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Lagrange multipliers in innite dimensional

spaces, examples of application

A. Bersani, F. dell'Isola, P. Seppecher

1 Synonyms

Innite-dimensional constrained mechanical systems

2 Denitions

The Lagrange multipliers method is used in Mathematical Analysis, in Mechan- ics, in Economics and in several other elds, to deal with the search of the global maximum or minimum of a function, in presence of a constraint. The usual technique, applied to the case of nite-dimensional systems, transforms the constrained optimization problem into an unconstrained one, by means of the introduction of one or more multipliers and of a suitable Lagrangian function,to be optimized. In Mechanics, several optimization problems can be applied to innite-dimensional systems. Lagrange multipliers method can be applied also to these cases.

3 Introduction

In this entry we show that the theorem of Lagrange multipliers in innite di- mensional systems [1] can be a very powerful tool for dealing with constrained problems also in innite dimensional spaces. This tool is powerful but must be used carefully. As penalization is often invoked as an intuitive and numerically ecient approach to constrained problems, we also show that the penalization approach may present the same drawbacks as a rough application of Lagrange multipliers method does. To that aim we describe two examples issued from continuum mechanics [2]. The rst one goes back to Lagrange himself [3]: an incompressible uid, or an incompressible linear elastic material is a material whose displacement eld is constrained to be divergence-free. We show how easily the Lagrange multipliers method gives the system of balance equations and introduces the pressure as a supplementary unknown of the problem. The approach through penalization gives the same balance equations and the displacement eld be- comes divergence-free at the limit only. It is not very dicult to understand 1 why the pressure which is not independent from the displacement until the limit, becomes an independent quantity at the limit. Hence both approaches are ecient for understanding what an incompressible material is and how it behaves. The second example requires us to be more cautious. It consists in consider- ing that second gradient materials are nothing else than micromorphic materials subjected to the constraint that the micro-deformation coincides with the gradi- ent of the displacement eld. We show that a careful application of the Lagrange multipliers method leads to the correct system of equilibrium equations and we warn against its possible erroneous applications. We show that the penalization method, when over-interpreted, may lead to the same errors: one cannot infer from the admissible boundary conditions for a micromorphic material what can be the boundary conditions for the constrained model, that is for the second gradient model. The discrepancy between results obtained by studying directly the critical points of a second gradient energy and those obtained by a rough limit of micromorphic models has led to some confusion which still survives.

4 Incompressible materials

Consider an elastic material contained in a bounded Lipschitz domain . Its elastic energy density depends on the rst gradient of the displacement eldu.

The global energy has the form

1

L(u) :=Z

`(u(x);ru(x))dH3 The material is said to be linearly incompressible

2ifusatises almost every-

where in the condition div(u) = 0:(1) Assuming a quadratic growth at innity forLwith respect to its second argument, the natural functional framework is the Sobolev spaceH1( ;R3). In the sequelL2andH1stand for the Sobolev spacesL2( ;R3) andH1( ;R3). The goal is to write the partial dierential equation that any smooth critical pointuof the global energy must solve. We thus assume that bothuand`are of classC2.

4.1 Application of Lagrange Multiplier Theorem

It is well known that the divergence operator is surjective fromH1ontoL2. Indeed, for anyf2L2, it is enough to remind that the Dirichlet problem v=fon withv= 0 on the boundary admits a solution vand thus that1

HereH3stands for the Lebesgue measure onR3.

2Rubber is an example of an incompressible elastic material. Incompressibility is also of

high importance in hydrodynamics: in that caseumust be interpreted as the velocity eld andL(u) as the associated dissipation. 2 f= div(rv). Duality in the Hilbert spaceL2is trivial: Lagrange multiplier theorem states the existence of aL2functionpsuch that any critical pointu ofLunder the constraint (1) satises Z (@1`h+@2` rhpdiv(h))dH3= 0:(2) or Z (@1`h+ (@2`pId) rh)dH3= 0 for any smooth test functionh. Here the partial derivatives of`are taken at (u(x);ru(x)).

