Axiomatic spanning conditions clarified, recent history added with comparisons, details added to proofs, notation simplified, generalization of main result to elliptic measures deferred to a separate paper
We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of competing "surfaces," which span a given bounding set in some ambient space, one with minimal anisotropically weighted area. In particular, rectifiability of a candidate minimizer is proved without the assumption of quasiminimality. Our ambient spaces are a class of Lipschitz neighborhood retracts which includes manifolds with boundary and manifolds with certain singularities. Our competing surfaces are rectifiable sets which satisfy any combination of general homological, cohomological or homotopical spanning conditions. An axiomatic spanning criterion is also provided. Our boundaries are permitted to be arbitrary closed subsets of the ambient space, providing a good setting for surfaces with sliding boundaries.
$C^*$-algebras of transmission problems and elliptic boundary value problems with shift operators
Institute of Analysis, Leibniz University Hannover（汉诺威莱布尼茨大学分析研究所）；Ishlinsky Institute for Problems in Mechanics RAS（俄罗斯科学院伊什林斯基力学问题研究所）；Peoples’ Friendship University of Russia (RUDN University)（俄罗斯人民友谊大学）
