Calculus of variations with symmetric forms

Pati, Ashis Kumar (2022) Calculus of variations with symmetric forms. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Ashis Kumar Pati (12IP021)) 12IP021.pdf.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this thesis, we discuss the direct methods of calculus of variations in the framework of symmetric forms. We begin by introducing the relevant algebraic and analytic notions of symmetric forms. We then introduce the appropriate concepts of convexity for symmetric forms, namely the notions of ∨k-quasiconvexity and ∨k-rank one convexity. We explore the relations between these different notions of convexity. It is shown that convexity implies ∨k-quasiconvexity which, in turn, implies ∨k-rank one convexity. We settle Morrey’s Conjecture for the case of symmetric forms except when n = 2. While our proof is inspired by that of Sver`ak in the classical case, the construction of the subspace is little tricky here. Furthermore, we show that the class of ∨k-rank one affine functions and affine functions are all the same in the framework of symmetric forms. Having established the framework, we proceed to the study of the minimization problems. We show that ∨k-quasiconvexity of f is necessary for the weak lower semicontinuity of the functional. The sufficiency is also established, albeit for the case when k = 2. We conclude the discussion with a result on the existence of minimizers.

Actions (login required)

View Item