Elliptic systems and material interpenetration
EELLIPTIC SYSTEMS AND MATERIAL INTERPENETRATION
GIOVANNI ALESSANDRINI AND VINCENZO NESI
Abstract.
We classify the second order, linear, two by two systems for whichthe two fundamental theorems for planar harmonic mappings, the Rad´o–Kneser–Choquet Theorem and the H. Lewy Theorem, hold. They are thosewhich, up to a linear change of variable, can be written in diagonal form with the same operator on both diagonal blocks. In particular, we prove that theaforementioned Theorems cannot be extended to solutions of either the Lam`esystem of elasticity, or of elliptic systems in diagonal form, even with justslightly different operators for the two components. Introduction “A basic requirement of continuum mechanics is that interpenetration of matterdoes not occur, i.e. that in any deformed configuration the mapping giving theposition u ( x ) of a particle in terms of its position x in the reference configurationbe invertible”. J. M. Ball [5].It is then a natural question to ask what systems of equations, among thoseused as models for elastostatics, give rise to invertible solutions when reasonableboundary conditions are prescribed.In this note we shall prove by an example that the Lam`e system of isotropic,linearized elasticity in the plane, with constant Lam`e coefficients, may lead tophysically unacceptable solutions, because interpenetration of matter occurs. Letus recall here that the same phenomenon was previously found by Fosdick andRoyer–Carfagni [12] for a more involved anisotropic linear system, by elaboratingon an example due to Lekhnitskii [13]. In higher dimensions similar phenomenaoccur. From the mathematical point of view a basic example is due to De Giorgi[10]. In all these examples, however, a basic common feature is the presence ofsome sort of point singularity in the solution itself (in dimension greater than two)or at least in the gradient in any dimension. Such a singularity can only be presentwhen the coefficients of the elliptic system are irregular, for instance discontinuous.Our examples are different from those previously known in several ways, but themost crucial difference is that we choose smooth and in fact constant coefficientsin our systems. Therefore our examples, besides bringing a new argument to themany already well known, see for instance Ciarlet [8, p. 286] and the referencestherein, about the limitations of linearized elasticity, shed some new light on thetightness of certain classical properties enjoyed by harmonic mapping showing thatthey cannot be easily extended even within the class of constant coefficient systems.We now recall some fundamental properties of planar harmonic mappings, seethe book of Duren [11] for a very broad treatment of this subject.
The first author was supported in part by MiUR, PRIN no. 2006014115.The second author was supported in part by MiUR, PRIN no. 2006017833. a r X i v : . [ m a t h . A P ] M a y GIOVANNI ALESSANDRINI AND VINCENZO NESI
We begin with the classical theorem of Rad´o, Kneser and Choquet. This theoremwhich was first stated by Rad´o [22], proved by Kneser [16] immediately after andthen independently rediscovered by Choquet [7], remains the basic unequalled resultof invertibility for mappings solving an elliptic system of equations.Let B be an open disk in the plane, let Φ : ∂B → γ ⊂ R be a homeomorphismof ∂B onto a simple closed curve γ . Let u ∈ C ( B, R ) ∪ C ( B, R ) be the solutionto the Dirichlet problem(1.1) (cid:26) ∆ u = 0 , in B,u = Φ , on ∂B. Let D be the bounded region such that ∂D = γ . The Rad´o–Kneser–ChoquetTheorem states the following. Theorem 1.1. If D is convex, then u is a homeomorphism of B onto D . The main reasons why this theorem remains substantially unequalled are(i) no analogue of this theorem holds true in dimension three or higher, as it wasshown by a striking example by Laugesen [17], see also Melas [21],(ii) the convexity assumption on the target domain D is optimal. In fact, it isknown since Choquet [7], that if D is not convex then there exists homeomorphismsΦ : ∂B → ∂D , for which the injectivity of u , the solution to (1.1), fails. See also[2] for a thorough investigation of this issue.Nevertheless various kinds of generalizations of the Rad´o–Kneser–Choquet The-orem have been obtained. Regarding harmonic mappings between manifolds, seeSchoen and Yau [23] and Jost [15]. For mappings u whose components solve a linearelliptic equation, let us mention Bauman, Marini and Nesi [6] and also [1], [3]. It isworth pointing out that, in these last two papers the Rad´o–Kneser–Choquet The-orem has been extended to linear elliptic equations in divergence form with merelybounded measurable coefficients. For quasilinear equations of the p -Laplacian typesee [4].As a remarkable special case of our Theorem 1.4, which will be stated here below,we prove that no analogue of the Rad´o–Kneser–Choquet Theorem holds when thediagonal Laplacian system is replaced by a Lam`e system with constant moduli ofthe following form µ div(( ∇ u ) T + ∇ u ) + λ ∇ (div u ) = 0 . More precisely, we have the following.
