Olivier Kneuss

The Pullback Equation for Differential Forms

Introduction.- Part I Exterior and Differential Forms.- Exterior Forms and the Notion of Divisibility.- Differential Forms.- Dimension Reduction.- Part II Hodge-Morrey Decomposition and Poincare Lemma.- An Identity Involving Exterior Derivatives and Gaffney Inequality.- The Hodge-Morrey Decomposition.- First-Order Elliptic Systems of Cauchy-Riemann Type.- Poincare Lemma.- The Equation div u = f.- Part III The Case k = n.- The Case f x g > 0.- The Case Without Sign Hypothesis on f.- Part IV The Case 0 <= k <= n-1.- General Considerations on the Flow Method.- The Cases k = 0 and k = 1.- The Case k = 2.- The Case 3 <= k <= n-1.- Part V Holder Spaces.- Holder Continuous Functions.- Part VI Appendix.- Necessary Conditions.- An Abstract Fixed Point Theorem.- Degree Theory.- References.- Further Reading.- Notations.- Index.

Some Remarks on the Lie Derivative and the Pullback Equation for Contact Forms

Abstract Given the contact forms f and g, and the 1-form h, we discuss the existence of a vector field u verifying ℒ u ⁢ ( f ) = d ⁢ ( u ⁢ ⌟ ⁢ f ) + u ⁢ ⌟ ⁢ d ⁢ f = h .

Divisibility in Grassmann algebra

{\mathcal{L}_{u}(f\/)=d(u\,\lrcorner\,f\/)+u\,\lrcorner\,df=h.}

Some new results on differential inclusions for differential forms

This is closely related to the pullback equation, where we seek for a diffeomorphism φ satisfying φ ∗ ⁢ ( f ) = g .

The Equation divu = f

{\varphi^{\ast}(f\/)=g.}

The Hodge–Morrey Decomposition

Given a k-form f and an l-form g with 0 ≤ l ≤ k, we give necessary and sufficient conditions for the existence of a (k − l)-form u verifying

Hölder Continuous Functions

In this article we study some necessary and sufficient conditions for the existence of solutions in W-0(1,infinity) (Omega; Lambda(k)) of the differential inclusion d omega is an element of E a.e. in Omega where E subset of Lambda(k+1) is a prescribed set.

The Case 3 ≤ k ≤ n−1

which is constantly used in Chapter 10. Of course, most of the results can be found in Chapter 8. However, the proofs are much more elementary in this case and, in most cases, do not require the sophisticated machinery of Hodge–Morrey decomposition. They use only standard properties of the Laplacian. Therefore, for the convenience of the reader, we have gathered and proved the results in the present chapter.

Exterior Forms and the Notion of Divisibility

We recall, from Chapter 2, some notations that we will use throughout the present chapter. As usual, when necessary, we identify in a natural way 1-forms with vectors in ℝn.