Konečný, Milan

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Remarks to weakly continuous inverse operators and an application in hyperelasticity

Institute of Mathematics AS CR, v. v. i.

Acta Mathematica et Informatica Universitatis OstraviensisMilan KonečnýRemarks to weakly continuous inverse operators and an application in hyperelasticityActa Mathematica et Informatica Universitatis Ostraviensis, Vol. 7 (1999), No. 1, 39--46Persistent URL: http://dml.cz/dmlcz/120546Terms of use:© University of Ostrava, 1999Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czActa Mathematica et Informatica Universitatis Ostraviensis 7 (1999) 39-46 39 Remarks to weakly continuous inverse operators and an application in hyperelasticity Milan Konecny Abstract: In this paper we study weakly continuous inverse operators and application in nonlinear elasticity. The first part studies properties of weakly continuous operators. The main result is theorem with conditions for weakly continuity of inverse nonlinear operator. The second part contains results, which can be applied to nonlinear differential operator of hyperelasticity. Key Words: nonlinear functional analysis, variational calculus, optimization, weakly con-tinuous operators, nonlinear elasticity, hyperelasticity. Mathematics Subject Classification: 47H30 1. Introduc t ion The paper deals with weak continuity of inverse nonlinear operator in reflexive Banach spaces. This property is very useful in optimal control. Continous linear operator in Banach spaces is weakly continous, but there are some problems with weak continuity of continuous nonlinear operator. In section 2, we introduce basic notions for spaces and continuity - especially SC-locally weak continuity in x0 and we study basic properties weak and SC - locally weak continuity. We formulate conditions, which imply the main result - SC-locally weak continuity for nonlinear invers operator A~l in reflexive Banach space. Section 3 contains an application in mechanics of continua in hyperelasticity. 2. Abstrac t results 2.1. Notation Let X, Y be Banach spaces and X* ,Y* theirs duals. Let Bx(xQ) — {x e X, \\x -— x0\\x < r} and Bx(xQ) be its closure in X. Let D be a locally convex topological vector space and D* its dual with "weak - star" topology. Let A : X —> Y be an operator acting from the space X to Y. We shall denote the strong convergence in X (\\un ~~ u\\x ""* 0) by un ""* u m -̂*i w e a ^ convergence in X((b, un ~ u) —> 0, Vb E £ X*) by un —- u in X and weak* convergence in D*((u*n — u*, b) -» 0, Vb £ D) by u*~^u* in D* . 40 Milan Konecny Definition, (various types of continuity of operators) Operator A : X -> Y is - continuous iff xn —> x in X => Axn —> Ax in Y , - strongly continuous iff x n —- x in K => Axn —> Ax in y , - weakly continuous iff x n —-  x in X => Axn —- Ax in y , - SC locally weakly continuous in xQ £ K iff there exist r, s > 0 such that holds and V{x n } n l 1 C B ^ ( x J (xn - x in X -» Axn - ylz in Y ) , - demicontinuous iff x n —> x in K => Axn —- Ax in y . Futher operator A : X —> £>* is - (S,HJ*) continuous iff x n -> x in K => Axn--^Ax in D* , - (ut,ut*) continuous iff x n —- x in K ---> Axn---*Ax in D* . Remark. It would be more precise to speak about sequentially weakly continuos operators instead of weakly continues ones. 2.2. Weakly and SC-locally weakly cont inuous operators In this section we shall formulate basic properties of weakly and SC-locally weakly continuous operators x0 . Lemma (Properties of weakly continuous operators) (i) The set of weakly continuous operators A : X —> Y forms a linear space, i.e. if AX,A2 are weakly contiuous and c € R then A1 + A2 and cAl are also weakly continuous, (ii) Linear continuous operators A : X -> Y are weakly continuous. Lemma (Properties SC-locally weakly continuous operators in point x0) The set of SC-locally weakly continuous operators in x0 A : X -> Y forms a linear space, i.e. if AX,A2 are SC-locally weakly continuous in x0 and c G R then Al -f- A2 and cAx are also SC-locally weakly continuous in x0 . Remarks to weakly continuous inverse operators in reflexive Banach spaces . . . 