Theorem 1.2.
Let µ, λ ∈ R with µ > and µ + λ > . There exist a disk B ⊂ R ,a bounded convex domain D ⊂ R and a smooth diffeomorphism Φ : ∂B → ∂D , sothat the unique solution u ∈ W , ( B, R ) to (cid:26) µ div(( ∇ u ) T + ∇ u ) + λ ∇ (div u ) = 0 , in B,u = Φ , on ∂B, is not a homeomorphism of B onto D . Note that the natural unknown for the Lam`e system, should be the displacement field δ , rather than the deformation field u . However, due to the fact the the identitymapping I is also a solution of the Lam`e system, we have trivially that δ solves theLam`e system if and only u = I + δ solves the same system. This is the reason whyin Theorem 1.2 it is understood that the solution u is representing a deformationfield. ATERIAL INTERPENETRATION 3
In Remark 2.3, we shall see that the solution fails to be an homeomorphism in avery strong way, in fact u maps B onto a domain larger than D , moreover it foldsitself along a curve, across which the orientation is reversed.Our next Theorem is a far reaching generalization of the previous one. It showsthat the Rad´o–Kneser–Choquet Theorem holds only if one deals with elliptic sys-tems of diagonal form with the same scalar elliptic operator on both components .We need some definitions. Consider a constant coefficients second order ellipticsystem of the form(1.2) (cid:26) div( A ∇ u + B ∇ u ) = 0 , div( C ∇ u + D ∇ u ) = 0 . where A, B, C and D are 2 × u and u are real valued functions. we say that the system (1.2) is elliptic if it satisfiesthe Legendre–Hadamard condition(1.3) η Aξ · ξ + η η ( B + C ) ξ · ξ + η Dξ · ξ > , for every ξ, η ∈ R \{ } . This condition is weaker than the strong convexity condition, namely the positivityof the 4 × M = (cid:18) A BC D (cid:19) and, as it is well known, ellipticity (1.3) is the same as rank one convexity of thequadratic form associated to M , see [9, Theorem 5.3]. We note that there is no lossof generality in assuming that the matrices A, B, C and D are symmetric. Definition 1.3.
We shall say that the system (1.2) is equivalent to the system (cid:26) div( A (cid:48) ∇ u + B (cid:48) ∇ u ) = 0 , div( C (cid:48) ∇ u + D (cid:48) ∇ u ) = 0if there exists a non-singular 2 × (cid:18) α βγ δ (cid:19) such that(1.4) (cid:18) A BC D (cid:19) = (cid:18) α Id β Id γ Id δ Id (cid:19) (cid:18) A (cid:48) B (cid:48) C (cid:48) D (cid:48) (cid:19) . Theorem 1.4.
Let the ellipticity condition (1.3) be satisfied. The following alter-native holds.Eitheri) the system (1.2) is equivalent to the system (1.5) (cid:26) div( A ∇ u ) = 0 , div( A ∇ u ) = 0 , orii) there exist a disk B ⊂ R , a bounded convex domain D ⊂ R and a smooth dif-feomorphism Φ : ∂B → ∂D , so that the unique solution u = ( u , u ) ∈ W , ( B, R ) to div( A ∇ u + B ∇ u ) = 0 , in B, div( C ∇ u + D ∇ u ) = 0 , on B,u = Φ , on ∂B. is not a homeomorphism of B onto D . GIOVANNI ALESSANDRINI AND VINCENZO NESI
The above results show that one of the most basic properties enjoyed by planarharmonic mappings cannot be extended to other elliptic systems in the plane. It isthen natural to ask similar questions for another fundamental property of injectiveharmonic mapping. A benchmark of the theory is a result of H. Lewy [18] provingthat harmonic homeomorphisms are, in fact, diffeomorphisms. More precisely wehave the following result.
