ((u)):

(21)

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Then we have [(a(x)u; x + b(x))]; x = [g(x)]; x = −f (a(x)u; x + b(x)) = g(‘) = g‘

∀x ∈ I;

at x = ‘:

(22)

This shows that for every solution  of (17), u(x) de ned by (18) is indeed a solution of the (BVP). Since DW c () is a monotone mapping, the solutions of the algebraic equation (17) are well de ned for the given g(x) ∈ C 1 (I ). Hence, for each solution  of (17), u is uniquely de ned by (18). Conversely, if u(x) is a solution of the (BVP), then u has to satisfy the geometricconstitutive equation (21) and the equilibrium conditions (22). For any given  ∈ Sa , solving the equilibrium equation in (22), we have au; x + b(x) = g(x)=(x):

(23)

Substituting this into the inverse constitutive equation (20) leads to the dual Euler– Lagrange equation (17). Since the constitutive relation =((u)) is one-to-one, one of the solution  of the algebraic equation (17) satis es the inverse constitutive equation (20). For this , solving (23) for u gives the formulation (18). Hence the solution u(x) of the (BVP) can be written in the form (18). The proof for the rest of the theorem will be given in the following sections. Remark 1. For Dirichlet boundary value problems, the integral constant g‘ in (16) can be determined by the boundary condition u(‘) = u‘ . For example, let us consider an elastic string with an original length ‘. Suppose that the string is xed at its two ends and is subjected to a distributed transverse load f(x). The feasible space Ua is Ua = {u ∈ W1;1 (0; ‘) | u(0) = 0; u(‘) = 0}: The Euler equation for the nonlinear variational problem Z Z q 2 k0 1 + u; x d x − fu d x → min ∀u ∈ Ua I

I

is a well-known nonlinear dierential equation in dierential geometry :   d  k0 u; x  q + f = 0 ∀x ∈ (0; ‘); dx 1 + u2 ;x

where k0 ¿ 0 is a√material constant. Let  = (u) = 1 + u;2x ¿ 0. The stored energy density W () = k0  is a strictly concave function. In nite deformation theory,  is the so-called right Cauchy–Green strain. Its dual variable k0  = DW () = √ ¿ 0 2  is the second Piola–Kirchho stress. The Legendre-conjugate of W is the so-called complementary energy density W c () = −k02 =(4), which is also a strictly concave function. Since a = 2; b = 0; c = 1 in this problem, the dual Euler–Lagrange equation

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q (17) has a unique solution (x) = 12 k02 − g2 (x): By Theorem 1, the analytic solution for this elastic string is then Z ‘ Z x F(t) + g‘ q dt; F(x) = f(x) d x: u(x) = 0 x k02 − (F(t) + g‘ )2 If f(x) = −1; ‘ = 1; F(x) = x − 1, the boundary condition u(1) = 0 gives g‘ = 1=2: The Einstein’s relativistic theory is an analogue of this static deformation problem : Z Z q −m0 c2 1 − (u; t =c)2 dt − fu dt → min; I

I

where m0 ¿ 0 is the mass of a particle at rest, and c is the speed p of light. In terms of velocity v = u; t , the kinetic energy density We (v) = −m0 c2 1 − v2 =c2 is strictly convex. If v = 0, this leads to the well-known Einstein’s equation E = −We = m0 c2 . 2 2 However, in terms of the nite √ “velocity measure” : =(u)=1−u; t =c ¿ 0, the kinetic 2 energy density W ()=−m0 c  is convex for  ¿ 0. The complementary q kinetic energy

density W c () = −m20 c4 =(4) is convex for all  ¡ 0 and  = − 12 c g2 + m20 c2 is the unique solution of the dual Euler–Lagrange equation (17). Thus, R for any given external force eld f(t) and integral constant gT ∈ R such that g(t) = f(t) dt + gT , the general analytic solution for Einstein’s theory is Z t cg(t) q dt + u(t0 ): u(t) = t0 g(t)2 + m20 c2

Remark 2. Results proposed in Theorem 1 can be easily generalized into nonconvex variational inequality problems with inequality constraints in Ua . For example, if Ua is a convex cone : Ua = {u ∈ W;1 (I ) | u(x) ≥ 0 ∀x ∈ (0; ‘); u(0) = 0};

(24)

then the variational inequality (VI ) :

(25)

P(u; v) ≥ 0 ∀v ∈ Ua

leads to the so-called nonlinear complementarity problem (NCP) : [(b + au; x )D W ((u))]; x + f(x) ≤ 0

∀x ∈ I;

u(x)[[(b + au; x )D W ((u))]; x + f(x)] = 0 (b + au; x )D W ((u)) = g‘

∀x ∈ I;

at x = ‘:

(26) (27) (28)

In this case, the function g in (17) and in (18) should be replaced by Z g(x) ≤

x

‘

f(t) dt + g‘ :

(29)

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Fig. 1. Singular algebraic curve for the dual Euler–Lagrange equation.

For every solution  of the dual Euler–Lagrange equation (17), if the u(x) de ned by (18) is in Ua , then this u(x) is a solution of the (NCP). A comprehensive study on variational inequalities and free boundary value problems is given in [20]. Duality theory for nonlinear complementarity problems and variational inequalities in nite deformation obstacle problems were discussed in [27,66]. Remark 3. In algebraic geometry, the dual Euler–Lagrange equation (17) is the socalled singular algebraic curve in –g space ( = 0 is on the curve, see [53, p. 99]). Since DW c is monotone, if the continuous function g(x) keeps the same sign on I , each root (x) of (17) will not change its sign on I (see Fig. 1). In Theorem 1, it is required that (x) 6= 0. Actually,  = 0 is a solution of the dual Euler–Lagrange equation (17) if and only if g(x) = 0. If this is a case, we have D W (u(x)) = 0

∀x ∈ I;

D W (u(x))|x=l = 0:

Since D W ((u)) is an one-to-one mapping on E, this homogeneous problem has usually either trivial solution or in nite number of solutions. For example, if W is a quadratic function : W () = 12 E2 , then W c () =  − W () = (1=2E)2 is also a quadratic function. In this case, the dual Euler–Lagrange equation (17) is a cubic algebraic equation   1  +  = g2 (x) ∀x ∈ I: (30) 2a2 E Making a change of variables, the singular cubic curve can be given by the well-known Weierstrass equation y2 = x3 + x2 + x + Á. If we let Cns be a set consisting of non-singular points on the curve, then Cns is an Abelian group. This fact in algebraic geometry is very important in understanding the following triality theorem. For quadratic stored energy W , we have the following result.

