We shall consider weak solutions of nonlinear elliptic boundary value problems and initial-boundary value problems for semilinear and nonlinear parabolic differential equations with certain nonlocal terms.  We shall prove theorems on the number of solutions and find multiple solutions. These statements are based on arguments for fixed points of some real functions and operators, respectively, and existence-uniqueness theorems on partial differential equations (without functional terms).

The logarithmic norm was introduced in 1958 for matrices, and for the purpose of estimating growth rates in initial value problems. Since then, the concept has been extended to nonlinear maps, differential operators and function spaces. There are applications in operator equations in general, including evolution equations as well as boundary value problems. The logarithmic norm is the extremal value of a quadratic form.

In this talk we outline how logarithmic norms of differential operators can be computed, and how they are related to variational calculus and ellipticity. Thus, while one typically seeks the minimizing function in a variational problem, the logarithmic norm is the corresponding extremal value of the functional associated with a particular symmetrized differential operator. There are also connections to eigenvalue problems for selfadjoint and non-selfadjoint operators. This will also be illustrated with an application to classical singular problems, such as the Bessel equation, as well as to the biharmonic operator.