An essentially unique homeomorphic solution to the Beltrami equation with measurable coefficients was found in 1930s by Morrey. The most well-known proof from 1960s uses theory of Calderón–Zygmund and singular integral operators $$L^p(\mathbb {C})$$ . We will present an alternative method solve using Hodge star operator standard elliptic PDE theory. also discuss a different prove regularity sol...

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold and prove they converge to their differential-geometric analogues as the triangulation becomes small. The first such result is for a cochain cup product converging to the wedge product on differential forms. Moreover, we show any extension of this product to a C∞algebra also converges to ...

requires that a normal vector field n(p) be defined a.e. p ∈ ∂A. In this paper we give a new proof and extension of this theorem by replacing n with a limit ⋆∂A of 1-dimensional polyhedral chains taken with respect to a norm. The operator ⋆ is a geometric dual to the Hodge star operator and is defined on a large class of k-dimensional domains of integration A in n-space the author calls chainle...

Let X = G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.

The calculus of generalized p-forms,−1 p n, is developed further. On an n-dimensional manifold with metric, M, a (Hodge) star operator, inner product, co-differential and Laplacian operators are introduced. The inner product and Lie derivative with respect to vector fields on M are defined. Applications are made to Hamiltonian mechanics, field theories and Einstein’s vacuum equations. PACS numb...

We present several results on the geometry of the quantum projective plane CPq. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding ‘monopoles’ and ‘instanton’ charges; complete diagonalization of gauged Laplacians on these line bundles; quantum...

We give a systematic account of the exterior algebra of forms on q-Minkowski space, introducing the q-exterior derivative, q-Hodge star operator, q-coderivative, q-LaplaceBeltrami operator and the q-Lie-derivative. With these tools at hand, we then give a detailed exposition of the q-d’Alembert and q-Maxwell equation. For both equations we present a q-momentum-indexed family of plane wave solut...