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Fig. 2. Graphs of h() = 2a2 (=E + ) and the criteria of uniqueness.

Corollary 1. Suppose that the stored energy W ((u)) = 12 E( 12 u;2x + bu; x + c)2 is a ∗ quadratic function of  = (u). For the given f ∈ L (I ) and boundary data, if g(x) 2 2 has extremum values gmin and gmax on I such that gmin ≤ g2 (x) ≤ gmax ∀x ∈ [0; ‘], then there exists two constants  g 2=3  g 2=3 min max and max = 1:5 : (31) min = 1:5 E E If  = b2 =2 − c ∈ (−∞; min ); the solution u(x) deÿned by (18) is a unique global minimizer of the variational problem (P); if  ∈ (min ; max ), then there exists a subdomain Is ⊂ I such that on Is ; the problem may have three solutions; if  ∈ (max ; +∞); the problem may have three solutions on the whole domain. If 1 (x) ¿ 0 ¿ 2 (x) ¿ 3 (x) ∀x ∈ I; then u1 (x) is a global minimizer; u2 (x) is a local minimizer and u3 (x) is a local maximizer of the double well energy P(u). Proof. Here we only prove the existence and uniqueness. The extremum properties will be proved by the triality theorem in the next section. Let h() = 22 ( E1  + ). Its graph is shown in Fig. 2. If  ≤ 0, the algebraic equation (30) has only one real solution for any give g(x) 6= 0 (Fig. 2(b)). In this case, the total potential energy P is strictly convex. If  ¿ 0, h() has a local maximum hmax = 8E 2 3 =27 at  = −2E=3, and h() ≤ hmax ∀ ≤ 0. Since g : I → R is a continuous function, for the given load 2 2 and a maximum value gmax on the closed f ∈ L (I ), g2 (x) has a minimum value gmin 2 2 domain [0; ‘]. The conditions hmax =gmin and hmax =gmax give the critical values of min and max de ned in (31). So if  ∈ (−∞; min ), the algebraic equation (30) has only one real solution 1 ¿ 0. However if  ∈ (min ; max ), there exists a subdomain Is ⊂ I such that g2 (x) ¡ hmax ∀x ∈ Is . In this case, Eq. (30) has three real solutions i (x); i = 1; 2; 3

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in Is , and one of them must be positive, the rest two should be negative. If  ¿ max , the algebraic equation (30) has three real roots everywhere in (0; ‘), which give three solutions ui (i = 1; 2; 3) of the problem. 3. Canonic dual transformation and complementary variational principle To solve nonconvex variational problems via the so-called canonic dual transformation method discussed in this section, the most important step is to choose the appropriate geometrical measure  = (u) such that the complementary function W c () of W () can be easily obtained by the classical Legendre transformation W c () = sta{ − W () | ∀ ∈ E};

(32)

and the following canonic duality relations hold:  = DW () ⇔  = DW c () ⇔  = W () + W c ():

(33)

Assume that the nonlinear operator  : U → E is Gˆateaux dierentiable. The directional derivative of (u) at u in the direction of u ∈ U can be written as  (u;  u) = t (u)u; where t : U → E is the Gˆateaux derivative of . The following decomposition is crucial important in duality analysis [35]: (u) = t (u)u + c (u);

(34)

where c =  − t is the so-called complementary geometrical operator. For the quadratic operator (u) = 12 a u;2x + bu; x + c, we have t (u)u = (b + au; x )u; x ;

c (u) = c − 12 au; x u; x :

Using integration by parts, Z  i = (b + au ; x )u; x  d x ht (u)u; I

= (au ; x + b)(x)u|‘x=0 −  + J@I ; = hu; ∗t (u)i

(35)

Z I

[(b + au ; x )]; x u d x

 is a formal adjoint of t (u)  de ned by where J@I = (au ; x + b)(x)u|‘x=0 , ∗t (u)  = −[(b + au ; x )]; x ∗t (u)

∀x ∈ I:

(36)

For the (BVP), Ju =g‘ u(‘)−g(0)u0 . Then the equilibrium equation (12) in (BVP) can be split into the universal tri-canonical equations in fully nonlinear systems [34,35] : 1: Geometrical equation:  = (u); 2: Constitutive equation :  = DW (); 3: Equilibrium equation : f =

∗t (u):

(37)

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By the general theory given in [26,29], the total complementary energy P d : Sa → R for the mixed boundary value problem (BVP) can be given by Z (38) P d () = − W c () d x − G ∗ (); ∗

I

where G is the so-called pure complementary gap function, de ned by G ∗ () = sta{h; (u)i − hf; ui − g‘ u(‘) | ∀u ∈ Ua }:

(39) ∗

For the quadratic operator (9), if Ua is de ned by (8), G can be easily obtained as  Z  2 b b2 g (x) +  − g(x) d x − g(0)u0 ;  = − c: (40) G ∗ () = 2a a 2a I ∗

For a given f ∈ L (I ) and boundary data, g ∈ C 1 (I ) is well-de ned by (16). Then the dual extremum problem associated with (BVP) can be proposed as d ) (Psta

P d () → sta ∀ ∈ Sa :

