Topics in the calculus of variations

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This report originated from a series of lectures at the East China Institute of Technology, Nanjing, in September 1992. Its aim is to familiarize beginning research students with selected problems in variational calculus, focusing on two fundamental principles: the existence of (generalized) solutions and the related question of regularity. The first chapter explores variational integrals for vector-valued functions, building on Morrey's pioneering work from the 1930s. It emphasizes problems with nonlinear side conditions imposed on admissible comparison functions. Special cases discussed include mappings constrained to lie within a Riemannian manifold, potentially with boundary, and functions requiring a strictly positive Jacobian. The variational integrals addressed are typically nonquadratic concerning the gradient, leading to degenerate elliptic systems. Key applications include p-harmonic maps between Riemannian manifolds, systems of degenerate variational inequalities, and model problems in nonlinear elasticity. While working within appropriate Sobolev spaces typically allows for the establishment of generalized solutions, this often results in a priori discontinuous functions.