Hence, in the sense of distributions on

1`H3j div((@2`pId)H3j ) = 0:

Now let us use the divergence theorem

3: div('H3j ) = div(')H3j (n')H2j@ (3) wherenstands for the outward normal to@ andH2j@ stands for the restriction to the boundary@ of the two-dimensional Hausdor measure. We get (@1`div((@2`pId))H3j +n(@2`pId)H2j@ = 0: In mechanics the quantities@1`and@2`pIdare respectively denotedfand and interpreted as thebulk external forceand thestress tensor. Using this notation, previous equation reads (f+ div())H3j +nH2j@ = 0:

As the measuresH3j

,H2j@ are orthogonal (or mutually singular) [5], this equa- tion splits in f+ div() = 0H3a.e. in n= 0H2a.e. on@ We recover the standard equilibrium equations where the stress tensor=

2`pIdinvolves explicitly the Lagrange multiplierp. Its oppositepis

interpreted as anunknown pressurewhich has to be determined together with the equilibrium displacementuby using the equilibrium equations together with the constraint div(u) = 0.3 Indeed, for anyC1function'and any smooth test function , D div('H3j ); E =D 'H3j ;r E =Z ' r dH3=Z div(') dH3Z (n') dH2: 3

4.2 Approach through penalization

An intuitive way for dealing with incompressibility is to penalize the fact that div(u) does not vanish. Looking for minimizers of the total energy, one decides to look for a sequence of approximate minimizers by considering, instead of the original potential subjected to the constraint, the original Lagrangian to which one adds a penalization term: for instance L "(u) =Z (`(u(x);ru(x)) +"1(div(u))2)dH3:(4) This corresponds to a slightly compressible elastic material. Denotingp":=

2"1div(u), any criticalusatises, for any smooth test functionh,Z

(@1`h+@2` rhp"div(h))dH3: This equation is identical to (2). It then leads to the same system of equilibrium equations. The dierence is that now the stress tensor is directly related tou through the so-calledconstitutive equation=@2l+ 2"1div(u)Id. When passing to the limit"!0, div(u) tends to zero, the system of equilibrium equations is preserved but the constitutive equation is partially lost as"1div(u) is an undetermined form.

5 Second gradient as constrained generalized model

Consider an elastic material contained in a bounded simply connected domain with piecewiseC1boundary. In the sequelL2andHsstand for the Sobolev spacesL2( ;R3) andHs( ;R3). The elastic material is said to be asecond gradient material(or equivalently astrain-gradientone) if its elastic energy density depends on the gradient and the second gradient of the displacement eldu. The global energy has the form Z l(u(x);ru(x);rru(x))dH3 Such a model and particularly the associated boundary conditions present some diculties of interpretation (see for instance [9, 10, 11]). That is why many researchers prefer to see this model as a special case of a generalized continuum [8]. An elastic material is said to be amicromorphic generalized continuumif its elastic energy density depends on an extra matrix valued kinematic quantityv and on the rst gradients ofuandv. The global elastic energy has the form

L(u;v) :=Z

`(u(x);v(x);ru(x);rv(x))dH3 It is clear that, when such a model is constrained in such a way thatvhas to coincide with the rst gradient ofu, that is if

C(u;v) :=v ru= 0;

4 then one recovers the previous second gradient model. This indirect approach does coincide with the direct approach under the express condition that theLagrange Multiplier Theoremis precisely respected. A rough application of Lagrange Multiplier Method may lead and have led to misunderstanding boundary conditions. In the sequel we assume that the considered energies have quadratic growth at innity so the Sobolev spacesL2( ;R3),Hs( ;R3) orHs( ;R33) are the natural framework. We simply denoteL2andHsthese spaces as the tensorial nature of the involved functions is clear from the context. We are thus looking for smooth critical points of the functionalL(u;v) de- ned onH1H1under the constraintC(u;v) = 0. It is obvious that this problem can be simplied by setting

4l(u;;) :=`(u;;;) and by looking for

the critical points of

L(u;v) :=Z

l(u(x);ru(x);rv(x))dH3 under the same constraint.