(41)

d

The stationary condition P (;  ) = 0 ∀ ∈ Sa leads to the dual Euler–Lagrange equation 1  + ] = g2 (x) ∀x ∈ I: (42) 2 [DW c () 2a The following theorem plays a fundamental role for obtaining the general analytic solution of the (BVP). ∗

Theorem 2. For any arbitrarily given f(x) ∈ L (I ), u is a critical point of P if and  is a critical point of P d ; and only if the associated  = D W ((u)) P(u)  = P d (): 

(43)

 is the associated dual Proof. Suppose that u is a critical point of P;  = D W ((u)) variable, then we must have   = −[(b + au ; x )]  ; x = f(x) ∀x ∈ I: ∗t (u) Solving for u; x gives   1 g(x) −b : u ; x (x) = a (x) 

(44)

 is equiSince W is convex in  = (u),  by (33), the constitutive law  = D W ((u) valent to  (u)  = DW c (): Substituting (44) into this dual constitutive law leads the dual Euler–Lagrange equation (42). So  must be a critical point of P d .  = DW c ()  into the dual Conversely, if  is a critical point of P d , substituting (u) Euler–Lagrange equation (42), we have ±(au ; x + b) = g(x): Taking derivative with respect to x on both sides and choosing a right sign, we have the equilibrium equation. So the u associated with  is a critical point of P.

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D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Since W () is convex, by the equivalent relations in (33), we have W ((u))  = (  u)  − W c ():  Substituting this into P(u)  and using the operator decomposition =t +c , we obtain Z Z  d x − fu d x − g‘ u(‘) P(u)  = h(u);  i  − W c () I I Z ∗ c   − fi − W ()  d x − G(u;  )  + u0 g(0); (45) = hu;  t (u) I

where

Z   i  = G(u;  )  = h−c (u);

I

 1 2 au − c  d x 2 ;x

(46)

is the so-called complementary gap function introduced by Gao-Strang (1989) in nite deformation theory. Substituting the equilibrium conditions and Eq. (44) into Eq. (45)  we have P(u)  = P d (). Remark. This theorem shows that the critical points of P and P d are solutions of the (BVP). Eq. (43) shows that there is no duality gap between the primal and the dual variational problems. For general nonlinear Gˆateaux dierentiable operator  : U → E, if the directional derivative of the complementary operator c is symmetrical, i.e., c (u; v)=c (v; u); such that the dual functional P d can be well de ned via nonlinear Lagrangian (see the following sections), then Theorem 2 is also hold. For n-dimensional problems, a method for obtaining the dual energy was discussed in [28–34]. 4. Lagrangian and triality theory In order to study the extremum properties of the general analytic solution, we need to know the convexity of the primal and dual functionals. Duality theory in general parametrical variational problems was studied in [29]. In this section, we assume that (A1):

 : U → E is convex, Gˆateaux dierentiable;

(47)

(A2):

W : E → R is convex; lower semicontinuous (l:s:c):

(48)

In the total potential energy P, if we replace W ((u)) by the Legendre transformation (u) − W c (), the nonlinear Lagrangian L(u; ) is then obtained: Z Z c (49) L(u; ) = [(u) − W ()] d x − fu d x − g‘ u(‘): I

I

A point (u;  )  ∈ U × S is said to be a critical point of L if  )  = 0 ⇒ ∗t (u)   − f = 0; Du L(u;

(50)

 )  = 0 ⇒ (u)  − DW c ()  = 0: D L(u;

(51)

Here Du , D denote the partial Gateaux derivatives on U and S, respectively. By the equivalent relations (33) we know that the critical points of L solve the (BVP).

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In convex variational problems, if  is linear, the Lagrangian is usually a saddle functional, i.e. L(u;  ) ≤ L(u;  )  ≤ L(u; ) 

∀(u; ) ∈ U × S:

(52)

But in geometrically nonlinear, nonconvex variational problems, we need the following de nitions. Deÿnition 1. A point (u;  )  is said to be a super-critical (or @+ -critical) point of L if L(u;  ) ≤ L(u;  )  ≥ L(u; )  ∀(u; ) ∈ U × S:

(53)

A point (u;  )  is said to be a sub-critical (or @− -critical) point of L if L(u;  ) ≥ L(u;  )  ≤ L(u; ) 

∀(u; ) ∈ U × S:

(54)

According to nonsmooth analysis, for any given real-valued function F : U → R, the sub-dierential and super-dierential of F at u ∈ U are de ned by @− F(u) = {u∗ ∈ U∗ | F(v) − F(u) ≥ hu∗ ; v − ui ∀v ∈ U}; @+ F(u) = {u∗ ∈ U∗ | F(v) − F(u) ≤ hu∗ ; v − ui ∀v ∈ U}; respectively. In convex analysis, @− is simply written as @. The super-dierential @+ is also called the over-dierential, written as @ (see [2]). It is easy to check that  = inf P(u) if and @+ F = −@− (−F). So u is a solution of the primal problem, i.e., P(u) only if 0 ∈ @− P(u).   is a global maximizer of P d if and only if 0 ∈ @+ P d ().  According to [26,29], we have the following result. Theorem 3. Suppose that (u;  )  is a critical point of L. Then we have 1: (u;  )  is a saddle point of L if and only if  );  0 ∈ @− u L(u;

0 ∈ @+  ):   L(u;

(55)

If the conditions in (A1) and (A2) hold; then (55) is true if and only if (x)  ≥0 ∀x ∈ I . 2: (u;  )  is a super-critical point of L if and only if  );  0 ∈ @+ u L(u;

0 ∈ @+  ):   L(u;

(56)

Under assumptions (A1) and (A2); the relations in (56) hold if and only if (x)  ≤ 0 ∀x ∈ I . 3: (u;  )  is a sub-critical point of L if and only if  );  0 ∈ @− u L(u;

0 ∈ @−  ):   L(u;

(57)

Moreover; if (A1) holds and W : E → R is concave; then (57) holds if and only if (x)  ≥ 0 ∀x ∈ I .

