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2017 | OriginalPaper | Chapter

A Duality Approach in Some Boundary Value Problems

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Abstract

We describe several results from the literature concerning approximation procedures for variational boundary value problems, via duality techniques. Applications in shape optimization are also indicated. Some properties are quite unexpected and this is an argument that the present duality approach may be of interest in a large class of problems.

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Literature
1.
go back to reference Arnăutu, V., Langmach, H., Sprekels, J., Tiba, D.: On the approximation and optimization of plates. Numer. Funct. Anal. Optim. 21(3–4), 337–354 (2000)CrossRefMATHMathSciNet Arnăutu, V., Langmach, H., Sprekels, J., Tiba, D.: On the approximation and optimization of plates. Numer. Funct. Anal. Optim. 21(3–4), 337–354 (2000)CrossRefMATHMathSciNet
2.
go back to reference Barbu, V., Precupanu, T.: Convexity and optimization in Banach spaces. Sijthoff & Noerdhoff, Alphen aan de Rijn (1978)MATH Barbu, V., Precupanu, T.: Convexity and optimization in Banach spaces. Sijthoff & Noerdhoff, Alphen aan de Rijn (1978)MATH
3.
go back to reference Beirao da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual elements methods. Math. Models Methods Appl. Sci. 23, 114–214 (2013) Beirao da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual elements methods. Math. Models Methods Appl. Sci. 23, 114–214 (2013)
5.
6.
go back to reference Chenais, D., Zerner, M.: Numerical methods for elliptic boundary value problems with singular dependence on a small parameter, necessary conditions. Comput. Methods Appl. Mech. Eng. 115(1–2), 145–163 (1994)CrossRef Chenais, D., Zerner, M.: Numerical methods for elliptic boundary value problems with singular dependence on a small parameter, necessary conditions. Comput. Methods Appl. Mech. Eng. 115(1–2), 145–163 (1994)CrossRef
7.
go back to reference Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH
8.
go back to reference De Szoeke, R.A., Samelson, R.M.: The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. 32, 2194–2203 (2002)CrossRefMathSciNet De Szoeke, R.A., Samelson, R.M.: The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. 32, 2194–2203 (2002)CrossRefMathSciNet
9.
go back to reference Gao, D.Y.: Nonlinear elastic beam theory with application in contact problems and variational approaches. Mech. Res. Commun. 23, 11–17 (1996)CrossRefMATHMathSciNet Gao, D.Y.: Nonlinear elastic beam theory with application in contact problems and variational approaches. Mech. Res. Commun. 23, 11–17 (1996)CrossRefMATHMathSciNet
10.
go back to reference Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer, New York (2008)MATH Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer, New York (2008)MATH
11.
go back to reference Gomes, D.A.: Duality principles for fully nonlinear elliptic equations. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 61, pp. 125–136. Springer, Berlin (2005) Gomes, D.A.: Duality principles for fully nonlinear elliptic equations. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 61, pp. 125–136. Springer, Berlin (2005)
12.
go back to reference Harvey, F.R., Lawson, H.B. Jr.: Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J. Differ. Geom. 88, 395–482 (2011)CrossRefMATHMathSciNet Harvey, F.R., Lawson, H.B. Jr.: Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J. Differ. Geom. 88, 395–482 (2011)CrossRefMATHMathSciNet
14.
go back to reference Krell, S., Manzini, G.: The discrete duality finite volume method for the stokes equations on 3-d polyhedral meshes. SIAM J. Numer. Anal. 50(2), 808–837 (2012)CrossRefMATHMathSciNet Krell, S., Manzini, G.: The discrete duality finite volume method for the stokes equations on 3-d polyhedral meshes. SIAM J. Numer. Anal. 50(2), 808–837 (2012)CrossRefMATHMathSciNet
16.
go back to reference Merluşcă, D.: A duality algorithm for the obstacle problem. Ann. Acad. Rom. Sci. Ser. Math. Appl. 5(1–2), 209–215 (2013)MATHMathSciNet Merluşcă, D.: A duality algorithm for the obstacle problem. Ann. Acad. Rom. Sci. Ser. Math. Appl. 5(1–2), 209–215 (2013)MATHMathSciNet
17.
go back to reference Merluşcă, D.: A duality-type method for the obstacle problem. An. Şt. Univ. “Ovidius”, Constanţa 21(3), 181–195 (2013) Merluşcă, D.: A duality-type method for the obstacle problem. An. Şt. Univ. “Ovidius”, Constanţa 21(3), 181–195 (2013)
18.
go back to reference Merluşcă, D.: A duality-type method for the fourth order obstacle problem. U.P.B. Sci. Bull., Ser. A 76(2), 147–158 (2014) Merluşcă, D.: A duality-type method for the fourth order obstacle problem. U.P.B. Sci. Bull., Ser. A 76(2), 147–158 (2014)
19.
go back to reference Neittaanmäki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems. Theory and Applications. Springer, New York (2006)MATH Neittaanmäki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems. Theory and Applications. Springer, New York (2006)MATH
20.
go back to reference Nguyen, V.P., Rabczak, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008)CrossRefMATHMathSciNet Nguyen, V.P., Rabczak, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008)CrossRefMATHMathSciNet
21.
go back to reference Sprekels, J., Tiba, D.: A duality approach in the optimization of beams and plates. SICON 37(2), 486–501 (1998/1999) Sprekels, J., Tiba, D.: A duality approach in the optimization of beams and plates. SICON 37(2), 486–501 (1998/1999)
22.
go back to reference Sprekels, J., Tiba, D.: Optimal design of mechanical structures. In: Imanuvilov, O., Leugering, G., Triggiani, R., et al. (eds.) Control Theory of Partial Differential Equations. Pure and Applied Mathematics, vol. 242, pp, 259–271. Chapman and Hall/CRC, Boca Raton (2005) Sprekels, J., Tiba, D.: Optimal design of mechanical structures. In: Imanuvilov, O., Leugering, G., Triggiani, R., et al. (eds.) Control Theory of Partial Differential Equations. Pure and Applied Mathematics, vol. 242, pp, 259–271. Chapman and Hall/CRC, Boca Raton (2005)
23.
go back to reference Sprekels, J., Tiba, D.: Extensions of the control variational method. Control Cybern. 40(4), 1099–1108 (2011)MATHMathSciNet Sprekels, J., Tiba, D.: Extensions of the control variational method. Control Cybern. 40(4), 1099–1108 (2011)MATHMathSciNet
24.
go back to reference Tiba, D.: A duality approximation of some nonlinear PDE’s. Ann. Acad. Rom. Sci. Ser. Math. Appl. 8(1), 68–77 (2016)MATHMathSciNet Tiba, D.: A duality approximation of some nonlinear PDE’s. Ann. Acad. Rom. Sci. Ser. Math. Appl. 8(1), 68–77 (2016)MATHMathSciNet
Metadata
Title
A Duality Approach in Some Boundary Value Problems
Author
Dan Tiba
Copyright Year
2017
DOI
https://doi.org/10.1007/978-3-319-64489-9_20

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