Vector variational problem with knitting boundary conditions
Authors:
Graça Carita, Vladimir V. Goncharov and Georgi V. Smirnov
Journal:
Quart. Appl. Math. 75 (2017), 249-265
MSC (2010):
Primary 49J45, 74B20, 92C50
DOI:
https://doi.org/10.1090/qam/1457
Published electronically:
October 5, 2016
MathSciNet review:
3614497
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Additional Information
Abstract: We consider a variational problem with a polyconvex integrand and nonstandard boundary conditions that can be treated as minimization of the strain energy during the suturing process in plastic surgery. Existence of minimizers is proved and necessary optimality conditions are discussed.
References
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- E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), no. 3, 329–371. MR 1385074, DOI https://doi.org/10.1007/BF01235534
- Emilio Acerbi and Nicola Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), no. 2, 125–145. MR 751305, DOI https://doi.org/10.1007/BF00275731
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- John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI https://doi.org/10.1007/BF00279992
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- Pietro Celada and Gianni Dal Maso, Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 6, 661–691 (English, with English and French summaries). MR 1310627, DOI https://doi.org/10.1016/S0294-1449%2816%2930173-1
- Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
- S. Clain, G. J. Machado, R. M. S. Pereira, and G. Smirnov, Soft tissue modelling for analysis of errors in breast reduction surgery, $11^{\mathrm {th}}$ World Congress on Computational Mechanics - WCCM XI, Barcelona July 2014; 1678–1687 (2014).
- Sergio Conti and Camillo De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3, 521–549. MR 2020859
- Bernard Dacorogna, Direct methods in the calculus of variations, 2nd ed., Applied Mathematical Sciences, vol. 78, Springer, New York, 2008. MR 2361288
- Bernard Dacorogna and Paolo Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 393–396 (French, with English summary). MR 1071650
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- Peter Deuflhard, Martin Weiser, and Stefan Zachow, Mathematics in facial surgery, Notices Amer. Math. Soc. 53 (2006), no. 9, 1012–1016. MR 2256584
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- M. Focardi, N. Fusco, C. Leone, P. Marcellini, E. Mascolo, and A. Verde, Weak lower semicontinuity for polyconvex integrals in the limit case, Calc. Var. Partial Differential Equations 51 (2014), no. 1-2, 171–193. MR 3247385, DOI https://doi.org/10.1007/s00526-013-0670-0
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- A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski. MR 528295
- Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916
- D. Lopes, S. Clain, R.M.S. Pereira, G.J. Machado, G. Smirnov, and I. Vasilevsky, Numerical simulation of breast reduction with a new knitting condition, Int. J. Numer. Meth. Biomed. Engng. (2016), publ. online in Wiley Online Library, DOI: 10.1002/cnm.2796.
- Jan Malý, Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 4, 681–691. MR 1237608, DOI https://doi.org/10.1017/S0308210500030900
- Paolo Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 391–409 (English, with French summary). MR 868523
- Charles B. Morrey Jr., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53. MR 54865
- S. Müller, Tang 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), no. 2, 217–243 (English, with English and French summaries). MR 1267368, DOI https://doi.org/10.1016/S0294-1449%2816%2930193-7
- Y. Payan, Soft Tissue Biomechanical Modelling for Computer Assisted Surgery, Springer, Heidelberg, 2012.
- E. Sifakis, J. Hellrung, J. Teran, A. Oliker, and C. Cutting, Local flaps: a real-time finite element based solution to the plastic surgery defect puzzle, Studies in Health and Tech. Inform. 142, 313-318 (2009).
- G. Smirnov and V. Sá, Simulação numérica da cirurgia plástica da mama, Proc. of the CMNE/CILAMCE, Porto, 13-15 de Junho (2007).
- Yu. G. Reshetnyak, Stability theorems for mappings with bounded excursions, Siberian Math. J. 9, 499-512 (1968).
- O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, $6^{\mathrm {th}}$ edition, Elsevier, Amsterdam, 2005.
References
- Emilio Acerbi, Giuseppe Buttazzo, and Nicola Fusco, Semicontinuity and relaxation for integrals depending on vector valued functions, J. Math. Pures Appl. (9) 62 (1983), no. 4, 371–387 (1984). MR 735930
- E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), no. 3, 329–371. MR 1385074, DOI https://doi.org/10.1007/BF01235534
- Emilio Acerbi and Nicola Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), no. 2, 125–145. MR 751305, DOI https://doi.org/10.1007/BF00275731
- N. Ayache and H. Delingette (Eds.), Surgery Simulation and Soft Tissue Modelling, Lecture Notes in Computing Science, Springer, Berlin, 2003.
