Bianca Stroffolini

Nonlinear Hodge Theory on Manifolds with Boundary(

SummaryThe 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 ℒp 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.

A VERSION OF THE HOPF-LAX FORMULA IN THE HEISENBERG GROUP

ABSTRACT We consider Hamilton-Jacobi equations in the , where is the Heisenberg group and denotes the horizontal gradient of u. We establish uniqueness of bounded viscosity solutions with continuous initial data . When the hamiltonian H is radial, convex and superlinear the solution is given by the Hopf-Lax formula where the Lagrangian L is the horizontal Legendre transform of H lifted to by requiring it to be radial with respect to the Carnot-Carathéodory metric.

Convex functions on Carnot groups

We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.

Degree formulas for maps with nonintegrable Jacobian

This paper arose from a discussion sparked between the authors after the lecture of Louis Nirenberg at the Conference in Naples on June 1, 1995. He presented a joint work with Haim Brezis [BN] on the degree theory for VMO (vanishing mean oscillation) mappings f : X → Y between n-dimensional smooth manifolds. Their results include a variety of discontinuous maps. We soon realized that we can contribute to their work by studying some Orlicz– Sobolev classes weaker than W (X,Y ). Our approach relies on new estimates for the Jacobians [IS], [GIM] and most recent improvements [I] concerning nonlinear commutators. Also L-Hodge theory [S], [ISS] plays a crucial role in this paper. Let us begin with the well known formula for the degree of a C-map f : X → Y :

Partial Regularity for Minimizers of Quasi-convex Functionals with General Growth

We prove a partial regularity result for local minimizers of quasiconvex variational integrals with general growth. The main tool is an improved A-harmonic approximation, which should be interesting also for classical growth.The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for

Non-stationary flows of asymptotically Newtonian fluids

\mathbb{P }_1

Everywhere regularity of functionals with φ-growth

conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the