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A symmetry problem in the calculus of variations

Graziano Crasta — 2006

Journal of the European Mathematical Society

We consider the integral functional J ( u ) = Ω [ f ( | D u | ) u ] d x , u W 0 1 , 1 ( Ω ) , where Ω n , n 2 , is a nonempty bounded connected open subset of n with smooth boundary, and s f ( | s | ) is a convex, differentiable function. We prove that if J admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball.

Geometric constraints on the domain for a class of minimum problems

Graziano CrastaAnnalisa Malusa — 2003

ESAIM: Control, Optimisation and Calculus of Variations

We consider minimization problems of the form min u ϕ + W 0 1 , 1 ( Ω ) Ω [ f ( D u ( x ) ) - u ( x ) ] d x where Ω N is a bounded convex open set, and the Borel function f : N [ 0 , + ] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω and the zero level set of f , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

Extremal selections of multifunctions generating a continuous flow

Alberto BressanGraziano Crasta — 1994

Annales Polonici Mathematici

Let F : [ 0 , T ] × n 2 n be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if F satisfies the following Lipschitz Selection Property: (LSP) For every t,x, every y ∈ c̅o̅F(t,x) and ε > 0, there exists a Lipschitz selection ϕ of c̅o̅F, defined on a neighborhood of (t,x), with |ϕ(t,x)-y| < ε, then there exists a measurable selection f of ext F such that, for every x₀, the Cauchy problem ẋ(t) = f(t,x(t)), x(0) = x₀, has a unique Carathéodory solution, depending...

Geometric constraints on the domain for a class of minimum problems

Graziano CrastaAnnalisa Malusa — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We consider minimization problems of the form min u ϕ + W 0 1 , 1 ( Ω ) Ω [ f ( D u ( x ) ) - u ( x ) ] d x where Ω N is a bounded convex open set, and the Borel function f : N [ 0 , + ] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of and the zero level set of , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

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