Displaying similar documents to “Convex approximation of an inhomogeneous anisotropic functional”

Convex approximations of functionals with curvature

Giovanni Bellettini, Maurizio Paolini, Claudio Verdi (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We address the numerical minimization of the functional F v = Ω D v + Ω μ v d H n - 1 - Ω x v d x , for v B V Ω ; - 1 , 1 . We note that F can be equivalently minimized on the larger, convex, set B V Ω ; - 1 , 1 and that, on that space, F may be regularized with a sequence { F ϵ ( v ) = Ω ϵ 2 + D v 2 + Ω μ v d H n - 1 - Ω x v d x } ϵ of regular functionals. Then both F and F ϵ can be discretized by continuous linear finite elements. The convexity of the functionals in B V Ω ; - 1 , 1 is useful for the numerical minimization of F . We prove the Γ - L 1 Ω -convergence of the discrete functionals to F and present a few numerical examples. ...

An elementary proof of Marcellini Sbordone semicontinuity theorem

Tomáš G. Roskovec, Filip Soudský (2023)

Kybernetika

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The weak lower semicontinuity of the functional F ( u ) = Ω f ( x , u , u ) d x is a classical topic that was studied thoroughly. It was shown that if the function f is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on W 1 , p ( Ω ) . However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.

On the eigenvalues of an elliptic operator a x , H u

Sergio Campanato (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let Ω be a bounded open convex set of class C 2 . Let a x , H u be a non linear operator satisfying the condition (A) (elliptic) with constants α , γ , δ . We prove that a number λ 0 is an eigenvalue for the operator a x , H u if and only if the number α λ is an eigen-value for the operator Δ u . If λ 0 , the two systems a x , H u = λ u and Δ u = α λ u have the same solutions. In particular, also the eventual eigen-values of the operator a x , H u should all be negative. Finally, we obtain a sufficient condition for the existence of solutions u H 2 H 0 1 Ω ...

On some nonlinear partial differential equations involving the 1-Laplacian

Mouna Kraïem (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let Ω be a smooth bounded domain in N , N > 1 and let n * . We prove here the existence of nonnegative solutions u n in B V ( Ω ) , to the problem ( P n ) - div σ + 2 n Ω u - 1 sign + ( u ) = 0 in Ω , σ · u = | u | in Ω , u is not identically zero , - σ · n u = u on Ω , where n denotes the unit outer normal to Ω , and sign + ( u ) denotes some L ( Ω ) function defined as: sign + ( u ) . u = u + , 0 sign + ( u ) 1 . Moreover, we prove the tight convergence of u n towards one of the first eingenfunctions for the first 1 - Laplacian Operator - Δ 1 on Ω when n goes to + .

Nonvariational basic parabolic systems of second order

Sergio Campanato (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Ω is a bounded open set of R n , of class C 2 and T > 0 . In the cylinder Q = Ω × 0 , T we consider non variational basic operator a H u - u / t where a ξ is a vector in R N , N 1 , which is continuous in ξ and satisfies the condition (A). It is shown that f L 2 Q the Cauchy-Dirichlet problem u W 0 2 , 1 Q , a H u - u / t = f in Q , has a unique solution. It is further shown that if u W 0 2 , 1 Q is a solution of the basic system a H u - u / t = 0 in Q , then H u and u / t belong to H l o c 1 Q . From this the Hölder continuity in Q of the vectors u and D u are deduced respectively when n 4 and...