Theorem 1.5. (H. Lewy.) Lut u = ( u , u ) : B → R be a harmonic mapping. If u is invertible, then (1.6) det Du (cid:54) = 0 for every ( x, y ) ∈ B. Also in this case the validity is limited to two dimensions. J. C. Wood [24] founda third degree polynomial harmonic mapping which provides a counterexample indimension three. On the positive side, Hans Lewy [19] recognized that, in threedimensions, if u is the gradient of an harmonic function and it is a homeomor-phism, then it is a diffeomorphism. This result was extended to any dimension inthe remarkable paper by Gleason and Wolff [14]. In a different direction, severalgeneralizations of Lewy’s Theorem have been achieved in dimension two when thecomponents of u satisfy the same linear elliptic equation of the form div( σ ∇ u i ) = 0.For the case of sufficiently smooth σ see [6]. When σ is allowed to be discontinuous,weak forms of Lewy’s Theorem have been obtained in [1] and [3]. A version for p -Laplacian type equations can be found in [4].In the next theorem, we show, by means of examples, that Theorem 1.5 cannotbe extended to an arbitrary elliptic system with constant coefficients, unless, again,the systems has the special form (1.5). Theorem 1.6.
Let the ellipticity condition (1.3) be satisfied. The following alter-native holds.Eitheri) the system (1.2) is equivalent to the system (1.5) orii) there exists a polynomial solution to (cid:26) div( A ∇ u + B ∇ u ) = 0 , in B, div( C ∇ u + D ∇ u ) = 0 , on B, which is a homeomorphism of a closed disk B onto u ( B ) and such that in the centerof the disk, denoted by O , we have det Du ( O ) = 0 . Remark . It may be evident that the Rad´o–Kneser–Choquet and the H. LewyTheorems continue to hold for any system of the form (1.5), since it can be elemen-tarily reduced to a Laplacian diagonal system via a linear change of the independentcoordinates. It is although rather remarkable that Theorems 1.4, 1.6 show that theRad´o–Kneser–Choquet and the H. Lewy Theorems do not extend to very slightperturbations of the Laplacian diagonal system such as, for instance, the followingone (cid:26) u xx + u yy = 0 , (1 + ε ) u xx + u yy = 0 , where ε is any positive number. ATERIAL INTERPENETRATION 5 Proofs
In what follows, when no ambiguity occurs, we shall identify points ( x, y ) ∈ R with column vectors (cid:18) xy (cid:19) . Also, for θ ∈ R , we shall denote c θ = cos θ , s θ = sin θ .For the proofs of Theorems 1.4 and 1.6 we shall make use of the following twopropositions, which we will prove at the end of this Section. Proposition 2.1.
Let the ellipticity condition (1.3) be satisfied. If the system (1.2)is not equivalent to (1.5), then there exists θ ∈ [0 , π ] and a quadratic polynomial p ( x, y ) = ( ax + 2 bxy + cy ) such that (cid:18) u u (cid:19) = (cid:18) c θ − s θ s θ c θ (cid:19) (cid:18) x + y p ( x, y ) (cid:19) is a solution to (1.2). Proposition 2.2.