41 2.3. SC-locally weak continuity of inverse operators We shall deal with SC -locally weak continuity of locally inverse operator to A in 0 y . A~l : Dom[A-1] C Y -> X, i.e. with the following property: Property LWCIO: (SC-locally weakly continuous inverse operator in 0Y) There exist r, s > 0 such that A'1 : BrY(0) C Y -> Bsx(0) C X is a weakly continuous operator i.e. V{yn}~= 1 C BrY(0) (yn - y in Y => A "1 ^ - A~ly in X) Assumptions: G-condition — general condition for spaces and operators. Let X, Y be reflexive Banach spaces and L, C be Banach spaces, D locally convex topologic vector space and D* dual to D with "weak - star" topology and Y «-> L be a continuous imbedding. Let us consider operators: p Л F Dom[A) C X -> Y,0X € Dom[.4], A ( 0 J = 0y C->L>* L->L>* C - condition — continuity of operators i) operator P : X -> C is strongly continuous i.e. z n - - a: in X => F(.xn) -> P(x) in C ii) operator .A : C —> L>* is (s,w*) - continuous i. e. un -> u in C => i4(izn)-^i4(u) in L>* iii) operator F : L —•> D* is (tD,Hj*) continuous 2/ n - - t / in L^F(Hn)-F(H) in D* EUS - condition — existence and uniqueness of solution to Ax = y 3r, s > 0 \/ye BTY (0) 3\x e Bsx (0) (Ax = y) EGE - condition — existence of generalized equation Ax = y in Y => A(Px) = F(H) in L>* RE - condition — regularity equation \/yЄBrү(0) \/xЄBsx(0) (A(Px) = F(y) in D* => (Ax = y) in Y) 42 Milan Konečný Theorem 1. (SC-locally weak continuity of A - 1 na -By (0)) Let G, C, EUS, EGE, RE — condition be satisfied. Then the operator A~l SC-locally weakly continuous in 0 y i.e. there exist r, s > 0 such that A'1 :Brү(0)cY^Bx{0)cX and V{3/„}~=1 C BrY(0) ( j , n - - y in Y =• A~lyn - A~ly in X ) . Proof: Let r, 5 > 0 be constants from EUS — the condition which implies existence of a locally inverse operator A~l A~l : BrY(0) C Y -> B£(0) C X. Now we will show that A~l is locally weakly continuous in 0 . Let {yn}n<L1 C C L?y(0) and yn —- H in F . Hence V is a reflexive Banach space and -By(O) C Y is a convex and closed set,which means also weakly closed set in Y. This implies that y G ££ (0 ) C Y. If xn = -4~1yn, then x n G B£(0) C X , (Ar n = yn). As K is a reflexive and H^(0) C X is weakly compact, then there exist a subsequence {xn.}<j±l such that xn. - - x £ B^(0) in K. Now we show that A~ly = ;r, (Ax = = y) and all subsequences {xn.}^Ll weakly converge to x in X. If Axn. = Hn for all j e N,then EGE - condition implies that A(Pxn.) = F(yn.). If xn, —-—- x e B^(0) in X and P is strongly continuous (C - condition ) implies that B(xn.) -> P(x) in C. Hyn.-xy'mY and Y is continuous imbedding to L, it implies yn. —- y in L. From C - condition implies that F(yn.)^F(y) in D*. The equations A((Pxn.) = F(yn.) and C - condition implies that A(Px) = F(y) in D*. RR- condition and y € -B^(0) C F and x G B^(0) in X implies that Ax = y in Y and A"1!/ = x. By contradiction we prove that all subsequences {xn.}fl1 converge to x. Let xn. —- x ^ x in X, then we can prove analogously as before, that Ax = u and EUS condition implies that x = x, which is a contradiction. rj 3. Application in hyperelasticity In the last section we use the main result to the proof the SC- weak continuity of inverse operator of hyperelasticity. In the first part of the section we will formulate G, C, EUS, EGE, RE conditions in hyperelasticity and sketch proofs. At the end we formulate Theorem 2 about SC - locally weak continuity for inverse operator to the hyperelasticity operator. Remarks to weakly continuous inverse operators in reflexive Banach spaces . . . 43 3.1 . Assumptions Let p > 3, n C R 3 , 3 n £ C 2 , S3 - be a space symmetri matrices of the type 3 x 3 G - condition - general condition for spaces and operators. Let x = VP = w2^(n)nwj'p(n) = (w2*(n)fn(w^p(n))\ Y - Lp(n) = = (L p (n) )3 , C = C ( n ) = (C7(n))3x3, L = L x (n) = (L^ft))3. All spaces have standard norms. D = V = (C^(Vt)f , F>* = D* = ( (Q° (n ) ) 3 )* . The topology is a product of standard topology of Co°(n). [1] Assumptions for hyperelasticity operator: Body is represented by a three dimen-sional bounded domain l l c R 3 , with the boundary dfl G C2. We assume that the material is homogeneous and isotropic and the reference configuration is a natural state. For modeling of the hyperelasticity operator we will use Green-Saint-Venant strain tensor E and second Piola-Kirchhoff stress tensor E. The relation between Green-Saint-Venant strain tensor E and displacement u is E(u) = ^ ( V u T + Vu + V u T • Vu) , (1) The response function £ = £(E), (2) which is a relation between Green-Saint-Venant strain tensor E and second Piola-Kirchhoff stress tensor E, has the follouing properties: i) The response function 27: W(0) C S 3 - > S 3 , W(0) is an open neighbourhood of zero in S3 . ii) The relation between stress and strain tensors is E = E(E) = X(trE)I + 2fiE + 0 ( | | E | | | 3 ) , (3) where A > 0, /i > 0 are Lame constants. iii) The body is a natural state, which means that 0 = E(E(0)). iv) The response function is twice continuously differentiable, i.e. E G C2. A is a hyperelasticity operator A : Dom[A] C V P 4 L p (n ) A : u 4 -d iv{( I + V u ) £ ( E ( u ) ) } P is a projection operator P : Vp -> c(n) P : U K V U A - operator of a generalized equation 44 Milan Konečný Á : C(fi) -> V A(v) : X> -> R A(v):(Dh-> /" ( I - f v ) Ť ( S ( v ) ) : V(Dci/i J n S : C(fi) -> C(H),S : VH-> I ( vT + v + v T -v) E(u) = S(Vu) F : L -> 79* F(y) : P -> R F(y) : (D K-> / y • (DO!