Abstract
New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form \({-Au \in \partial\varphi(u)}\) as well ass dissipative initial value evolutions of the form \({-\dot{u}(t) - Au(t) + \omega{u}(t) \in \partial\varphi(l,u(l))}\) where \({\varphi}\) is a convex potential on an infinite dimensional space, A is a linear operator and \({\omega}\) is any scalar. The framework developed in the above mentioned paper reformulates these problems as \({0\in \bar{\partial}L(u)}\) and \({\dot{u}(t) \in \bar{\partial}L(t,u(t))}\) respectively, where \({\bar{\partial}L}\) is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term.
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Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated.
Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.
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Ghoussoub, N., Tzou, L. Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations. Annali di Matematica 187, 323–352 (2008). https://doi.org/10.1007/s10231-007-0046-1
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DOI: https://doi.org/10.1007/s10231-007-0046-1