Calculus of Variations and Geometric Measure Theory

A. Figalli - W. Gangbo - T. Yolcu

A variational method for a class of parabolic PDEs

created by figalli on 24 Feb 2010
modified on 28 Jan 2011


Accepted Paper

Inserted: 24 feb 2010
Last Updated: 28 jan 2011

Journal: Ann. Scuola Norm. Sup. Pisa Cl. Sci.
Year: 2010


In this manuscript we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming the initial condition to be a density function, not necessarily smooth, but solely of bounded first moments and finite ``entropy'', we use a variational scheme to discretize the equation in time and construct approximate solutions. Then De Giorgi's interpolation method reveals to be a powerful tool for proving convergence of our algorithm. Finally we show uniqueness and stability in $L^1$ of our solutions.