
6. Variational problems

1. 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 first-order 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 ${\mathcal H}^m(F_1) \le \sum_{j=2}^k {\mathcal H}^m(F_j)$?
This question is related to the lower semicontinuity and existence for the Plateau problem.
• 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 $V$?
For $m=1$ the answer is yes. One may further ask whether $\Pi$ can be chosen to be one of the coordinate $m$-planes and if so, whether the Lipschitz constant can be chosen close to $1$. Note that $1$-Lipschitz graphs over coordinate $m$-planes are examples of minimizing sets.
• 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 measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?
The relative isoperimetric inequality is equivalent to the $(1,1)$-Poincaré inequality. Under certain geometric conditions on $X$ (such as a geometric version of Semmes’ “pencil of curves” condition), one knows that the answer to the above question is yes. Examples of spaces with such curves include the Euclidean spaces and the Heisenberg group. Can the Semmes pencil requirement be removed?

Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.