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September 2018, 38(9): 4509-4536. doi: 10.3934/dcds.2018197

On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile

Received  October 2017 Published  June 2018

Let $\Omega $ be a smooth bounded axisymmetric set in $\mathbb{R}^3$. In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis in [12,Section 6], a phenomenon of concentration of energy can occur preventing the strong convergence in $W^{1,2}(\Omega ,\mathbb{R}^3)$ of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where $\Omega $ is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are $W^{1,2}$-quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for 3D neo-Hookean materials.

Citation: Duvan Henao, Rémy Rodiac. On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4509-4536. doi: 10.3934/dcds.2018197
References:
[1]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1976/77), 337-4.3. doi: 10.1007/BF00279992.

[2]

J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306 (1982), 557-611. doi: 10.1098/rsta.1982.0095.

[3]

J. M. Ball, Minimizers and the Euler-Lagrange equations, In Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), 195 of Lecture Notes in Phys., pages 1-4. Springer, Berlin, 1984. doi: 10.1007/3-540-12916-2_47.

[4]

J. M. Ball, Progress and puzzles in nonlinear elasticity, In Poly-, Quasi- and Rank-one Convexity in Applied Mechanics, 516 of the CISM International Centre for Mechanical Sciences book series, pages 1-15. Springer, Vienna, 2010. doi: 10.1007/978-3-7091-0174-2_1.

[5]

J. M. Ball and F. Murat, $W^1,p$-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.

[6]

P. BaumanN. C. Owen and D. Phillips, Maximum principles and a priori estimates for a class of problems from nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 119-157. doi: 10.1016/S0294-1449(16)30269-4.

[7]

H. BrezisJ.-M. Coron and E. H. Lieb, Harmonic maps with defects, Comm. Math. Phys., 107 (1986), 649-705. doi: 10.1007/BF01205490.

[8]

H. Brezis and L. Nirenberg, Degree theory and BMO. Ⅱ. Compact manifolds with boundaries, Selecta Math. (N. S.), 2 (1996), 309-368. With an appendix by the authors and Petru Mironescu. doi: 10.1007/BF01587948.

[9]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.

[10]

G.-Q. ChenM. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math., 62 (2009), 242-304. doi: 10.1002/cpa.20262.

[11]

R. CoifmanP.-L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), 72 (1993), 247-286.

[12]

S. Conti and C. De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2 (2003), 521-549.

[13]

C. De Lellis and F. Ghiraldin, An extension of the identity $\textbf{Det} = \textbf{det}$, C. R. Math. Acad. Sci. Paris, 348 (2010), 973-976. doi: 10.1016/j.crma.2010.07.019.

[14]

J. Dieudonné, Treatise on Analysis. Vol. III, Academic Press, New York-London, 1972. Translated from the French by I. G. MacDonald, Pure and Applied Mathematics, Vol. 10-Ⅲ.

[15]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

[16]

M. GiaquintaG. Modica and J. Souček, Cartesian currents and variational problems for mappings into spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 393-485 (1990).

[17]

M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations. I, volume 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998. Cartesian currents. doi: 10.1007/978-3-662-06218-0.

[18]

M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations. II, volume 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998. Variational integrals. doi: 10.1007/978-3-662-06218-0.

[19]

R. HardtF.-H. Lin and C.-C. Poon, Axially symmetric harmonic maps minimizing a relaxed energy, Comm. Pure Appl. Math., 45 (1992), 417-459. doi: 10.1002/cpa.3160450404.

[20]

D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal., 197 (2010), 619-655. doi: 10.1007/s00205-009-0271-4.

[21]

D. Henao and C. Mora-Corral, Lusin's condition and the distributional determinant for deformations with finite energy, Adv. Calc. Var., 5 (2012), 355-409. doi: 10.1515/acv.2011.016.

[22]

F.-H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps, Ann. of Math. (2), 149 (1999), 785-829. doi: 10.2307/121073.

[23]

F.-H. Lin and T. Rivière, Energy quantization for harmonic maps, Duke Math. J., 111 (2002), 177-193. doi: 10.1215/S0012-7094-02-11116-8.

[24]

L. Martinazzi, A note on $n$-axially symmetric harmonic maps from $B^3$ to $S^2$ minimizing the relaxed energy, J. Funct. Anal., 261 (2011), 3099-3117. doi: 10.1016/j.jfa.2011.07.022.

[25]

D. Mucci, A variational problem involving the distributional determinant, Riv. Math. Univ. Parma (N.S.), 1 (2010), 321-345.

[26]

S. Müller, Higher integrability of determinants and weak convergence in $L^1$, J. Reine Angew. Math., 412 (1990), 20-34. doi: 10.1515/crll.1990.412.20.

[27]

S. Müller, Notes for the lectures partial differential equations and modelling, http://bolzano.iam.uni-bonn.de/~zwicknagl/pdem, 2013.

[28]

S. MüllerT. Qi and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 217-243. doi: 10.1016/S0294-1449(16)30193-7.

[29]

S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal., 131 (1995), 1-66. doi: 10.1007/BF00386070.