/i I n C - condition — continuity of operators i) Operator P = idoV , where V : V p -> ( W ^ r T l ) ) 3 * 3 , V : u -> Vu is linearly continuous and weakly continuous , id: (W1,p(fl)) —> ( C ( n ) ) 3 x 3 is strongly continuous from Rellich - Kondrasov embedding theorem and that implies P is strongly continuous. ii) u n -> u in ( C ( n ) )3 * 3 => S ( u n ) -> S(u) in (C( f i ))3 * 3 => => £ ( S ( u J ) -> £ (S(u ) ) in (C(f t ) ) 3 x 3 => => (I + u j £ ( S ( u J ) -> (I + u )£ (S (u ) ) in (CjJT))3*3 => => (I + u n ) r ( S ( u n ) ) 4 ( I + u)JC(S(u)) in H => => Vy> e £>( / (I + u j £ ( S ( u J ) : V^d/ i -> Jn -> / (I + u ) £ ( S ( u ) ) : V^d**)-* j n => A ( u J - A ( u ) in 19* iii) Yn ~" y ^ L => Vip € - 0 / ^ • <pd[x -> / n y • <pdfj. => F ( y n ) A F ( y ) in D* H FUS - condition — existence and uniqueness of solutions to Ax — y. The proof of this condition follows from local inversion theorem, and it can be found it in [1]. Under smoothness assumptions on response function £ , it can be shown that the nonlinear operator A : Dom[A] C V p —> L (fi) and A is continously Remarks to weakly continuous inverse operators in reflexive Banach spaces . . . 45 differentiable . The key result there is that Sobolev spaces WliP(fl) forms an algebra for p > 3 and continuously differentciable response function E maps W 1 , p ( f t ) to W 1 , p ( f t ) . The assumption that the reference configuration is in natural state implies that A(0X) = 0y . The equation A'(Ox)u = / is precisely boundary value problem of linearized elasticity that implies that A'(0X) is an isomorphism between the spaces V p and L (ft) . All assumptions of local inversion theorem are fulfilled and this implies existence of continuous locally inverse operator. EGE - condition — existence of generalized equation. It is the application of Green's formulas. RE - condition — regularity equation. u G W2 'p(f t ) => Vu G W1-P(ft) Vu G W x ' p(f t ) and E(E(u)) G W1 'p(f t) => (I + Vu)27(E(u)) G W 1 , p ( f t ) We can use Green's formulas and basic properties of distributions which imply RE - condition. 3.2. The main result. Now we can formulate the main result in application. Theorem 2. The inverse hyperelasticity operator is SC-locally weakly continuous in 0y Proof: It is the application of Theorem 1. Remarks. This property is very useful in optimal control. Acknowledgment. I would like to thank to MUDr. Lenka Karpetova for her motivation to finish this paper. References [1] Ciarlet, P. G. (1988), Mathematical Elasticity, Vol I: Three Dimensional Elastic-ity, Studies in Mathematics and its Applications 20, North Holland, Amsterdam. [2] Francu, J. (1990), Monotone operators, Appl. Math., Praha. [3] Francu, J. (1994), Weakly continuous operators.Aplication to differential equa-tions., Applications of mathematics, CSAV Praha [4] Giaquinta, M., Modica, G. and Soucek, J.(1998), Cartesian Currents in the Calculus of Variations I, Springer, Berlin [5] Giaquinta, M., Modica, G. and Soucek, J.(1998), Cartesian Currents in the Calculus of Variations II, Springer, Berlin [6] Necas, J. (1983), Introduction to the Theory of Nonlinear Elliptic Equations, TEUBNER-TEXTE zur Mathematik, B. G. Teubner Verlagsgesellschaft, Leipzig. [7] Zeidler, E. (1986), Nonlinear Functional Analysis and its Applications, Springer-Verlag, New York. 46 Milan Konecny [8] Zeidler, E. (1995), Applied Functional analysis - Applications to Mathematical Physics, Springer-Verlag, New York. [9] Zeidler, E. (1995), Applied Functional analysis - Main Principles and Their Applications, Springer-Verlag, New York. Author's address: Milan Konecny, Department of Mathematics, University of Ostrava, Brafova 7, Czech Republic, E-mail: konecnyOosu.cz Received: August 1, 1999 

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Remarks to weakly continuous inverse operators and an application in hyperelasticity

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