Unveiling the Power of Nonlinear Dirichlet Forms

Beginning in the 60s, Rockafellar and others [BR65, Mor63, Roc70a, Roc70b, RW98] introduced and studied multivalued operators and subgradients of convex functionals. In fact, it is easy to show that the subgradient ∂Eb of Eb is equal to B. Hence, there is a direct connection between Eb,B and the semigroup S generated by B, without mentioning the original bilinear form. Studying bilinear forms by studying the energy has a major advantage. While bilinear forms are always associated with linear operators, subgradients of arbitrary, not necessarily quadratic, energies are not. This approach led to a new way of investigating a large class of nonlinear problems. In the 60s and 70s Brezis, Crandall, Pazy and others developed a theory of nonlinear accretive operators and nonlinear semigroups, first on Hilbert spaces [Lio69, BP72, Kat67, Bre73] and later on also on Banach spaces [CL71, CP72]. Surprisingly this theory closely resembles the linear theory sketched previously. Among other results, they showed that a proper, convex and lower semicontinuous map E : H → (−∞, ∞] on a Hilbert space H admits a m-accretive subgradient ∂E, which in turn generates a semigroup R of Lipschitz continuous contractions such that t → Rtu0 is the unique mild solution of the abstract Cauchy problem ∂tu + ∂Eu =0,