1174

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Proof. By de nition, if (u;  )  is a saddle point of L, then  );  0 ≤ L(u; )  − L(u;  )  ∀u ∈ U; ⇒ 0 ∈ @− u L(u; 0 ≥ L(u;  ) − L(u;  ) 

 ):  ∀ ∈ S; ⇒ 0 ∈ @+  L(u;

Similarly for super- and sub-critical points. Under the assumption (A1), for any given  ∈ S, the Lagrangian L : U → R is convex if (x) ≥ 0, and concave if (x) ≤ 0. If (u;  )  is a critical point of L, we have L(u; )  ≥ L(u;  )  i  ≥ 0 ∀x ∈ I; L(u; )  ≤ L(u;  )  i  ≤ 0 ∀x ∈ I: If W is convex, its conjugate W c , de ned by W c () = sup { − W ()} ∈E

is convex, l.s.c. Hence, (u;  )  is a saddle point if and only if  ≥ 0; (u;  )  is a super-critical point if and only if  ≤ 0. However, if W () is concave, then its conjugate W c , de ned by W c () = inf { − W ()} ∈E

is concave and upper semicontinuous (u.s.c.). (u;  )  is a sub-critical point if and only if  ≥ 0. Let Su ⊂ Ua × Sa be the so-called equilibrium admissible space: Su = {(u; ) ∈ Ua × Sa | hDu L(u; ); vi = 0 ∀v ∈ Ua }:

(58)

If Su is not empty, the pure dual functional P d : Sa → R can be de ned by P d () = sta L(u; ): u∈Ua

(59)

We have the following complementary variational principles Theorem 4. Suppose that the assumptions in (A1) and (A2) hold. For any u ∈ Ua ; P(u) = sup L(u; ): ∈S

For a given  ∈ Sa such that (u; ) ∈ Su ; then   inf L(u; ) if (x) ≥ 0;  u∈U a d P () =   sup L(u; ) if (x) ≤ 0:

(60)

(61)

u∈Ua

Proof. By the convexity of W (), we have W ∗∗ ((u)) = sup {(u) − W c ()} = W ((u)). So P(u) = sup L(u; ) holds on Ua . For any  ∈ Sa , if the equilibrium admissible space Su is not empty, then there exists  ; u) = 0 ∀u ∈ Ua . If (x) ≥ 0; L : U → R is convex. Then u a u ∈ Ua such that L(u;

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

1175

should be the solution of inf u L(u; ) and P d () = L(u;  ) = inf u L(u; ). If (x) ≤ 0, then L : U → R is concave. In this case, u should be the solution of supu L(u; ) and P d () = supu L(u; ). This theorem shows the extremality relations among the primal, dual functionals and the nonlinear Lagrangian L(u; ). Lemma 1. If (u;  )  is any one of the saddle point; the super- or sub-critical points of L; and L is partially Gˆateaux dierentiable at (u;  );  then (u;  )  must be a critical  respectively; then point of L. If P and P d are Gˆateaux dierentiable at u and ; DP(u)  = 0; DP d ()  = 0 and  P(u)  = L(u;  )  = P d ():

(62)

A detailed proof of this lemma is given in general parametric variational problems in [29]. Let  u) = 0 ∀u ∈ Ua }; Uc = {u ∈ Ua | P(u;  ) = 0 ∀ ∈ Sa }: Sc = { ∈ Sa | P d (; For any given critical point u ∈ Uc and  ∈ Sc , we let Ub and Sb be their (open) neighborhoods such that on Ub and Sb , u and  are the only one critical point of P and P d , respectively. Then the triality extremum theorem proposed recently in [26,29] also hold for any convex geometrical operator . ∗

Theorem 5. Suppose that; in addition to (A1) and (A2); for the given f ∈ L (I ) and boundary conditions; Uc and Sc are not empty. For any u ∈ Uc and associated  = D W ((u))  ∈ Sc ; if (x)  ¿ 0 ∀x ∈ I; then  P(u)  = inf P(u) = sup P d () = P d (): u∈Ub

∈Sb

(63)

However; if (x)  ¡ 0 ∀x ∈ I; then we have either  P(u)  = inf P(u) = inf P d () = P d ()

(64)

 P(u)  = sup P(u) = sup P d () = P d ():

(65)

u∈Ub

∈Sb

or u∈Ub

∈Sb

If  : U → E is strictly convex and Sc contains only one strictly positive critical point ;  the primal variational problem (P) has at most one solution. Proof. From Lemma 1 we know that (u;  )  must be a critical point of L and P(u)  =  If  ¿ 0, by Theorem 3 (u;  )  is a saddle point of L on Ub × Sb . For L(u;  )  = P d (). any given u ∈ Ua ; L : S → R is concave, and  ≥ L(u;  )  ∀u ∈ Ua : P(u) = sup L(u; ) = L(u; ) 

1176

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

By Lemma 1, L(u;  )  = P(u)  ≤ P(u) ∀u ∈ Ua . So the critical point u minimizes (P) on Ub . Since the  ¿ 0; L : Ub → R is convex, then by Theorem 4,  ) ≤ L(u;  )  P d () = inf L(u; ) = L(u; u d

 = P ()

∀ ∈ Sa :

This shows that the critical point  ∈ Sb maximizes P d and Lemma 1 gives that P(u)=   P d (). If  ¡ 0, (u;  )  should be a super-critical point of L on Ub × Sb , and  = L(u;  )  = sup L(u; sup L(u; )  ) ∀(u; ) ∈ Ub × Sb : u



(66)

If u maximizes P on Ub , then P(u)  = sup P(u) = sup sup L(u; ); u

u∈Ub



 = sup sup L(u; ) = sup P ∗ () = P d (): 

u

∈Sb

(67)

