Quasiconvexity at the boundary and a simple variational formulation of Agmon's condition

We study the question of positivity of quadratic funtionals Q(phi) = integr al(Omega)C(0)(x)[V phi(x), del phi(x)]dx which typically arise as the secon d variation at a critical point u of a functional. For interior points x(1) is an element of Omega rank-one convexity of C-0(x(1)) is a necessary cond ition for u to be a local minimizer. For boundary points x(2) is an element of partial derivative Omega where phi is allowed to vary freely the strong er 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 C-0(x(2)) 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 esta blished. Several applications in elasticity theory are treated.