scholarly journals Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus

2019 ◽  
Author(s):  
Thomas Mollenhoff ◽  
Daniel Cremers
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Variational Problems ◽  
Geometric Measure ◽  
Discrete Exterior Calculus ◽  
Exterior Calculus ◽  
Natural Formulation

scholarly journals LOCAL SET APPROXIMATION: MATTILA–VUORINEN TYPE SETS, REIFENBERG TYPE SETS, AND TANGENT SETS

2015 ◽  
Vol 3 ◽  
Author(s):  
MATTHEW BADGER ◽  
STEPHEN LEWIS
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Singular Part ◽  
Variational Problems ◽  
Geometric Measure ◽  
Model Sets ◽  
Tangent Sets ◽  
Asymptotic Geometry ◽  
Set Approximation ◽  
Type Sets

We investigate the interplay between the local and asymptotic geometry of a set $A\subseteq \mathbb{R}^{n}$ and the geometry of model sets ${\mathcal{S}}\subset {\mathcal{P}}(\mathbb{R}^{n})$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^{n}$ ($n\geqslant 4$) has upper Minkowski dimension at most $n-4$.

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