Notions of affinity in calculus of variations with differential forms

Ext-int.\ one affine functions are functions affine in the direction of one-divisible exterior forms, with respect to exterior product in one variable and with respect to 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

Exterior convexity and classical calculus of variations

We study the relation between various notions of exterior convexity introduced in Bandyopadhyay-Dacorogna-Sil \cite{BDS1} with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in Bandyopadhyay-Dacorogna-Sil \cite{BDS1}.Comment: The original publication is available at www.esaim-cocv.org https://www.esaim-cocv.org/articles/cocv/abs/2016/02/cocv150007/cocv150007.htm

Differential inclusions for differential forms

We study necessary and sufficient conditions for the existence of solutions in $$W^{1,\infty}_0 (\Omega; \Lambda^{k}(\mathbb{R}^n))$$ of the problem $${\rm d}\omega (x)\in E,\quad \text{a.e. in }\Omega$$ where $$E\subseteq\Lambda^{k+1}(\mathbb{R}^{n})$$ is a given set. Special attention is given to the case of the curl (i.e. k = 1), particularly in dimension 3. Some applications to the calculus of variations are also state

Notions of affinity in calculus of variations with differential forms

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

Some linear and nonlinear problems involving differential forms

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

Microfinance in the Improvement of Living Standard and GNH

This paper mainly aims to extend the philosophy of capability development at the micro level for achieving individual happiness as a part of a community through social transformation and to achieve happiness at individual and community level. One of the main indicators for Gross National Happiness is living standard, and this paper will examine briefly the role of microfinance in India, Bhutan, Bangladesh, Nepal, and Pakistan in transforming the lives and social behaviour of the poor people of this part of the world. This paper will try to establish the strong linkage between microfinance and capability building through the process of social transformation by improvement of living standard for arriving at Gross National Happiness

THE PULLBACK EQUATION FOR DEGENERATE FORMS

Abstract. We discuss the existence of a diffeomorphism ϕ: R n → R n such that ϕ ∗ (g) = f, where f, g: R n → Λ k are closed differential forms and 2 ≤ k ≤ n − 1