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Critical nonlinear elliptic equations with singularities and cylindrical symmetry

Marino Badiale, Enrico Serra (2004)

Revista Matemática Iberoamericana

Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in RN with cylindrical symmetry and with singularities on a whole subspace of RN. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.

Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case

Gilles A. Francfort, Nam Q. Le, Sylvia Serfaty (2009)

ESAIM: Control, Optimisation and Calculus of Variations

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.

Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case

Gilles A. Francfort, Nam Q. Le, Sylvia Serfaty (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.

Differentiability for minimizers of anisotropic integrals

Paola Cavaliere, Anna D'Ottavio, Francesco Leonetti, Maria Longobardi (1998)

Commentationes Mathematicae Universitatis Carolinae

We consider a function u : Ω N , Ω n , minimizing the integral Ω ( | D 1 u | 2 + + | D n - 1 u | 2 + | D n u | p ) d x , 2 ( n + 1 ) / ( n + 3 ) p < 2 , where D i u = u / x i , or some more general functional with the same behaviour; we prove the existence of second weak derivatives D ( D 1 u ) , , D ( D n - 1 u ) L 2 and D ( D n u ) L p .

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