Erik Duse: Generic ill-posedness of the energy-momentum equations in vectorial calculus of variations

Abstract: This talk is about the energy-momentum equations in calculus of variations. These arises from inner-variations, i.e. domain variations, as opposed to outer variations that gives rise to the Euler-Lagrange equations. For vectorial problems in calculus of variations, i.e., minimization problems involving vector valued functions, the energy-momentum equations become important, especially in combination with the Euler-Lagrange equations. In recent work I proved that the energy-momentum equations are generically ill-posed with respect to Dirichlet data using convex integration theory. A similar result for the Euler-Lagrange equations was proven by Sverak and Müller in 2003. Besides reporting on this work I will explain various fundamental notions for vectorial problems like quasiconvexity and rank-one convexity.