5.1 Application of Lagrange Multiplier Theorem

As our functionals are dened on the Hilbert spaceH1H1, assumed to be dif- ferentiable, and as the constraintC(u;v) = 0 is linear, it is enough for applying Lagrange Multiplier Theorem [1] to check that the rangeFofCis a Banach space and to identify its dual. ActuallyFcoincides with the Hilbert spaceHcurlwhich has been extensively studied [4, 6]. This space is dened as the set of all functions inL2( ) whose curl (in the sense of distributions on ) belongs toL2( ). As curl(ru) = 0, curl(C(u;v)) = curl(v)2L2( ) and so the inclusionFHcurlis obvious. For proving the converse inclusion, let us considerw2Hcurland let us use Helmholtz decomposition of curl(w)2L2(see theorem 4.2 of [6]) and write curl(w) =r+ curl(v) with2H10( ) andv2H1( ). Obviously2 H 10( ) satises = 0 and thus, owing to the maximum principle, it vanishes. So curl(wv) = 0 in the sense of distributions which is equivalent to the orthogonality ofwvwith the curl of every smooth test function that is with every divergence-free smooth function. Hence (see [4] page 217) (wv)2L2( is a gradient:wv=ruwithu2H1( ). We havew=v+ru=C(u;v)2F. The simplest way for identifying the dualFconsists in taking advantage of the Hilbert structure ofHcurland applying Riez Theorem: to any element ofFis associated a functionfinHcurlsuch that, for anyw2Hcurl, h;wi=R (fw+ curl(f)curl(w))dH3. As smooth functions on are dense (Theorem 2 p. 204 of [4]) and continuously embedded inHcurl, is completely determined by its values for such smooth functionsw. For such4 Note that@1l=@1`,@2l=@2`+@3`and@3l=@4`where@ifstands for the partial derivative of a functionfwith respect to itsi-th variable. The densitiesland`are assumed to be of classC1. 5 functions, one can integrate by parts in the sense of distributions on and get h;wi=hf+curl(curl(f));wi. Any element ofFappears to be a distribution of order one on So the theorem of Lagrange multipliers implies the existence, at any critical point (u;v), of a distribution5 on such that, for any pair of smooth test functions (h;k), Z (@1lh(x) +@2l rh(x) +@3l rk(x))dH3+D ;(k rh)E = 0 (5) where the partial derivatives oflare taken at (u(x);ru(x);rv(x)). Using H 3j := 1

H3, the restriction to

of the three-dimensional Hausdor measure, this condition can be rewritten in terms of distributions on D 1lH3j ;hE +D 2lH3j ;rhE +D 3lH3j ;rkE +D ;(k rh)E = 0 or D (@1lH3j div(@2lH3j )div());hE +D (div(@3lH3j ) + );kE = 0:

Hence, still in the sense of distributions on

1lH3j div(@2lH3j )div() = 0;and div(@3lH3j ) + = 0: Eliminating the Lagrange multiplier is straightforward. We get 1lH3j div(@2lH3j ) + div(div(@3lH3j )) = 0: Now we make use twice of the divergence theorem (3). We obtain successively (@1ldiv(@2l))H3j +n@2lH2j@ + div(div(@3l)H3j )div(n@3lH2j@ ) = 0; (@1ldiv(@2l)+div(div(@3l)))H3j +n(@2ldiv(@3l))H2j@ div(n@3lH2j@ ) = 0: We apply now a more general version of Stokes theorem, valid on a sub-manifold with boundary 6: div('H2j@ ) = div==('==)H2j@ 'H1j@@ (n')Dnj@ 5 A frequent error consists in introducing the Lagrange multiplier as a function. We will soon see that a part of is a measure concentrated on the boundary of the domain. Note that the constraintv=ruhas some implications on the boundary: owing to this constraint ruhas, likev, a trace on the boundary of which is not natural inH1(

6Indeed, for anyC1function'and any smooth test function ,

D div('H2j@ ); E =Dquotesdbs_dbs14.pdfusesText_20

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