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Regularity of differential forms minimizing degenerate elliptic functionals. (English) Zbl 0776.35006
We prove global regularity for vector-valued differential \(m\)-forms \(\omega=(\omega^ 1,\dots,\omega^ N)\) defined on a Riemannian manifold \(M\), which are solutions of the system \(\delta(\rho(|\omega|^ 2)\omega)=0\), \(d\omega=0\), where \(\rho(Q)\simeq Q^{(p-2)/2}\) with \(p>1\), and which satisfy the Neumann or Dirichlet condition \(\rho(|\omega|^ 2)\omega^ \perp=0\) or \(\omega^ T=0\) on the boundary \(\partial M\). We then prove partial regularity up to the boundary for the minimizers \(u\) of the related variational integral \(\int k(x,u,| Du|)dx\), which have constant or free boundary values. [See also the review of the paper with the same title in Bonner Mathematische Schriften, 199. Bonn: Univ. Bonn, Math.-Naturwiss. Fak., Diss. (1989; Zbl 0679.35016).
Reviewer: C.Hamburger (Bonn)

MSC:
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
58A15 Exterior differential systems (Cartan theory)
35J70 Degenerate elliptic equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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