6. Variational problems

Inner variations
Consider the inner variational equation for planar maps $$ (f_z\overline{f_{\overline{z}}})_{\overline{z}} = 0. $$ Locally, this equation can be reduced to a firstorder equation $f_zf_{\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \in M$, with $M$ a certain set of $2\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.
The equation $$ (f_z\overline{f_{\overline{z}}})_{\overline{z}} = 0. $$ expresses the stationarity of the Dirichlet energy $\int Df^2$ under inner variations of $f$ (precomposition with diffeomorphisms).Problem 6.1.
[Leonid Kovalev] Is $Df$ continuous? 
Variational problems for dilatation functionals
Problem 6.2.
[Melkana Brakalova] Study extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter. 
Hausdorff measures in $\ell^p_n$
Let ${\mathcal H}^m$ denotes the Hausdorff $m$measure corresponding to the norm in $\ell^p_n$.Problem 6.3.
[Thierry De Pauw] Let $P$ be a polyhedral $m$cycle in $\ell^p_n$, $p\ne 2$, with faces $F_1,\ldots,F_k$. Is ${\mathcal H}^m(F_1) \le \sum_{j=2}^k {\mathcal H}^m(F_j)$? 
Compact deformations of minimizing sets
Problem 6.4.
[Thierry De Pauw] An $m$rectifiable closed set $S \subset V = \ell^\infty_n$ is called minimizing if for every Lipschitz map $f:V\to V$ such that $f$ is the identity off a cube $C$ and $f(C)\subset C$, ${\mathcal H}^m(C\cap S) \le {\mathcal H}^m(f(C\cap S))$. Let $S$ be minimizing. Is it true that for ${\mathcal H}^m$almost every $x \in S$, there exists a neighborhood $U$ of $x$ such that $S\cap U$ is a Lipschitz graph over an $m$dimensional subspace $\Pi$ of $V$? 
Isoperimetric inequalities in metric measure spaces
Problem 6.5.
[Nageswari Shanmugalingam] Can the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality $$ \min \{ \mu (E\cap B), \mu (B \setminus E) \} \le C r P(E,\lambda B), $$ where $E \subset X$ is a Borel set and $B \subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measuretheoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?
Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.