×

Anti-self-dual Lagrangians: variational resolutions of non-self-adjoint equations and dissipative evolutions. (Lagrangiens anti-autoduaux : Résolution variationnelle d’équations non-autoadjointes et de systèmes d’évolution dissipatifs.) (English) Zbl 1170.35308

Summary: We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional \(I\), but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler-Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.

MSC:

35A15 Variational methods applied to PDEs
35F10 Initial value problems for linear first-order PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
58E30 Variational principles in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] Auchmuty, G., Saddle points and existence-uniqueness for evolution equations, Differential Integral Equations, 6, 1161-1171 (1993) · Zbl 0813.35026
[2] Auchmuty, G., Variational principles for operator equations and initial value problems, Nonlinear Analysis, Theory, Methods and Applications, 12, 5, 531-564 (1988) · Zbl 0658.47016
[3] Barbu, V., Optimal Control of Variational Inequalities, Research Notes in Mathematics, vol. 100 (1984), Pitman · Zbl 0574.49005
[4] Bardos, C., Problèmes aux limites pour les equations aux dérivées partielles du premier ordre a coefficients réels ; Théorèmes d’approximation ; Application à l’équation de transport, Ann. Sci. École Norm. Sup. (4), 3, 185-233 (1970) · Zbl 0202.36903
[5] Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Preprint, 2004; Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Preprint, 2004 · Zbl 1107.74028
[6] Brezis, H., Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam-London · Zbl 0252.47055
[7] Brezis, H.; Ekeland, I., Un principe variationnel associé à certaines equations paraboliques. Le cas independant du temps, C. R. Acad. Sci. Paris Sér. A, 282, 971-974 (1976) · Zbl 0332.49032
[8] Brezis, H.; Ekeland, I., Un principe variationnel associé à certaines equations paraboliques. Le cas dependant du temps, C. R. Acad. Sci. Paris Sér. A, 282, 1197-1198 (1976) · Zbl 0334.35040
[9] Ghoussoub, N., A variational principle for non-linear transport equations, Comm. Pure Appl. Anal., 4, 4, 735-742 (2005) · Zbl 1089.35014
[10] N. Ghoussoub, Anti-selfdual Hamiltonians: Variational resolution for Navier-Stokes equations and other nonlinear evolutions, Comm. Pure Applied Math. (2005) 25 pp., in press; N. Ghoussoub, Anti-selfdual Hamiltonians: Variational resolution for Navier-Stokes equations and other nonlinear evolutions, Comm. Pure Applied Math. (2005) 25 pp., in press
[11] N. Ghoussoub, A class of selfdual partial differential equations and its variational principles (2005), in preparation; N. Ghoussoub, A class of selfdual partial differential equations and its variational principles (2005), in preparation
[12] Ghoussoub, N.; McCann, R., A least action principle for steepest descent in a non-convex landscape, Contemp. Math., 362, 177-187 (2004) · Zbl 1084.37060
[13] N. Ghoussoub, A. Moameni, On the existence of Hamiltonian paths connecting Lagrangian submanifolds (2005), submitted for publication; N. Ghoussoub, A. Moameni, On the existence of Hamiltonian paths connecting Lagrangian submanifolds (2005), submitted for publication · Zbl 1192.37089
[14] N. Ghoussoub, A. Moameni, Selfdual variational principles for periodic solutions of Hamiltonian and other dynamical systems, Comm. Partial Differential Equations (2006), in press; N. Ghoussoub, A. Moameni, Selfdual variational principles for periodic solutions of Hamiltonian and other dynamical systems, Comm. Partial Differential Equations (2006), in press · Zbl 1130.35008
[15] N. Ghoussoub, A. Moameni, Selfduality and periodic solutions of certain Schrödinger equations and infinite dimensional Hamiltonian systems (2006), in preparation; N. Ghoussoub, A. Moameni, Selfduality and periodic solutions of certain Schrödinger equations and infinite dimensional Hamiltonian systems (2006), in preparation
[16] Ghoussoub, N.; Tzou, L., A variational principle for gradient flows, Math. Ann., 30, 3, 519-549 (2004) · Zbl 1062.35008
[17] Ghoussoub, N.; Tzou, L., Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations, Ann. Mat. Pura Appl., 30 (2005), pp
[18] N. Ghoussoub, L. Tzou, Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows, Calc. Var. Partial Differential Equations (2006) 28 pp., in press; N. Ghoussoub, L. Tzou, Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows, Calc. Var. Partial Differential Equations (2006) 28 pp., in press · Zbl 1134.49029
[19] Jost, J., Riemannian Geometry and Geometric Analysis (2002), Springer University Text · Zbl 1034.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.