Calculus of Variations for Differential Forms

Abstract

In this thesis we study calculus of variations for differential forms. In the first part we develop the framework of direct methods of calculus of variations in the context of minimization problems for functionals of one or several differential forms of the type, $$ \int_{\Omega} f(d\omega), \quad \int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m}) \quad \text{ and } \int_{\Omega} f(d\omega, \delta\omega).$$ We introduce the appropriate convexity notions in each case, called ext. polyconvexity, ext. quasiconvexity and ext. one convexity for functionals of the type $\int_{\Omega} f(d\omega),$ vectorial ext. polyconvexity, vectorial ext. quasiconvexity and vectorial ext. one convexity for functionals of the type $\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})$ and ext-int. polyconvexity, ext-int. quasiconvexity and ext-int. one convexity for functionals of the type $\int_{\Omega} f(d\omega, \delta\omega).$ We study their interrelationships and the connections of these convexity notions with the classical notion of polyconvexity, quasiconvexity and rank one convexity in classical vectorial calculus of variations. We also study weak lower semicontinuity and weak continuity of these functionals in appropriate spaces, address coercivity issues and obtain existence theorems for minimization problems for functionals of one differential forms.\smallskip In the second part we study different boundary value problems for linear, semilinear and quasilinear Maxwell type operator for differential forms. We study existence and derive interior regularity and $L^{2}$ boundary regularity estimates for the linear Maxwell operator $$ \delta (A(x)d\omega) = f$$ with different boundary conditions and the related Hodge Laplacian type system $$ \delta (A(x)d\omega) + d\delta\omega = f,$$ with appropriate boundary data. We also deduce, as a corollary, some existence and regularity results for div-curl type first order systems. We also deduce existence results for semilinear boundary value problems \left\lbrace \delta ( A (x) ( d\omega ) ) + f( \omega ) = \lambda\omega \text{ in } \Omega, \nu \wedge \omega = 0 \text{ on } \partial\Omega. \right. Lastly, we briefly discuss existence results for quasilinear Maxwell operator \delta ( A ( x, d \omega ) ) = f , with different boundary data.