Unveiling the Power of Nonlinear Dirichlet Forms
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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...
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,
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