So sup P(u) = sup P d () as we can take supremum in either order on Ub × Sb . If u is a local minimizer of P on Ub , we need to prove that the associated  minimizes P d on Sb . Since  is a critical point of P d , it must be either a local extremum point or a local saddle point of P d . If  is a local maximizer, then by Eq. (67) the associated u should be a local maximizer of P. This against our assumption. If  is a local saddle point of P d and it maximizes P d in the direction o ∈ Sb such that P d ()  = sup P d (o + Â) ∀ ≥ 0. The solution Âo of this problem should satisfy  = o + Âo . Substituting  = o +  into (67) we get the contradiction too. So the critical point  must be a local minimizer of P d .  then L : S → R is If Sc contains the only one strictly positive critical point , strictly concave. Since D W ((u)) is one-to-one, (u) is strictly convex, then L(u; ) has only one saddle point, and P : U → R is strictly convex. Therefore, the primal problem has at most one solution. Combining Theorems 4 and 5, we have the following interesting result. Corollary 2. Suppose that; in addition to (A1) and (A2); (u;  )  is a critical point of  ).  If (x)  ≥ ∀x ∈ I; then L; Ub × Sb is a neighborhood of (u;  )  = sup inf L(u; ): inf sup L(u; ) = L(u;

u∈Ub ∈Sb

∈Sb u∈Ub

(68)

However; if (x)  ¡ 0 ∀x ∈ I; then either  )  = inf sup L(u; ) inf sup L(u; ) = L(u;

(69)

 )  = sup sup L(u; ): sup sup L(u; ) = L(u;

(70)

u∈Ub ∈Sb

∈Sb u∈Ub

or u∈Ub ∈Sb

∈Sb u∈Ub

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

1177

Fig. 3. Graph of P d () (solid line) and graph of P(u) (dashed line).

Let us show this interesting result by the very simple nonconvex optimization problem: P(u) = 12 E( 12 u2 − )2 − gu → min

∀u ∈ R:

Let  = (u) = 12 u2 − ; W () = 12 E2 ;  = DW = E. Then the dual function is   g2 1 2 d  +  − : P () = − 2E 2 For the given data  and g, the graphs of P and P d are shown in Fig. 3. By Corollary 1 we know that for a given  ¿ min , the dual Euler–Lagrange equation (30) has three roots 1 ¿ 0 ¿ 2 ¿ 3 , corresponding to three critical points of P d (see Fig. 3(a)). Then 1 is a global maximizer of P d ; u1 = g=1 is a global minimizer of P: P d takes local minimum and local maximum at points 2 and 3 , respectively. However, u2 = g=2 is a local minimizer of P, while u3 = g=3 is a local maximizer. If  ¡ min , the problem has only one global minimizer (see Fig. 3(b)). 5. Application in nonsmooth bifurcation problem Let us consider the equilibrium problem of a one-dimensional elastoplastic bar, which is xed at x = 0, free at x = l, and subjected to the distributed load f(x) ∀x ∈ I = (0; l). The displacement u(x) is a real-valued function. The Cauchy–Green strain  is de ned by the geometrical nonlinear equation  = (u) = 12 (u;2x − 1):

(71)

So  is a quadratic operator with a = 1; b = 0; c = −1=2 = −. Let b ¿ 0 be the elastic strain limit. If  ≤ b , the bar is in elastic deformation state, while  ¿ b the bar is in

1178

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Fig. 4. Discontinuous constitutive relation.

Fig. 5. Nonsmooth W and smooth W c .

plastic state. Suppose that the constitutive relation is picewise linear (see Fig. 4), ( if  ≤ b ; E1  (72) = E2 ( − b ) + d if  ¿ b ; where E1 ; E2 and d are material constants. Then stored energy W is a nonsmooth function (see Fig. 5a) ( We () = 12 E1 2 if  ≤ b ; (73) W () = Wp () = 12 E1 b2 + 12 E2 ( − b )2 + d ( − b ) if  ¿ b : Since W () is convex, its Legendre conjugate can be easily obtained as  c W () = 2E1 1 2 if  ≤ b ;    1 c 2 1 c if b ¡  ≤ d ; (74) W () = W2 () = 2E1 b + b ( − b )    W c () = 1 2 +  ( −  ) + 1 ( −  )2 if  ¿  : b b d d 3 2E1 b 2E2 which is a smooth function of  (see Fig. 5b).

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

The inverse constitutive relation  = DW c is then a continuous function  1  if  ≤ b ;   E1 if b ¡  ≤ d ;  = b   1 b + E2 ( − d ) if  ¿ d :

1179

(75)

In terms of the in nitesimal strain e = u; x , the stored energy W ( 12 (e2 − 1)) is a double-well function of e. The feasible space Ua in this problem is a convex set Ua = {u ∈ W4;1 (0; ‘) | u(0) = 0}: Let Ie :={x ∈ I | (x) ≤ b };

Ip :={x ∈ I | (x) ¿ b }:

On Ua , the primal problem (1) can be written as     Z Z 1 1 1 2 1 2 u; x − dx + u; x − dx Wp P(u) = We 2 2 2 2 Ie Ip Z − f(x)u(x) d x → min ∀u ∈ Ua : I

(76)

This is a nonsmooth, nonconvex variational problem. Since the interface of the elastic region Ie and the plastic region Ip are unknown till the problem is solved, this problem is more dicult than a free boundary problem. However, by the theorems proved above, the analytic solution of this very dicult problem can be well determined. Since  = 12 (u;2x − 1) ≥ − 12 ∀u ∈ Ua , the ranges of the constitutive mapping  = DW are Re = [ − 12 E1 ; b ];

∀ ≤ b + 12 ;

Rp = (d ; +∞);

∀ ¿ b + 12 :

Then the dual feasible spaces can be given as Sa1 = { ∈ H(I ) | − 12 E1 ≤ (x) ≤ b ; (x) 6= 0 ∀x ∈ I }; Sa2 = { ∈ H(I ) | b ¡ (x) ≤ d ; (x) 6= 0 ∀x ∈ I };

