6. Variational problems
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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)? -
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 measure-theoretic 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.