- John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 0475169
- J. M. Ball and F. Murat, $W^{1,p}$-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), no. 3, 225–253. MR 759098, DOI https://doi.org/10.1016/0022-1236%2884%2990041-7
- A. Cardoso, G. Coelho, H. Zenha, V. Sá, G. Smirnov, and H. Costa, Computer simulation of breast reduction surgery, Aesth. Plast Elsevier Science Ltd (March 1979). Surg., 37, 68-76 (2013).
- Pietro Celada and Gianni Dal Maso, Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 6, 661–691 (English, with English and French summaries). MR 1310627
- Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Three-dimensional elasticity, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. MR 936420
- S. Clain, G. J. Machado, R. M. S. Pereira, and G. Smirnov, Soft tissue modelling for analysis of errors in breast reduction surgery, $11^{\mathrm {th}}$ World Congress on Computational Mechanics - WCCM XI, Barcelona July 2014; 1678–1687 (2014).
- Sergio Conti and Camillo De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3, 521–549. MR 2020859
- Bernard Dacorogna, Direct methods in the calculus of variations, 2nd ed., Applied Mathematical Sciences, vol. 78, Springer, New York, 2008. MR 2361288
- Bernard Dacorogna and Paolo Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 393–396 (French, with English summary). MR 1071650
- G. Dal Maso and C. Sbordone, Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z. 218 (1995), no. 4, 603–609. MR 1326990, DOI https://doi.org/10.1007/BF02571927
- G. De Philippis, S. Di Marino, and M. Focardi, Lower semi-continuity for non-coercive polyconvex integrals in the limit case, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 2, 243–264. MR 3475296, DOI https://doi.org/10.1017/S0308210515000438
- Peter Deuflhard, Martin Weiser, and Stefan Zachow, Mathematics in facial surgery, Notices Amer. Math. Soc. 53 (2006), no. 9, 1012–1016. MR 2256584
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- M. Focardi, N. Fusco, C. Leone, P. Marcellini, E. Mascolo, and A. Verde, Weak lower semicontinuity for polyconvex integrals in the limit case, Calc. Var. Partial Differential Equations 51 (2014), no. 1-2, 171–193. MR 3247385, DOI https://doi.org/10.1007/s00526-013-0670-0
- N. Fusco and J. E. Hutchinson, A direct proof for lower semicontinuity in $L^{1}$, SIAM J. Math. Anal. 23, 1081-1098 (1992).
- A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski. MR 528295
- Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916
- D. Lopes, S. Clain, R.M.S. Pereira, G.J. Machado, G. Smirnov, and I. Vasilevsky, Numerical simulation of breast reduction with a new knitting condition, Int. J. Numer. Meth. Biomed. Engng. (2016), publ. online in Wiley Online Library, DOI: 10.1002/cnm.2796.
- Jan Malý, Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 4, 681–691. MR 1237608, DOI https://doi.org/10.1017/S0308210500030900
- Paolo Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 391–409 (English, with French summary). MR 868523
- Charles B. Morrey Jr., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53. MR 0054865
- S. Müller, Tang 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), no. 2, 217–243 (English, with English and French summaries). MR 1267368
- Y. Payan, Soft Tissue Biomechanical Modelling for Computer Assisted Surgery, Springer, Heidelberg, 2012.
- E. Sifakis, J. Hellrung, J. Teran, A. Oliker, and C. Cutting, Local flaps: a real-time finite element based solution to the plastic surgery defect puzzle, Studies in Health and Tech. Inform. 142, 313-318 (2009).
- G. Smirnov and V. Sá, Simulação numérica da cirurgia plástica da mama, Proc. of the CMNE/CILAMCE, Porto, 13-15 de Junho (2007).
- Yu. G. Reshetnyak, Stability theorems for mappings with bounded excursions, Siberian Math. J. 9, 499-512 (1968).
- O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, $6^{\mathrm {th}}$ edition, Elsevier, Amsterdam, 2005.
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Additional Information
Graça Carita
Affiliation:
CIMA, Universidade de Évora, Rua Romão Ramalho 59, 7000-671, Évora, Portugal
MR Author ID:
944039
Email:
gcarita@uevora.pt
Vladimir V. Goncharov
Affiliation:
CIMA, Universidade de Évora, Rua Romão Ramalho 59, 7000-671, Évora, Portugal – and – Institute of System Dynamics and Control Theory, RAS, ul. Lermontov 134, 664033, Irkutsk
MR Author ID:
242721
Email:
goncha@uevora.pt
Georgi V. Smirnov
Affiliation:
Universidade do Minho, Braga, Portugal
MR Author ID:
232796
Email:
smirnov@math.uminho.pt
Keywords:
Calculus of variations,
polyconvex integrand,
coercivity assumptions,
trace operator,
knitting boundary conditions
Received by editor(s):
October 1, 2015
Published electronically:
October 5, 2016
Article copyright:
© Copyright 2016
Brown University
