  We are concerned with the Dirichlet energy of mappings defined on domains in
the complex plane. The motivation behind our questions, however, comes from
more general energy integrals of mathematical models of Hyperelasticity. The
Dirichlet Principle, the name coined by Riemann, tells us that the outer
variation of a harmonic mapping increases its energy. Surprisingly, when one
jumps into details about inner variations, which are just a change of
independent variables, new equations and related questions start to matter. The
inner variational equation, called the Hopf Laplace equation, is no longer the
Laplace equation. Its solutions are generally not harmonic; we refer to them as
Hopf harmonics. The natural question that arises is how does a change of
variables in the domain of a Hopf harmonic map affect its energy? We show,
among other results, that in case of a simply connected domain the energy
increases. This should be viewed as Riemann's Dirichlet Principle for Hopf
harmonics.
  The Dirichlet Principle for Hopf harmonics in domains of higher connectivity
is not completely solved. What complicates the matter is the insufficient
knowledge of global structure of trajectories of the associated Hopf quadratic
differentials, mainly because of the presence of recurrent trajectories.
Nevertheless, we have established the Dirichlet Principle whenever the Hopf
differential admits closed trajectories and crosscuts. Regardless of these
assumptions, we established the so-called Infinitesimal Dirichlet Principle for
all domains and all Hopf harmonics. Precisely, the second order term of inner
variation of a Hopf harmonic map is always nonnegative.
