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We study the $$L^p$$ L p -mean distortion functionals, $$\begin{aligned} \mathcal{E}_p[f] = \int \int \limits _{\mathbb {D}}{\mathbb {K}}^p(z,f) \; \mathrm{d}z,\quad f_{|{\mathbb {S}}}=f_0 \end{aligned}$$ E p [ f ] = ∫ ∫ D K p ( z , f ) d z , f | S = f 0 for Sobolev self homeomorphisms of the unit disk $${\mathbb {D}}$$ D with prescribed boundary values $$f_0:{\mathbb {S}}\rightarrow {\mathbb {S}}$$ f 0 : S → S and pointwise distortion function $${\mathbb {K}}={\mathbb {K}}(z,f)$$ K = K ( z , f ) . Here we discuss aspects of the existence, regularity and uniqueness questions for minimisers and discuss the diffeomorphic critical points of $$\mathcal{E}_p$$ E p presenting results we know and making some conjectures. Remarkably, smooth minimisers of the $$L^p$$ L p -mean distortion functionals have inverses which are harmonic with respect to a metric induced by the distortion of the mapping. From this we are able to deduce that the complex conjugate Beltrami coefficient of a smooth minimiser is locally quasiregular and we identify the quasilinear equation it solves. This has other consequences such as a maximum principle for the distortion.
Computational Methods and Function Theory – Springer Journals
Published: Apr 1, 2014
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