An examination of the symbiotic and productive relationship between
fully nonlinear partial differential equations and generalized
potential theories In recent years, there has evolved a symbiotic
and productive relationship between fully nonlinear partial
differential equations and generalized potential theories. This
book examines important aspects of this story. One main purpose is
to prove comparison principles for nonlinear potential theories in
Euclidian spaces straightforwardly from duality and monotonicity
under the weakest possible notion of ellipticity. The book also
shows how to deduce comparison principles for nonlinear
differential operators, by marrying these two points of view, under
the correspondence principle. The authors explain that comparison
principles are fundamental in both contexts, since they imply
uniqueness for the Dirichlet problem. When combined with
appropriate boundary geometries, yielding suitable barrier
functions, they also give existence by Perron’s method. There are
many opportunities for cross-fertilization and synergy. In
potential theory, one is given a constraint set of 2-jets that
determines its subharmonic functions. The constraint set also
determines a family of compatible differential operators. Because
there are many such operators, potential theory strengthens and
simplifies the operator theory. Conversely, the set of operators
associated with the constraint can influence the potential theory.
General
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