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We study minimisers of the $p$-conformal energy functionals, \[\mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \] defined forself mappings $f:\ID\to\ID$ with finite distortion and prescribed boundaryvalues $f_0$. Here \[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} =\frac{1+|\mu_f(z)|^2}{1-|\mu_f(z)|^2}\] is the pointwise distortion functionaland $\mu_f(z)$ is the Beltrami coefficient of $f$. We show that forquasisymmetric boundary data the limiting regimes $p\to\infty$ recover theclassical Teichm\"uller theory of extremal quasiconformal mappings (in part aresult of Ahlfors), and for $p\to1$ recovers the harmonic mapping theory.  Critical points of $\mathsf{E}_p$ always satisfy the inner-variationaldistributional equation \[ 2p\int_\ID\IK^p\;\frac{\overline{\mu_f}}{1+|\mu_f|^2}\varphi_\zbar \; dz=\int_\ID \IK^p\; \varphi_z\; dz,\quad\forall\varphi\in C_0^\infty(\ID ). \] We establish theexistence of minimisers in the {\em a priori} regularity class$W^{1,\frac{2p}{p+1}}(\ID)$ and show these minimisers have a pseudo-inverse - acontinuous $W^{1,2}(\ID)$ surjection of $\ID$ with $(h\circ f)(z)=z$ almosteverywhere. We then give a sufficient condition to ensure $C^{\infty}(\ID)$smoothness of solutions to the distributional equation. For instance$\IK(z,f)\in L^r_{loc}(\ID)$ for any $r>p+1$ is enough to imply the solutionsto the distributional equation are local diffeomorphisms. Further $\IK(w,h)\inL^1(\ID)$ will imply $h$ is a homeomorphism, and together these results yield adiffeomorphic minimiser. We show such higher regularity assumptions to benecessary for critical points of the inner variational equation.

The $L^p$ Teichm\"uller theory: Existence and regularity of critical points