Appl Math Optim 50:229–257 (2004)
2004 Springer Science+Business Media, Inc.
Variational Principle for General Diffusion Problems
Luca Petrelli and Adrian Tudorascu
Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213, USA
Communicated by D. Kinderlehrer
Abstract. We employ the Monge–Kantorovich mass transfer theory to study the
existence of solutions for a large class of parabolic partial differential equations.
We deal with nonhomogeneous nonlinear diffusion problems (of Fokker–Planck
type) with time-dependent coefﬁcients. This work greatly extends the applicability
of known techniques based on constructing weak solutions by approximation with
time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It
also generalizes previous results of the authors, where proofs of convergence in
the case of a right-hand side in the equation is given by these methods. To prove
the existence of weak solutions we establish an interesting maximum principle for
such equations. This involves comparison with the solution for the corresponding
homogeneous, time-independent equation.
Key Words. Anomalous diffusion, Diffusion equations, Optimal mass trans-
portation, Wasserstein distance, Discretized gradient ﬂow, Implicit schemes, Non-
homogeneous, Nonautonomous problem, Weak solution.
AMS Classiﬁcation. 35D05, 35G25, 46N10, 49J99.
In the present work we study general diffusion problems with drift and diffusion coefﬁ-
cients which may be explicitly time dependent. The boundary conditions considered here
This research was partially supported by NSF Grant DMS 0305794.