Let the ellipticity condition (1.3) be satisfied. If the system (1.2)is not equivalent to (1.5), then there exists θ ∈ [0 , π ] and a cubic polynomial q ( x, y ) = 12 (cid:18) a x bx y + cxy + d y (cid:19) such that (cid:18) u u (cid:19) = (cid:18) c θ − s θ s θ c θ (cid:19) (cid:18) x ( x + y ) q ( x, y ) (cid:19) is a solution to (1.2).Proof of Theorem 1.4. We assume that (1.2) is not equivalent to (1.5). We choose θ according to Proposition 2.1 and we write R θ = (cid:18) c θ − s θ s θ c θ (cid:19) . Being linear mappings solutions to (1.2), we have that also the following is solutionto (1.2) (cid:18) u u (cid:19) = R θ (cid:18) x + y − ky + p ( x, y ) (cid:19) where k (cid:54) = 0 is a constant to be determined later on.We choose B = (cid:110) ( x, y ) ∈ R : (cid:0) x − (cid:1) + y < (cid:111) . We have x = − x + y on ∂B. We now select Φ. We set M = (cid:18) k (cid:19) , Ψ( x, y ) = (cid:18) xy + k p ( x, y ) (cid:19) , Φ( x, y ) = R θ M Ψ( x, y ) . A straightforward calculation shows that when(2.1) k ≥ | b | Ψ is a homeomorphism of ∂B onto a closed convex curve Γ. Consequently Φ is alsoa homeomorphism of ∂B onto the closed convex curve γ = R θ M Γ. GIOVANNI ALESSANDRINI AND VINCENZO NESI
Let D be the bounded convex domain such that ∂D = γ . It is easy to check that D ⊂ R θ M S where S = (cid:40) ( v , v ) ∈ R : 1 − √ ≤ v ≤ √ (cid:41) . However u (0 ,
0) = − R θ M (cid:18) (cid:19) / ∈ R θ M S. (cid:3)
Remark . Note that we can compute the the Jacobian determinant of u in termsof the coefficients of p and obtain det Du = 2 kx + 2( b ( x − y ) + ( c − a ) xy ). Hencethe Jacobian determinant vanishes at (0 ,
0) and, in fact, it changes sign across its nodal line H = (cid:8) ( x, y ) ∈ R : kx + ( b ( x − y ) + ( c − a ) xy ) = 0 (cid:9) . which is always an hyperbola (unless a = b = c = 0 when it degenerates in a straightline). Note that kx + ( b ( x − y ) + ( c − a ) xy ) is positive in (1 ,
0) zero in (0 ,
0) andnegative in ( − ,
0) because of (2.1). See Figures 1 and 2 where the behaviour of u and its Jacobian determinant are depicted in the specific case of the Lam`e system. Proof of Theorem 1.6.
We assume again that (1.2) is not equivalent to (1.5). Wechoose θ and q according to Proposition 2.2. Being linear mappings solutions to(1.2), we have that also the following is solution to (1.2) (cid:18) u u (cid:19) = (cid:18) c θ − s θ s θ c θ (cid:19) (cid:18) x ( x + y ) y + q ( x, y ) (cid:19) . It is now easy to check that, once we have chosen the polynomial q according toProposition 2.2, the following two properties hold. First there exists a positiveradius r such that one has det Du ( x, y ) > x, y ) ∈ B r ( O ) \{ O } , where we haveset O = (0 , Du ( O ) = 0. Choose 0 < ρ < r , and denote byΦ = u (cid:12)(cid:12) ∂B ρ ( O ) . A very simple calculation shows that Φ maps ∂B in a one to one way onto a closedcurve γ provided one has2 + bρ > dρ > . Let D be the bounded domain such that ∂D = γ . We can now apply a topologicalresult of Meisters and Olech [20]. Indeed we have a smooth mapping u defined ona closed disk B and which is a local homeomorphism at each point of B , with thepossible exception of the point O only. Moreover, the restriction of u to ∂B is ahomeomorphism. Therefore the hypotheses of Theorem 1 in [20] (see also Corollary2) are satisfied and we can conclude that u is a homeomorphism of B ρ ( O ) onto D . (cid:3) Proof of Theorem 1.2.
It suffices to verify that the Lam`e system is not equivalentto any elliptic system of the form (1.5). In fact it can be rewritten in the form (1.2)with the following choices A = (cid:18) µ + λ µ (cid:19) , B = C (cid:18) µ + λ µ + λ (cid:19) , D = (cid:18) µ
00 2 µ + λ (cid:19) . ATERIAL INTERPENETRATION 7
Now
B, C and D can be scalar multiples of A only if µ + λ = 0, which contradictsthe assumption µ + λ > (cid:3) Remark . Note that our assumptions µ > µ + λ > λ = µ = 1. With this choice, condition (2.1) takes the form k ≥ (1+ √ λ +3 µλ + µ and, for the picture we have chosen the limiting value k = 2(1+ √ u and the u directions are scaled differently. -0.5 0.5 1 1.5-1-0.50.51 -0.5 0.5 1 1.5-7.5-5-2.52.557.5 Figure 1. ∂B and its image Φ( ∂B ). Figure 2.
