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• W95019045 abstract "Evolution Equations on Variable Domains: an Approach by Minimizing Movements * UGO GIANAZZA GIUSEPPE SAVARE 0. Introduction In a recent work ([15]) E. De Giorgi has proposed a very general method, the so-called Minimizing Movements method, which provides a unifying framework for different problems relative to Variational Calculus, Partial Differential Equations and Geometric Measure Theory. Here we consider a problem suggested in [15] concerning a parabolic equation on a non-cilindrical domain. The Minimizing Movement’s tool leads to study a time discretization of an associated penalized equation in a fixed domain, the discretization step and the penalizing term being related to each other. We study this problem in the framework of abstract evolution equations in Hilbert spaces (see [23], [24], [3], [7], [8], [21]) so that De Giorgi’s problem will be recovered as a special case; the same abstract setting can be applied to study parabolic equations on a fixed domain but with mixed (and varying) lateral boundary conditions and parabolic variational inequalities on variable convex sets. We study the convergence properties of the approximation procedure under general assumptions on the data and on the interplay between discretization and penalization, proving weak and strong convergence results depending on the regularity of the solution of the continuous problem. New regularity results for this solution are also given, with sharp error estimates in the energy norm. The plan of the work is the following: in Section 1 we introduce the notation and state the main results with the related applications; in Section 2 we prove the basic existence and convergence results; refinement of the regularity properties of the continuous solution are given in the next Section, and in the last one we prove the stronger convergence and continuity results with the error estimates. Pervenuto alla Redazione il 27 Marzo 1995. * This work has been partially supported by M.U.R.S.T. through 60% funds and by I.A.N." @default.
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• W95019045 date "1996-01-01" @default.
• W95019045 modified "2023-09-26" @default.
• W95019045 title "Abstract evolution equations on variable domains : an approach by minimizing movements" @default.
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