Abstract
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.
Highlights
We establish Lipschitz regularity of solutions of nonlinear first-order PDEs that arise from inner variation of numerous energy integrals
When dealing with mappings with nonnegative Jacobian the inner variation is necessary to preserve the sign of the Jacobian
We look for h in the form h(z) = 2z ψ(−2 log|z|), |z| 1 where ψ : [0, ∞) → [1, ∞) is a strictly increasing function with ψ(0) = 1
Summary
We establish Lipschitz regularity of solutions of nonlinear first-order PDEs that arise from inner variation of numerous energy integrals. 2. Inner-variational equations Let us consider the energy integral for mappings h : Ω → C (2.1). Given any test function η ∈ C0∞(Ω) and a complex parameter t , small enough so that the map z → z + t η(z) represents a diffeomorphism of Ω onto itself, consider the inner variation ht(z) = h(z + tη) and its energy. Let h ∈ Wlo1c,1(Ω) be an inner-stationary mapping for the energy integral (2.6) with E [h] < ∞, where F satisfies the conditions (2.7)– (2.11). Subject to homeomorphisms h : Ω −on−→to Ω∗ in the Sobolev space W 1,2(Ω) This case gains additional interest in Geometric Function Theory because the transition to the energy of the inverse mapping f = h−1 : Ω∗ o−n−t→o Ω results in the L p-norm of the distortion function.
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