[30]

J. Sivaloganathan and S. J. Spector, On the existence of minimizers with prescribed singular points in nonlinear elasticity, J. Elasticity, 59 (2000), 83-113. doi: 10.1023/A:1011001113641.

[31]

M. Struwe, Variational Methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, 2008. Applications to nonlinear partial differential equations and Hamiltonian systems.

[32]

L. G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, USA, 1975.

show all references

References:
[1]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1976/77), 337-4.3. doi: 10.1007/BF00279992.

[2]

J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306 (1982), 557-611. doi: 10.1098/rsta.1982.0095.

[3]

J. M. Ball, Minimizers and the Euler-Lagrange equations, In Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), 195 of Lecture Notes in Phys., pages 1-4. Springer, Berlin, 1984. doi: 10.1007/3-540-12916-2_47.

[4]

J. M. Ball, Progress and puzzles in nonlinear elasticity, In Poly-, Quasi- and Rank-one Convexity in Applied Mechanics, 516 of the CISM International Centre for Mechanical Sciences book series, pages 1-15. Springer, Vienna, 2010. doi: 10.1007/978-3-7091-0174-2_1.

[5]

J. M. Ball and F. Murat, $W^1,p$-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.

[6]

P. BaumanN. C. Owen and D. Phillips, Maximum principles and a priori estimates for a class of problems from nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 119-157. doi: 10.1016/S0294-1449(16)30269-4.

[7]

H. BrezisJ.-M. Coron and E. H. Lieb, Harmonic maps with defects, Comm. Math. Phys., 107 (1986), 649-705. doi: 10.1007/BF01205490.

[8]

H. Brezis and L. Nirenberg, Degree theory and BMO. Ⅱ. Compact manifolds with boundaries, Selecta Math. (N. S.), 2 (1996), 309-368. With an appendix by the authors and Petru Mironescu. doi: 10.1007/BF01587948.

[9]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.

[10]

G.-Q. ChenM. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math., 62 (2009), 242-304. doi: 10.1002/cpa.20262.

[11]

R. CoifmanP.-L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), 72 (1993), 247-286.

[12]

S. Conti and C. De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2 (2003), 521-549.

[13]

C. De Lellis and F. Ghiraldin, An extension of the identity $\textbf{Det} = \textbf{det}$, C. R. Math. Acad. Sci. Paris, 348 (2010), 973-976. doi: 10.1016/j.crma.2010.07.019.

[14]

J. Dieudonné, Treatise on Analysis. Vol. III, Academic Press, New York-London, 1972. Translated from the French by I. G. MacDonald, Pure and Applied Mathematics, Vol. 10-Ⅲ.

[15]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

[16]

M. GiaquintaG. Modica and J. Souček, Cartesian currents and variational problems for mappings into spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 393-485 (1990).

[17]

M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations. I, volume 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998. Cartesian currents. doi: 10.1007/978-3-662-06218-0.

[18]

M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations. II, volume 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998. Variational integrals. doi: 10.1007/978-3-662-06218-0.

[19]

R. HardtF.-H. Lin and C.-C. Poon, Axially symmetric harmonic maps minimizing a relaxed energy, Comm. Pure Appl. Math., 45 (1992), 417-459. doi: 10.1002/cpa.3160450404.

[20]

D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal., 197 (2010), 619-655. doi: 10.1007/s00205-009-0271-4.

[21]

D. Henao and C. Mora-Corral, Lusin's condition and the distributional determinant for deformations with finite energy, Adv. Calc. Var., 5 (2012), 355-409. doi: 10.1515/acv.2011.016.

[22]

F.-H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps, Ann. of Math. (2), 149 (1999), 785-829. doi: 10.2307/121073.

[23]

F.-H. Lin and T. Rivière, Energy quantization for harmonic maps, Duke Math. J., 111 (2002), 177-193. doi: 10.1215/S0012-7094-02-11116-8.

[24]

L. Martinazzi, A note on $n$-axially symmetric harmonic maps from $B^3$ to $S^2$ minimizing the relaxed energy, J. Funct. Anal., 261 (2011), 3099-3117. doi: 10.1016/j.jfa.2011.07.022.

[25]

D. Mucci, A variational problem involving the distributional determinant, Riv. Math. Univ. Parma (N.S.), 1 (2010), 321-345.

[26]

S. Müller, Higher integrability of determinants and weak convergence in $L^1$, J. Reine Angew. Math., 412 (1990), 20-34. doi: 10.1515/crll.1990.412.20.

[27]

S. Müller, Notes for the lectures partial differential equations and modelling, http://bolzano.iam.uni-bonn.de/~zwicknagl/pdem, 2013.

[28]

S. MüllerT. Qi and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 217-243. doi: 10.1016/S0294-1449(16)30193-7.

[29]

S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal., 131 (1995), 1-66. doi: 10.1007/BF00386070.

[30]

J. Sivaloganathan and S. J. Spector, On the existence of minimizers with prescribed singular points in nonlinear elasticity, J. Elasticity, 59 (2000), 83-113. doi: 10.1023/A:1011001113641.

[31]

M. Struwe, Variational Methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, 2008. Applications to nonlinear partial differential equations and Hamiltonian systems.

[32]

L. G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, USA, 1975.

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