# American Institute of Mathematical Sciences

June  2015, 35(6): 2701-2710. doi: 10.3934/dcds.2015.35.2701

## Implicit functions and parametrizations in dimension three: Generalized solutions

 1 Institute of Mathematics, Romanian Academy, P.O.BOX 1-764, 014700 Bucharest, Romania, Romania

Received  January 2014 Revised  March 2014 Published  December 2014

We introduce a general local parametrization for the solution of the implicit equation $f(x,y,z)=0$ by using Hamiltonian systems. The approach extends previous work of the authors and is valid in the critical case as well.
Citation: Mihaela Roxana Nicolai, Dan Tiba. Implicit functions and parametrizations in dimension three: Generalized solutions. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2701-2710. doi: 10.3934/dcds.2015.35.2701
##### References:
 [1] V. Barbu, Ecuatii Diferenţiale, Ed. Junimea, Iaşi, 1985. [2] G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Publ. of the Sc. Norm. Sup. 12, Pisa, 2009. [3] R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98, (1989), 511-547. doi: 10.1007/BF01393835. [4] K. Dobiasova, Parametrizing implicit curves, WDS'08 Proceedings of Contributed Papers, MATHFYZPRESS, (2008), 19-22. [5] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, 2009. doi: 10.1007/978-0-387-87821-8. [6] X.-S. Gao, Search methods revisited, Mathematics mechanization and applications, Academic Press, San Diego, (2000), 253-271. doi: 10.1016/B978-012734760-8/50011-9. [7] P. Hartman, Ordinary Differential Equations, J. Wiley & Sons, New York, 1964. [8] S. G. Krantz and H. R. Parks, The Implicit Function Theorem, Birkhäuser, Boston, 2002. doi: 10.1007/978-1-4612-0059-8. [9] P. Neittaanmaki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions, Inverse Problems, 25 (2009), 1-18. doi: 10.1088/0266-5611/25/5/055003. [10] P. Neittaanmaki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications, Springer Monographs in Mathematics. Springer, New York, 2006. [11] P. Neittaanmaki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 1-35. doi: 10.1088/0266-5611/28/9/093001. [12] P. Philip and D. Tiba, A penalization and regularization technique in shape optimization, SIAM J. Control Optim, 51 (2013), 4295-4317. doi: 10.1137/120892131. [13] J. Schicho, Rational Parametrizations of Algebraic Surfaces, Thesis, Kepler Univ. Linz, 1995. [14] J. A. Thorpe, Elementary Topics in Differential Geometry, Springer, New York, 1979. [15] D. Tiba, The implicit functions theorem and implicit parametrizations, Ann. Acad. Rom. Sci. Ser. Math. Appl., 5 (2013), 193-208; http://www.mathematics-and-its-applications.com [16] D. Wang, Irreducible decomposition of algebraic varieties via characteristic set method and Gröbner basis method, Comput. Aided Geom. Design, 9 (1992), 471-484. doi: 10.1016/0167-8396(92)90045-Q. [17] H. Yang, B. Jüttler and L. Gonzalez-Vega, An evolution-based approach for approximate parametrization of implicitly defined curves by polynomial parametric spline curves, Math. Comp. Sci., 4 (2010), 463-479. doi: 10.1007/s11786-011-0070-9. [18] E. Zuazua, Log-Lipschitz regularity and uniqueness of the flow for a field in $[W_{loc}^{n/p+1} (R^n)]^n$, C. R. Math. Acad. Sci. Paris, 335 (2002), 17-22. doi: 10.1016/S1631-073X(02)02426-3.

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##### References:
 [1] V. Barbu, Ecuatii Diferenţiale, Ed. Junimea, Iaşi, 1985. [2] G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Publ. of the Sc. Norm. Sup. 12, Pisa, 2009. [3] R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98, (1989), 511-547. doi: 10.1007/BF01393835. [4] K. Dobiasova, Parametrizing implicit curves, WDS'08 Proceedings of Contributed Papers, MATHFYZPRESS, (2008), 19-22. [5] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, 2009. doi: 10.1007/978-0-387-87821-8. [6] X.-S. Gao, Search methods revisited, Mathematics mechanization and applications, Academic Press, San Diego, (2000), 253-271. doi: 10.1016/B978-012734760-8/50011-9. [7] P. Hartman, Ordinary Differential Equations, J. Wiley & Sons, New York, 1964. [8] S. G. Krantz and H. R. Parks, The Implicit Function Theorem, Birkhäuser, Boston, 2002. doi: 10.1007/978-1-4612-0059-8. [9] P. Neittaanmaki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions, Inverse Problems, 25 (2009), 1-18. doi: 10.1088/0266-5611/25/5/055003. [10] P. Neittaanmaki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications, Springer Monographs in Mathematics. Springer, New York, 2006. [11] P. Neittaanmaki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 1-35. doi: 10.1088/0266-5611/28/9/093001. [12] P. Philip and D. Tiba, A penalization and regularization technique in shape optimization, SIAM J. Control Optim, 51 (2013), 4295-4317. doi: 10.1137/120892131. [13] J. Schicho, Rational Parametrizations of Algebraic Surfaces, Thesis, Kepler Univ. Linz, 1995. [14] J. A. Thorpe, Elementary Topics in Differential Geometry, Springer, New York, 1979. [15] D. Tiba, The implicit functions theorem and implicit parametrizations, Ann. Acad. Rom. Sci. Ser. Math. Appl., 5 (2013), 193-208; http://www.mathematics-and-its-applications.com [16] D. Wang, Irreducible decomposition of algebraic varieties via characteristic set method and Gröbner basis method, Comput. Aided Geom. Design, 9 (1992), 471-484. doi: 10.1016/0167-8396(92)90045-Q. [17] H. Yang, B. Jüttler and L. Gonzalez-Vega, An evolution-based approach for approximate parametrization of implicitly defined curves by polynomial parametric spline curves, Math. Comp. Sci., 4 (2010), 463-479. doi: 10.1007/s11786-011-0070-9. [18] E. Zuazua, Log-Lipschitz regularity and uniqueness of the flow for a field in $[W_{loc}^{n/p+1} (R^n)]^n$, C. R. Math. Acad. Sci. Paris, 335 (2002), 17-22. doi: 10.1016/S1631-073X(02)02426-3.
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