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We propose the use of algebras of generalized functions for the analysis ofcertain highly singular problems in the calculus of variations. After a generalstudy of extremal problems on open subsets of Euclidean space in this settingwe introduce the first and second variation of a variational problem. We thenderive necessary (Euler-Lagrange equations) and sufficient conditions forextremals. The concept of association is used to obtain connections to adistributional description of singular variational problems. We studyvariational symmetries and derive an appropriate version of N\"other's theorem.Finally, a number of applications to geometry, mechanics, elastostatics andelastodynamics are presented.

Foundations of the calculus of variations in generalized function algebras