Starting from the classic contraction mapping principle, we establish a general, flexible, variational setting that turns out to be applicable to many situations of existence in Differential Equations. This unifying feature is quite appealing and motivated our analysis. We show its potentiality with some selected examples including initial-value, Cauchy problems for ODEs; non-linear, monotone PDEs; linear and non-linear hyperbolic problems; and steady Navier–Stokes systems. Though the paper has the structure of a survey, we would like to explore in the future how this perspective could help in advancing for some new situations in PDEs.
Mots clés : Variational methods, Smooth functionals, Least-squares, ODEs and PDEs
@article{AMBP_2021__28_2_231_0,
author = {Pedregal, Pablo},
title = {On a general variational framework for existence and uniqueness in {Differential} {Equations}},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {231--256},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {28},
number = {2},
year = {2021},
doi = {10.5802/ambp.405},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ambp.405/}
}
TY - JOUR AU - Pedregal, Pablo TI - On a general variational framework for existence and uniqueness in Differential Equations JO - Annales mathématiques Blaise Pascal PY - 2021 SP - 231 EP - 256 VL - 28 IS - 2 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.405/ DO - 10.5802/ambp.405 LA - en ID - AMBP_2021__28_2_231_0 ER -
%0 Journal Article %A Pedregal, Pablo %T On a general variational framework for existence and uniqueness in Differential Equations %J Annales mathématiques Blaise Pascal %D 2021 %P 231-256 %V 28 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.405/ %R 10.5802/ambp.405 %G en %F AMBP_2021__28_2_231_0
Pedregal, Pablo. On a general variational framework for existence and uniqueness in Differential Equations. Annales mathématiques Blaise Pascal, Tome 28 (2021) no. 2, pp. 231-256. doi : 10.5802/ambp.405. http://www.numdam.org/articles/10.5802/ambp.405/
[1] Theoretical numerical analysis. A functional analysis framework, Texts in Applied Mathematics, 39, Springer, 2001
[2] Least-Squares Finite Element Methods, Applied Mathematical Sciences, 166, Springer, 2009
[3] Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, 2011 | DOI
[4] Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010
[5] Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001 | DOI
[6] Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer, 1984 | DOI
[7] Nonlinear vibrations and the wave equation, SpringerBriefs in Mathematics, Springer, 2018 | DOI
[8] A fully space-time least-squares method for the unsteady Navier–Stokes system, J. Math. Fluid Mech., Volume 23 (2021) no. 4, 102, 38 pages | MR | Zbl
[9] Analysis of continuous -least-squares methods for the steady Navier–Stokes system, Appl. Math. Optim., Volume 83 (2021) no. 1, pp. 461-488 | DOI | MR | Zbl
[10] About least-squares type approach to address direct and controllability problems, Evolution equations: long time behavior and control (London Mathematical Society Lecture Note Series), Volume 439, Cambridge University Press, 2018, pp. 118-136 | MR | Zbl
[11] A variational approach to dynamical systems and its numerical simulation, Numer. Funct. Anal. Optim., Volume 31 (2010) no. 7-9, pp. 814-830 | DOI | MR | Zbl
[12] The mountain pass theorem: theme and variations, Differential equations (S ao Paulo, 1981) (Lecture Notes in Mathematics), Volume 957, Springer, 1982, pp. 237-271 | MR | Zbl
[13] The three-dimensional Navier–Stokes equations. Classical theory., Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, 2016 | DOI
[14] Nonlinear PDEs. A dynamical systems approach, Graduate Studies in Mathematics, 182, American Mathematical Society, 2017
[15] Navier–Stokes equations. Theory and numerical analysis, American Mathematical Society, 2001 (Reprint of the 1984 edition)
[16] Exploring ODEs, Other Titles in Applied Mathematics, 157, Society for Industrial and Applied Mathematics, 2018
[17] A compact course on linear PDEs, Unitext, 126, Springer, 2020 | DOI
Cité par Sources :
