Nonconvex Problems of the Calculus of Variations and Differential Inclusions

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property.The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower Semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group.Maximum Principles are implied for the relaxed solution in the case of nonexistence of minimizers and for minimizing solutions of the Euler-Lagrange system of PDE.[W 1,q g (Ω)] N E and 0-minimizers correspond to solutions of (1.3). Our viewpoint of the Maximum Principle in the vector case is based on the observation that the scalar inequalities sup Ω u ≤ max ∂Ω u, inf Ω u ≥ min ∂Ω u when N = 1 can be recast as u(Ω) ⊆ min ∂Ω u, max ∂Ω u . When N ≥ 1, the appropriate vectorial extension is the so-called Convex Hull Property(1.4) u(Ω) ⊆ co u(∂Ω)

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Nonconvex Problems of the Calculus of Variations and Differential Inclusions

Cited by 6 publications

References 58 publications

Stability Properties of the Notions and Existence via Approximation

Stability Properties of the Notions and Existence via Approximation

Strong compactness in Sobolev spaces

Maximum Principles for vectorial approximate minimizers of nonconvex functionals

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