Quasiconvexity at the Boundary and a Simple Variational Formulation of Agmon's Condition

AbstractWe study the question of positivity of quadratic funtionals $$Q(\phi ) = \int {_\Omega C_0 (x)[\nabla \phi (x),\nabla \phi (x)]dx}$$ which typically arise as the second variation at a critical point u of a functional. For interior points x1∈ Ω rank-one convexity of C0(x1) is a necessary condition for u to be a local minimizer. For boundary points x2∈ ∂ Ω where ϕ is allowed to vary freely the stronger condition of quasiconvexity at the boundary is necessary. For quadratic functionals this condition is roughly equivalent to rank-one convexity and Agmon's condition. We derive an equivalent condition on C0(x2) which is purely algebraic; and, moreover, it is variational in the sense that it can be formulated in terms of positive semidefiniteness of Hermitian matrices. A connection to the solvability of matrix-valued Riccati equations is established. Several applications in elasticity theory are treated.