Left: circles C r of varying radii and the nodal line ofthe Jacobian (an hyperbola) drawn within B . Right: the images U ( C r ). Proof of Proposition 2.1.
We look for θ ∈ [0 , π ] and a quadratic polynomial p ( x, y ) = ( ax + 2 bxy + cy ) such that(2.2) (cid:18) u u (cid:19) = (cid:18) c θ − s θ s θ c θ (cid:19) (cid:18) x + y p ( x, y ) (cid:19) is a solution to (1.2).Since the Hessian matrices of u i are constant, the system (1.2) is equivalent tothe following two equations ( − s θ F + c θ G ) abc = Y GIOVANNI ALESSANDRINI AND VINCENZO NESI where F = (cid:18) a a a c c c (cid:19) , G = (cid:18) b b b d d d (cid:19) , and Y = (cid:18) Y Y (cid:19) is a vector of known data, possibly depending on θ . Given θ ,the above system has at least one solution if the rank of − s θ F + c θ G is two andconsequently we find a solution of the form (2.2) to (1.2). If this were not the case,then for every θ there exists φ such that(2.3) c φ ( − s θ A + c θ B ) + s φ ( − s θ C + c θ D ) = 0 . Choosing θ = 0 yields that B and D are linearly dependent. Similarly, by choosing θ = π , we get that A and C are linearly dependent. Recalling that, by ellipticity(1.3), A and D are positive definite, and hence nontrivial, we obtain that B = σD and C = γA for suitable constants σ, γ ∈ R . Plugging these linear dependenciesinto (2.3) and using once more ellipticity, we obtain that also D is a scalar multipleof A . In conclusion B, C and D are scalar multiples of A . (cid:3) Proof of Proposition 2.2.
We look for θ ∈ [0 , π ] and a cubic polynomial q ( x, y ) = 12 (cid:18) a x bx y + cxy + d y (cid:19) such that (cid:18) u u (cid:19) = (cid:18) c θ − s θ s θ c θ (cid:19) (cid:18) x ( x + y ) q ( x, y ) (cid:19) is a solution to (1.2). This time the Hessian matrices of u i are of the form xH + yH for suitable constant matrices H and H and thus (1.2) imposes the following fourconditions. ( − s θ F + c θ G ) abcd = Y where F = a a a a a a c c c c c c , G = b b b b b b d d d d d d , and Y ∈ R is a data vector. We make use of the following linear algebra fact.Given any two 2 × M = (cid:18) m m m m (cid:19) and S = (cid:18) s s s s (cid:19) , if we have det m m m m m m s s s s s s = 0 , and either M or S is positive definite, then M and S are linearly dependent. Thisfact may be verified in many ways, for instance with the aid of Gaussian elimination. ATERIAL INTERPENETRATION 9
Assume that ( − s θ F + c θ G ) is singular for every θ , then we must have that the matrix( − s θ C + c θ D ) is a scalar multiple of ( − s θ A + c θ B ) at least for all those θ for which( − s θ A + c θ B ) is positive definite. Equivalently, switching the roles of A, B and
D, C , if ( − s θ C + c θ D ) is positive definite, then ( − s θ A + c θ B ) is a scalar multiple of( − s θ C + c θ D ). Recalling that the matrices A, D are positive definite by ellipticity,and choosing θ = 0 , π , we deduce B = σD and C = γA for suitable constants σ, γ ∈ R . Moreover, it is evident that there exists an open interval I containing π for which ( − s θ A + c θ B ) remains positive definite as long as θ ∈ I . Thus, forall θ ∈ I , there exists φ such that (2.3) holds true, and from now we can arguesimilarly as in the proof of Proposition 2.1. (cid:3) References [1] G. Alessandrini and V. Nesi,
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E-mail address : [email protected] Dipartimento di Matematica, La Sapienza, Universit`a di Roma, Italy
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