Sa3 = { ∈ H(I ) | d ¡ (x) ¡ + ∞; (x) 6= 0 ∀x ∈ I }: For the homogeneous mixed boundary conditions u0 = 0; gl = 0, and the coecients a = 1; b = 0; c = − 12 , the dual functional P d can be simply written as  Z  1 g2 (x) d c d x; (77) W () +  + P () = − 2 2 I where g(x) = {gi (x)} is a continuous function de ned by Z gi (x) = −f(x) d x + di ; x ∈ Ii ; i = 1; 2; 3:

(78)

1180

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

The domains Ii and the integral constants di are determined by the dual-Euler–Lagrange equations   1  + 12 = g12 ; ∀ ∈ Sa1 ; (79) 22 E1 2

2

2

2





1 b + 2



= g22 ;

∀ ∈ Sa2 ;

1 ( − d ) + b + E2

 1 2

= g32 ;

(80) ∀ ∈ Sa3 :

For  = b and  = d , Eq. (80) gives p p gb = b 2b + 1; gd = d 2b + 1:

(81)

(82)

For the given external load f(x) and boundary conditions, the domains Ii (i = 1; 2; 3) can be well determined by the function g(x): I1 = {x ∈ I | |g(x)| ≤ gb };

I2 = {x ∈ I | gb ¡ |g(x)| ≤ gd };

I3 = {x ∈ I | |g(x)| ¿ gd }:

(83)

Since the bar is free at x = ‘, we have g‘ = 0, and Z l f(s) ds; x ∈ [xb ; L]; g1 (‘) = 0: g1 (x) = x

The interface xb is well determined by Z l f(x) d x = gb ⇒ xb : g1 (xb ) = xb

Similarly, the conditions g2 (xb ) = gb ; g3 (xd ) = gd lead to the integral constants d2 = gb and d3 = gd , and Z xb f(s) ds + gb ; x ∈ [xd ; xb ]; g2 (x) = x

Z g3 (x) =

x

xd

f(s) ds + gd ;

x ∈ [0; xd ]:

The condition g2 (xd ) = gd leads to the interface xd Z xb f(x) d x + gb − gd = 0 ⇒ xd : xd

In the subdomain I1 , the system is controlled by Eq. (79). Since a = 1, |g1 |max = gb , by Corollary 1, if =

1 2

≥ 1:5(gb =E1 )2=3 = 1:5(2b3 + b2 )1=3 ⇒ b ≤ 16 ;

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

1181

then in I1 , Eq. (79) may have three solutions (i.e., elastic buckling), depending on g(x). By Theorem 1, the analytic solution in I1 is Z x g1 (s) d x + u1 (xb ); x ∈ I1 ; (84) u1 (x) = xb (s) where u1 (xb ) will be determined by the interface condition. In the domain I2 , the dual Euler–Lagrange equation (80) is linear. Hence in I2 = (xd ; xb ), the solution is unique p (85) u2 (x) = sgn[g2 ] 2b + 1(x − xb ) + u1 (xb ); x ∈ I2 : In the domain I3 , g3min = gd , if  = b + 12 − d =E2 ≤ 1:5(gd =E2 )2=3 , the algebraic equation (81) has a unique root , which leads to the analytic solution Z x g3 (s) ds; x ∈ [0; xd ]: (86) u3 (x) = 0 (s) However, if b + 12 − d =E2 ¿ 1:5(gd =E2 )2=3 , the solution  may not be unique. In this case, if for a given f(x) such that the dual Euler–Lagrange equation (81) has three real roots (x), then the system is in plastic buckling state. The interface condition u3 (xd ) = u2 (xd ) gives u1 (xb ). Duality theory for 3-D nonsmooth Hencky plasticity was discussed in [36]. A general closed-form solution for elasto-plastic materials with hardening eects is proposed recently in [33]. 6. Applications in general nonconvex variational problems The canonic dual transformation method can be easily generalized to solve many nonconvex problems with dierent types of the geometrical operator  and stored energy W . Here we only consider two cases. 6.1. Nonconvex problems with multi-well energy If the geometrical operator  : U → E ⊂ Lÿ is a general nonlinear, nonconvex mapping, we can continually use the nonlinear dual transformation method discussed in Section 3, such that the general stored energy W can be written as W ((u)) = Wn (n (n−1 (: : : (1 (u)) : : :)));

(87)

where k (k−1 ) is either strictly convex or concave function of k−1 and we write Wk (k ) = k+1 (k );

k = 1; : : : ; n:

Then the sequential constitutive mappings DWk (k ) are one-to-one such that the canonic duality relations hold k = DWk (k ) ⇔ k = DWkc (k ) ⇔ Wk (k ) + Wkc (k ) = k k :

1182

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Without loss of generality, we assume that 1 (u) is quadratic function of 0 = u; x and we write 1 (u; x ) = (1) (u) = 12 a0 u;2x + b0 u; x + c0 ;

a0 (x) ¿ 0:

Then we have 0 = DWn (n )DWn−1 (n−1 ) : : : DW1 (1 ) = n n−1 : : : 1 (a0 0 + b0 ):

@1 @0

(88)

The geometrical operator  : U → E in this problem is a sequential compositions of mappings k : Ek−1 → Ek ; (k = 1; · · · ; n), E0 = U; En = E n (u) = [(n) ◦ (n−1) ◦ · · · ◦ (1) ](u); its Gˆateaux derivative is n (u; v) = Dn (u)v = n−1 !(1) t (u)v;

k = Dk+1 (k ):

So the trio-canonical equations (37) can be written as n = [(n) ◦ (n−1) ◦ · · · ◦ (1) ](u); k = DWk (k );

1 = (1) (u) = 12 a0 u;2x + b0 u; x + c0 ; (89)

k = 1; : : : ; n

∗

f = [Dn (u)] n = −[n · · · 1 (a0 u; x + b0 )]; x : Let  = {1 ; : : : ; n }. Since k+1 (k ) = Wk (k ) = k k − Wkc (k ); the generalized Lagrangian associated with the primal problem (P) can be written as Z Z c L(u; ) = [n n (n−1 ) − Wn (n )] d x − fu d x Z =

