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Concept# Differential form

Summary

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx is an example of a 1-form, and can be integrated over an interval [a, b] contained in the domain of f:
:\int_a^b f(x),dx.
Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S:
:\int_S (f(x,y,z),dx\wedge dy + g(x,y,z),dz\wedge dx + h(x,y,z),dy\wedge dz).
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space. In general, a k-form is an object that may be integrated over

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Saugata Bandyopadhyay, Swarnendu Sil

Ext-int. one affine functions are functions affine in the direction of one-divisible exterior forms with respect to the exterior product in one variable and with respect to the interior product in the other. The purpose of this article is to prove a characterization theorem for this class of functions, which plays an important role in the calculus of variations for differential forms.

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 \emph{ext. polyconvexity}, \emph{ext. quasiconvexity} and \emph{ext. one convexity} for functionals of the type $\int_{\Omega} f(d\omega),$ \emph{vectorial ext. polyconvexity}, \emph{vectorial ext. quasiconvexity} and \emph{vectorial ext. one convexity} for functionals of the type $\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})$ and \emph{ext-int. polyconvexity}, \emph{ext-int. quasiconvexity} and \emph{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 \begin{align*} \left\lbrace \begin{gathered} \delta ( A (x) ( d\omega ) ) + f( \omega ) = \lambda\omega \text{ in } \Omega, \ \nu \wedge \omega = 0 \text{ on } \partial\Omega. \end{gathered} \right. \end{align*} Lastly, we briefly discuss existence results for quasilinear Maxwell operator \begin{align*} \delta ( A ( x, d \omega ) ) = f , \end{align*} with different boundary data.

In this thesis, we study some linear and nonlinear problems involving differential forms. We begin by studying the problem of pullbacks which asks the following question: for two given differential forms, if one is the pullback of the other via a diffeomorphism satisfying some given condition. For volume forms, this problem was studied by Dacorogna-Moser giving a necessary and sufficient condition for the existence of the diffeomorphism with precise regularity. Our goal is to extend this result for general k-forms. We have obtained some necessary and sufficient conditions for two-forms and for some special classes of k-forms with sharp regularity. Then we turn our attention to the problem of differential inclusions involving differential forms. Although for zero-forms, the problem has been extensively studied, essentially nothing was known for higher forms including the curl operator. In this direction, we have obtained some necessary and some sufficient conditions for general k-forms unifying the study of the different cases. Moreover, we show that these necessary and sufficient conditions coincide for k = 1, solving the case of curl operator fairly completely. Besides these problems, we have studied some domain invariance property of the weighted-homogenous and non-homogenous Hardy constants as well. We have showed that the Hardy constant corresponding to these classes of inequalities enjoy, to some extent, the same domain invariance property as that of the Hardy constant corresponding to the standard Hardy's inequality.