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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: duzaar@math.fau.de
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