I

I

I

c [1 (u)n ! − Wnc (n ) − n Wn−1 (n−1 )

−··· −

n ! c W (1 )] d x − 1 1

Z I

fu d x:

(90)

The pure gap function for the quadratic operator (1) is Z  [n !(1) (u) − fu] d x|u ∈ Ua G ∗ () = sta Z  =

I

I

 g2 b0 g + 0 n ! − dx; 2a0 n ! a0

where 0 = b20 =a0 − c0 . The dual functional is  Z  n ! c c (n−1 ) + · · · + W1 (1 ) d x − G ∗ (): P d () = − Wnc (n ) + n Wn−1 1 I

(91)

(92)

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

The dual Euler–Lagrange equations are   2a0 n2 ! DW1c (1 ) + 0 = g2 ;

1183

(93)

c (k+1 ) = k DWkc (k ) − Wkc (k ); DWk+1

k = 1; : : : ; n − 1:

(94)

Since DWkc (k ) is one-to-one, for each k , k+1 can be determined uniquely by k via Eq. (94). By Theorem 1, for each solution  = {1 ; : : : ; n } of this algebraic system, the analytic solution is  Z x g b0 d x + u0 : (95) − u(x) = a0 n ! a0 0 If W is the nonconvex function with at most 2n-wells  2
2 W ((u)) = 12 an 12 an−1 (· · · 12 a1 ( 12 a0 u;2x − 1 )2 · · · − n−1 )2 − n ;

(96)

we can always choose the coecients b0 = c0 = 0 = 0, 0 = u; x , bk = −ak k ; ck = 1 2 (k) is an quadratic operator 2 ak k ; (k = 1; : : : ; n), such that the geometrical operator  of k−1 : 2 + bk−1 k−1 + ck−1 = 12 ak−1 (k−1 − k )2 : k = (k) (k−1 ) = 12 ak−1 k−1

Then for each k, the stored energy Wk and its Legendre-conjugate are quadratic functions: Wk (k ) = 12 ak (k − k )2 = k+1 (k ); Wkc (k )

1 2 =  + k k 2ak k

k = 1; : : : ; n;

and the canonic duality relations can be written as k = DWk (k ) = ak (k − k ); For n = 2, we have P2d (1 ; 2 ) = −

Z  Z

−

I

I

k = DWkc (k ) =

1 2  + 2 2 + 2 a2 2

g2 d x: 2a0 1 2

The dual Euler–Lagrange equations are   1 1 + 1 = g2 ; 2a0 12 22 a1 2 =

a2 2  − 2 a2 : 2a1 1



1 k + k : ak

1 2  + 1 1 2a1 1

 dx (97)

(98) (99)

1184

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Fig. 6. Lagrangian L2 for 3-well energy: 1 = 2 = 1; a0 = a2 = 1; a1 = 1:5.

For the given parameters, the graphs of g(1 ) (Fig. 6) show that this system has at most seven roots for 1 . The 2 is uniquely de ned by 1 . By Theorem 1, the analytic solution is Z g(x) d x + u0 : u(x) = a0 1 2 Let !2  2 1 1 1 a1 a0 u2 − 1 − 2 − gu: P2 (u) = a2 2 2 2 The nonlinear Lagrangian and dual function are 1 1 2 1  − 2 2 − 2 12 − 1 1 2 − gu; L2 (u; 1 ) = a0 u2 2 1 − 2 2a2 2 2a1   g2 1 2 1 2 + 2 2 + 2 12 + 1 1 2 − ; P2d (1 ) = − 2a2 2a1 2a0 1 2

(100) (101)

where 2 =a2 12 =(2a1 )−2 a2 . For the given parameters, the graph of L2 (u; 1 ) is shown in Fig. 6. Graphs of P2 , P2d and algebraic curves of the equation (98) are shown in Fig. 7. 6.2. Nonconvex dynamics: saddle energy W In nonconvex dynamic systems, the geometrical mapping  = {(t) ; (s) } : U → E = Et × Es is usually a vector-valued dierential operator such that the stored energy

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Fig. 7. P2 , P2d and algebraic curves for multi-well energies.

1185

1186

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

function W ((u)) = W (v(u); (u)) is a so-called action density. Very often, W (v; ) = K(v) − U ();

(102)

where K(v) is the kinetic energy density, and U () is the stored potential energy. In Newtonian systems, the time geometrical operator (t) is linear and the kinetic energy is usually a quadratic function of the velocity v = (t) (u) = u; t . However, the potential U ((u)) is often a nonconvex function of u. By the canonic dual transformation, we assume that  = (u) = {v(u); (u)} ∈ Et × Es ⊂ Lÿ (It ; R2 ) is Gˆateaux dierentiable such that W : Et × Es → R is a saddle functional W (∗; ) : Et → R is convex; l:s:c ∀ ∈ Es ;

(103)

W (v; ∗) : Es → R is concave; u:s:c ∀v ∈ Et :

Then the constitutive mapping ={p; }=DW () ∈ Et∗ ×Es∗ is a vector-valued operator: p = Dv W (v; );

 = D W (v; ):

(104)

The Legendre–Fenchel conjugate of W can be obtained by W c () = sta” { · ” − W (”)} = sup inf {pv +  − W (v; )} = W ∗ (p; ):

(105)



v

On the given time-space domain It = (0; T ) × (0; L), the least action principle leads to a nonconvex variational problem: Z Z (106) (P) : P(u) = W ((u)) dIt − fu dIt → min ∀u ∈ Ua : It

It

The critical condition leads to the nonlinear partial dierential equation [t(t) (u)]∗ Dv W ((u)) − [t(s) (u)]∗ D W ((u)) = f:

