Geodesic
Weighted Inertia-Dissipation-Energy Functionals for Semilinear Equations
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 1-27.

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We address a global-in-time variational approach to semilinear PDEs of either parabolic or hyperbolic type by means of the so-called Weighted Inertia-Dissipation-Energy (WIDE) functional. In particular, minimizers of the WIDE functional are proved to converge, up to subsequences, to weak solutions of the limiting PDE. This entails the possibility of reformulating the limiting differential problem in terms of convex minimization. The WIDE formalism can be used in order to discuss parameters asymptotics via $\Gamma$-convergence and is extended to some time-discrete situation as well. 
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Liero, Matthias; Stefanelli, Ulisse. Weighted Inertia-Dissipation-Energy Functionals for Semilinear Equations. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 1-27. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_1_a0/

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