Dirac-harmonic equations for differential forms. (English) Zbl 1325.58011
The authors introduce Dirac-harmonic equations for differential forms. This is a variant of so-called \(A\)-harmonic equations \(d^*A(x, d\omega)=0\) where \(\omega\) is a differential form, \(d\) the exterior differential operator and \(d^*\) its formal adjoint,and \(A\) an approbiate operator, e.g. \(A(x,\xi)=\xi|\xi|^p\). The basic idea of the authors is that in certain applications one would like to have a similar equation \(d^*A(x, D\omega)=0\) where \(D=d+d^*\) is the Hodge-Dirac operator. The main part of the paper is devoted to prove estimates for solutions \(\omega\) of the Dirac-harmonic equation such as Caccioppoli-type, Poincaré, Sobolev and weak reverse Hölder inequalities.
Reviewer: Nadine Große (Leipzig)
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J60 | Nonlinear elliptic equations |
| 35B45 | A priori estimates in context of PDEs |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
| 47J05 | Equations involving nonlinear operators (general) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 58A10 | Differential forms in global analysis |
| 35C99 | Representations of solutions to partial differential equations |
Keywords:
A-harmonic equations; Hodge-Dirac-operator; a priori estimates; Dirac-harmonic equation; differential forms; norm inequalitiesReferences:
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