(107)

The Lagrangian L : U × E → R in nonconvex dynamical systems can be proposed (see [31]) Z Z (108) L(u; ) = [ · (u) − W ∗ ()] dIt − fu dIt : It

Since W

∗

: Et∗

×

It

Es∗

→ R is also a saddle functional, we have always

inf sup inf L(u; p; ) = inf inf sup L(u; p; ) = inf inf sup L(u; p; ): u

p



u



p



u

p

(109)

If the action W can be written in (102), then the dual action P d : Et∗ × Es∗ → R can be written as Z (110) P d (p; ) = sta L(u; p; ) = [U ∗ () − K ∗ (p)] dIt + G ∗ (p; ); u∈Ua

It

where the gap function is de ned by the following stationary problem: Z G ∗ (p; ) = sta [p(t) (u) + (s) (u) − fu] dIt ; ∀u ∈ Ua : u

It

(111)

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

The classical Hamiltonian H : U × Et∗ → R is de ned by Z Z Hc (u; p) = inf L(u; p; e) = [K ∗ (p) + U ((s) (u))] dIt + fu dIt : e∈Es

It

It

1187

(112)

For linear operator (t) (u) = u; t , we have the classical Hamiltonian canonic forms (t) u = Dp Hc (u; p);

(t)∗ p = Du Hc (u; p):

(113)

(s)

Furthermore, if  (u) = u and U (u) is convex, then the classical Hamiltonian Hc is convex, and G ∗ (p; ) = pu|t=0;T

s:t:

 = p; t + f:

(114)

In this case, P d can be written in Z ∗ P (p) = [U ∗ (p; t + f) − K ∗ (p)] dIt :

(115)

Since the classical Lagrangian Lc : U × Et∗ → R Z Lc (u; p) = [pu; t − K ∗ (p) − U (u) − fu] dIt

(116)

It

It

is a super-critical point functional, the triality theory gives inf P(u) = inf sup Lc (u; p) = inf sup Lc (u; p) = inf P ∗ (p): u

u

p

p

u

p

(117)

This the Toland duality theory in convex Hamilton systems [59]. Duality and triality theory in large deformation structural dynamics has been established recently in [32]. Here we consider the following one-dimensional problem: #  2 Z " 1 2 1 2 (118) u −  − fu dt → min ∀u ∈ Ua : u; t − a P(u) = 2 2 I The kinetically admissible space Ua in this one-dimensional dynamical system is simply given as Ua = {u ∈ W4;1 (0; T )| u(0) = u0 ; u; t (0) = v0 }:

(119)

The critical condition of P leads to the well-known forced Dung equation u; tt = au( − 12 u2 ) − f(t);

∀t ∈ I = (0; T ); u ∈ Ua :

(120)

By introducing the nonlinear operator ” = (u) = {u; t ; 12 u2 − } the action density W (v; ) and its conjugate are saddle functions: 1 1 2 1 1 p − 2 : W ∗ (p; ) = W (v; ) = v2 − a2 ; 2 2 2 2a For any given p; , the pure gap function can be easily obtained    Z  1 2 u −  − fu dt pu; t +  G ∗ (p; ) = sta u 2 I  Z  (p; t + f)2 dt:  + = p(T )u(T ) − u02 − 2 I

(121) (122)

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Let  = −, and Et∗ = {p ∈ L2;1 (0; T )| p(0) = v0 }; Es∗ = { ∈ L2 (0; T )| − a ≤ (t) ¡ + ∞;

(t) 6= 0 ∀t ∈ (0; T );

(0) = a( 12 u02 − )}: So on the product space Et∗ × Es∗ , the dual action is well de ned:  Z  1 2 1 2 (p; t + f)2  +  + − p dt: P d (p; ) = p(T )u(T ) − u02 + 2 2 I 2a

(123)

The critical condition of P d leads to the following dual Dung system:   1 1 (p; t + f) + p = 0   ;t ∀t ∈ (0; T ): (124) a 2 2  ( + a) = (f + p; t ) 2 Numerical methods for solving this kind of dierential=algebraic systems have been well studied [6]. Theoretically speaking, for the same initial conditions, Dung equation (120) and its dual system (124) should have the same solution set. Numerically, the primal and dual Dung problems will give the complementary bounding approaches to the real solution. For the given data a = 1; f = 0; u0 = 0; v0 = 0:4, Fig. 8 shows that the dierences between the primal (solid line) and dual (dashed line) numerical solutions are vary with the parameters . Both primal and dual solutions very sensitive to the initial data. Detailed study on primal-dual approaches for solving nonhomogeneous Dung system and more very interesting results will be given in another paper. Since the action density is not convex, the classical least action principle is in fact a misnomer. The well-known mountain pass lemma (see [47]) is only a double-min type duality theory (i.e. Eq. (69)). The triality theory reveals a very interesting phenomenon in non-convex dynamic systems. 7. Conclusion Duality theory plays a fundamental role in natural phenomena. In geometrically linear convex systems, the symmetry between the primal and dual systems are amazingly beautiful (see, [42,54]). However, in fully nonlinear systems, the one-to-one symmetrical relations do not usually exist. The duality theory depends on the choice of the nonlinear operator  and the associated gap function. The key idea of the canonic dual transformation method is to choose the appropriate nonlinear operator  to ensure that the Legendre-conjugate of the stored energy W () can be easily determined so that the canonic duality relations hold. For a given mathematical physics problem, the choice of  may not be unique. But we can always use the sequential quadratic operators to approach any nonconvex stored energy. Since physical variables in natural systems appear always in pairs (i.e., one-to-one, see [45,61,62]), the canonic dual transformation method can be used to study wider classes of problems in engineering and science.

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Fig. 8. (a)  = 1; v0 = 0:4.

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1190

D.Y. Gao / Nonlinear Analysis 42 (2000) 1161 – 1193

Fig. 8. (b)  = 3; v0 = 0:4.

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