A VARIATIONAL APPROACH TO NON-LINEAR PARABOLIC EQUATIONS FRANK DUZAAR Abstract. In the talk a new purely variational approach to the existence problem of evolutionary partial differential equations associated to energy functionals of the form ˆ F (v) := f (Dv)dx, Ω will be explained. The integrand f : Rn → R is only assumed to be convex, while the initial and time independent lateral boundary datum uo satisfies the classical bounded slope bounded slope condition. If the integrand f is differentiable, the approach leads to a semi-classical solution u ∈ L∞ (ΩT ) ∩ C 0 ([0, T ]; L2 (Ω)) with Du ∈ L∞ (ΩT , Rn ) of the Cauchy Dirichlet problem ( ∂t u − div Df (Du) = 0 in ΩT , u = uo on ∂P ΩT . It will also be reported on previous results concerning gradient flows associated with functionals of linear growth from image restoration from the classical Calculus of Variations. In any case the approach guarantees the existence of global parabolic minimizers, in the sense that ˆ T ˆ ˆ T u∂t ϕ dx + F (u) dt ≤ F (u + ϕ) dt, 0 Ω 0 whenever T > 0 and ϕ ∈ C0∞ (Ω × (0, T )). The presented results are obtained in collaborations with Verena B¨ ogelein (Salzburg), Paolo Marcellini (Florence) and Stefano Signoriello (Erlangen). ¨ t Erlangen–Nu ¨ rnberg, CauerFrank Duzaar, Department Mathematik, Universita strasse 11, 91058 Erlangen, Germany E-mail address: [email protected] 1
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