The present study deals with the investigation of the minimization problem involving symmetric forms: \(\begin{aligned}\inf \left\{ \int _{\Omega }f\left( d_s \varphi (x)\right) \,dx:\varphi \in \varphi _0+W^{1,p}_{0}\left( \Omega ;\vee ^{k-1}({\mathbb {R}}^{n})\right) \right\} ,\end{aligned}\) where \(\Omega \subset {\mathbb {R}}^{n}\) is open, bounded, \(f:\vee ^{k}({\mathbb {R}}^{n})\rightarrow \bar{{\mathbb {R}}}\) and \(\varphi \) is a symmetric form and \(\varphi _{0}\in W^{1,p}\left( \Omega ;\vee ^{k-1}({\mathbb {R}}^{n})\right) \) . Symmetric forms come up naturally in geometry, in the form of Riemannian metrics, and in nonlinear elasticity when we talk about strain tensors. We discuss the direct methods in the calculus of variations in the framework of symmetric forms. We begin by introducing the notions of \(\vee ^{k}\) -quasiconvexity and \(\vee ^{k}\) -rank one convexity and study the inter relations. We settle Morrey’s Conjecture for the case of symmetric forms except when \(n=2\) . Furthermore, we show that the class of \(\vee ^{k}\) -rank one affine functions and affine functions are all the same. We prove the existence of minimizers for minimization problems involving symmetric forms.
