  We obtain Lipschitz regularity results for a fairly general class of
nonlinear first-order PDEs. These equations arise from the inner variation of
certain energy integrals. Even in the simplest model case of the Dirichlet
energy the inner-stationary solutions need not be differentiable everywhere;
the Lipschitz continuity is the best possible. But the proofs, even in the
Dirichlet case, turn out to relay on topological arguments. The appeal to the
inner-stationary solutions in this context is motivated by the classical
problems of existence and regularity of the energy-minimal deformations in the
theory of harmonic mappings and certain mathematical models of nonlinear
elasticity; specifically, neo-Hookian type problems.
