Regularity results for differential forms solving degenerate elliptic systems

We present a partial Hölder regularity result for differential forms solving degenerate systems $$d^\ast A(\cdot,\omega) = 0 \quad {\rm and} \quad d \omega = 0$$ on bounded domains in the weak sense. Here certain continuity, monotonicity, growth and structure condition are imposed on the coefficients, including an asymptotic Uhlenbeck behavior close to the origin. Pursuing an approach of Duzaar and Mingione (J Math Anal Appl 352(1):301–335, 2009), we combine non-degenerate and degenerate harmonic-type approximation lemmas for the proof of the partial regularity result, giving several extensions and simplifications. In particular, we benefit from a direct proof of the approximation lemma (Diening et al. 2010) that simplifies and unifies the proof in the power growth case. Moreover, we give the dimension reduction for the set of singular points.     

Convex functions on the Heisenberg group

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.Received: 5 July 2002, Accepted: 24 October 2002, Published online: 6 June 2003Mathematics Subject Classification (1991): 49L25, 35J70, 35J67, 22E30Guozhen Lu: First author supported by US NSF grant DMS-9970352Juan J. Manfredi: Second author supported by US NSF grant DMS-0100107Bianca Stroffolini: Third author was supported by G.N.A.M.P.A. and by the 2002 projectPartial Differential Equations and Control Theory     

Everywhere regularity of functionals with <Emphasis Type="Italic">φ</Emphasis>-growth

We prove C ^1,α -regularity for local minimizers of functionals with φ-growth, giving also the decay estimate. In particular, we present a unified approach in the case of power-type functions. Supported by PRIN Project: “Calcolo delle variazioni e Teoria Geometrica della Misura”.     

Elliptic Systems of PDE with BMO-coefficients

The present paper is concerned with elliptic systems with BMO coefficients: div(Au)=divF, where A(x) is a positive definite matrix whose entries are in BMO and F is a given matrix field in L ^p . We show existence and uniqueness of the distributional solution of class W ^1,p , provided p is close to 2, depending on the BMO norm of A. This result is achieved using a refinement of a classical Theorem of Coifman Rochberg and Weiss about commutators with BMO functions.     

The φ-harmonic approximation and the regularity of φ-harmonic maps

We extend the p-harmonic approximation lemma proved by Duzaar and Mingione for p-harmonic functions to φ-harmonic functions, where φ is a convex function. The proof is direct and is based on the Lipschitz truncation technique. We apply the approximation lemma to prove Hölder continuity for the gradient of a solution of a φ-harmonic system with critical growth.     

Nonlinear Hodge theory on manifolds with boundary

Summary The intent of this paper is first to provide a comprehensive and unifying development of Sobolev spaces of differential forms on Riemannian manifolds with boundary. Second, is the study of a particular class of nonlinear, first order, ellipticPDEs, called Hodge systems. The Hodge systems are far reaching extensions of the Cauchy-Riemann system and solutions are referred to as Hodge conjugate fields. We formulate and solve the Dirichlet and Neumann boundary value problems for the Hodge systems and establish the ℒ^p for such solutions. Among the many desirable properties of Hodge conjugate fields, we prove, in analogy with the case of holomorphic functions on the plane, the compactness principle and a strong theorem on the removability of singularities. Finally, some relevant examples and applications are indicated. Entrata in Redazione il 4 dicembre 1997. The first two authors were partially supported by NSF grants DMS-9401104 and DMS-9706611. Bianca Stroffolini was supported by CNR. This work started in 1993 when all authors were in Syracuse.