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• Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics  (1)
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Springer
Journal of elasticity 51 (1998), S. 23-41
ISSN: 1573-2681
Keywords: quasiconvexity at the boundary ; Agmon's condition ; rank-one convexity ; local minimizers ; coercivity of quadratic functionals ; null-Lagrangians ; matrix-valued Riccati equation ; elasticity theory.
Source: Springer Online Journal Archives 1860-2000
Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
Notes: Abstract We 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.
Type of Medium: Electronic Resource
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