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 & Continuous Dynamical Systems - A, 2015, 35 (6) : 2701-2710. doi: 10.3934/dcds.2015.35.2701
References:
[1]

V. Barbu, Ecuatii Diferenţiale,, Ed. Junimea, (1985).   Google Scholar

[2]

G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields,, Publ. of the Sc. Norm. Sup. 12, 12 (2009).   Google Scholar

[3]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Inventiones Mathematicae, 98 (1989), 511.  doi: 10.1007/BF01393835.  Google Scholar

[4]

K. Dobiasova, Parametrizing implicit curves,, WDS'08 Proceedings of Contributed Papers, (2008), 19.   Google Scholar

[5]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings,, Springer, (2009).  doi: 10.1007/978-0-387-87821-8.  Google Scholar

[6]

X.-S. Gao, Search methods revisited,, Mathematics mechanization and applications, (2000), 253.  doi: 10.1016/B978-012734760-8/50011-9.  Google Scholar

[7]

P. Hartman, Ordinary Differential Equations,, J. Wiley & Sons, (1964).   Google Scholar

[8]

S. G. Krantz and H. R. Parks, The Implicit Function Theorem,, Birkhäuser, (2002).  doi: 10.1007/978-1-4612-0059-8.  Google Scholar

[9]

P. Neittaanmaki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions,, Inverse Problems, 25 (2009), 1.  doi: 10.1088/0266-5611/25/5/055003.  Google Scholar

[10]

P. Neittaanmaki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications,, Springer Monographs in Mathematics. Springer, (2006).   Google Scholar

[11]

P. Neittaanmaki and D. Tiba, Fixed domain approaches in shape optimization problems,, Inverse Problems, 28 (2012), 1.  doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[12]

P. Philip and D. Tiba, A penalization and regularization technique in shape optimization,, SIAM J. Control Optim, 51 (2013), 4295.  doi: 10.1137/120892131.  Google Scholar

[13]

J. Schicho, Rational Parametrizations of Algebraic Surfaces, Thesis,, Kepler Univ. Linz, (1995).   Google Scholar

[14]

J. A. Thorpe, Elementary Topics in Differential Geometry,, Springer, (1979).   Google Scholar

[15]

D. Tiba, The implicit functions theorem and implicit parametrizations,, Ann. Acad. Rom. Sci. Ser. Math. Appl., 5 (2013), 193.   Google Scholar

[16]

D. Wang, Irreducible decomposition of algebraic varieties via characteristic set method and Gröbner basis method,, Comput. Aided Geom. Design, 9 (1992), 471.  doi: 10.1016/0167-8396(92)90045-Q.  Google Scholar

[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.  doi: 10.1007/s11786-011-0070-9.  Google Scholar

[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.  doi: 10.1016/S1631-073X(02)02426-3.  Google Scholar

show all references

References:
[1]

V. Barbu, Ecuatii Diferenţiale,, Ed. Junimea, (1985).   Google Scholar

[2]

G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields,, Publ. of the Sc. Norm. Sup. 12, 12 (2009).   Google Scholar

[3]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Inventiones Mathematicae, 98 (1989), 511.  doi: 10.1007/BF01393835.  Google Scholar

[4]

K. Dobiasova, Parametrizing implicit curves,, WDS'08 Proceedings of Contributed Papers, (2008), 19.   Google Scholar

[5]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings,, Springer, (2009).  doi: 10.1007/978-0-387-87821-8.  Google Scholar

[6]

X.-S. Gao, Search methods revisited,, Mathematics mechanization and applications, (2000), 253.  doi: 10.1016/B978-012734760-8/50011-9.  Google Scholar

[7]

P. Hartman, Ordinary Differential Equations,, J. Wiley & Sons, (1964).   Google Scholar

[8]

S. G. Krantz and H. R. Parks, The Implicit Function Theorem,, Birkhäuser, (2002).  doi: 10.1007/978-1-4612-0059-8.  Google Scholar

[9]

P. Neittaanmaki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions,, Inverse Problems, 25 (2009), 1.  doi: 10.1088/0266-5611/25/5/055003.  Google Scholar

[10]

P. Neittaanmaki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications,, Springer Monographs in Mathematics. Springer, (2006).   Google Scholar

[11]

P. Neittaanmaki and D. Tiba, Fixed domain approaches in shape optimization problems,, Inverse Problems, 28 (2012), 1.  doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[12]

P. Philip and D. Tiba, A penalization and regularization technique in shape optimization,, SIAM J. Control Optim, 51 (2013), 4295.  doi: 10.1137/120892131.  Google Scholar

[13]

J. Schicho, Rational Parametrizations of Algebraic Surfaces, Thesis,, Kepler Univ. Linz, (1995).   Google Scholar

[14]

J. A. Thorpe, Elementary Topics in Differential Geometry,, Springer, (1979).   Google Scholar

[15]

D. Tiba, The implicit functions theorem and implicit parametrizations,, Ann. Acad. Rom. Sci. Ser. Math. Appl., 5 (2013), 193.   Google Scholar

[16]

D. Wang, Irreducible decomposition of algebraic varieties via characteristic set method and Gröbner basis method,, Comput. Aided Geom. Design, 9 (1992), 471.  doi: 10.1016/0167-8396(92)90045-Q.  Google Scholar

[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.  doi: 10.1007/s11786-011-0070-9.  Google Scholar

[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.  doi: 10.1016/S1631-073X(02)02426-3.  Google Scholar

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