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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 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 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.
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