Geodesic
Parabolic equations and the bounded slope condition
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 355-379

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In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations

{∂tu−divDf(Du)=0in ΩT,u=uoon ∂PΩT.
The only assumptions needed are the convexity of the generating function f:Rn→R, and the classical bounded slope condition on the initial and the lateral boundary datum uo∈W1,∞(Ω). We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – uo could be any Lipschitz initial and boundary datum, vanishing at the boundary ∂Ω, and the boundary may contain flat parts, for instance Ω could be a rectangle in Rn.

DOI : 10.1016/j.anihpc.2015.12.005
Classification : 35A01, 35K61, 35K86, 49J40
Keywords: Existence, Parabolic equations, Bounded slope condition, Lipschitz solutions 
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     author = {B\"ogelein, Verena and Duzaar, Frank and Marcellini, Paolo and Signoriello, Stefano},
     title = {Parabolic equations and the bounded slope condition},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {355--379},
     publisher = {Elsevier},
     volume = {34},
     number = {2},
     year = {2017},
     doi = {10.1016/j.anihpc.2015.12.005},
     mrnumber = {3610936},
     zbl = {1372.35161},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.12.005/}
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Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo; Signoriello, Stefano. Parabolic equations and the bounded slope condition. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 355-379. doi: 10.1016/j.anihpc.2015.12.005

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