Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation

We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density $u$. In case of \emph{fast-decay} mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density $\rho$ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density $\rho$ allow us to motivate the aforementioned change of variable and to state the results in terms of the original density $u$ without prescribing any boundary conditions

Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity

We define a class of deformations in W^1,p(\u3a9,R^n), p>n 121, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality between the distributional determinant and the pointwise determinant of the gradient. Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W^1,p, and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity

Gamma-convergence of integral functionals on divergence free fields

We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Gamma-convergence. We prove that the Gamma-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Gamma-limit is also stable under volume constraint and various type of boundary conditions

On a nonlocal functional arising in the study of thin-film blistering

The energy of a Von Karman circular plate is described by a nonlocal nonconvex one-dimensional functional depending on the thickness epsilon. Here we perform the asymptotic analysis via Gamma-convergence as the parameter epsilon goes to zero

Erratum: "Asymptotic analysis of periodicaly-perforated nonlinear media" (Journal des Mathematiques Pures et Appliquees (2002) vol. 81 (439-451) 10.1016/S0021-7824(01)01226-0)

[No abstract available

Multiscale analysis by Gamma-convergence of a one-dimensional nonlocal functional related to a shell-membrane transition

We study the asymptotic behavior of one-dimensional functionals associated with the energy of a thin nonlinear elastic spherical shell in the limit of vanishing thickness ( proportional to a small parameter) epsilon and under the assumption of radial deformations. The functionals are characterized by the presence of a nonlocal potential term and defined on suitable weighted functional spaces. The shell-membrane transition is studied at three different relevant scales. For each we give a compactness result and compute the Gamma-limit. In particular, we show that if the energies on a sequence of configurations scale as epsilon(3/2), then the limit configuration describes a ( locally) finite number of transitions between the undeformed and the everted configurations of the shell. We also highlight a kind of "Gibbs phenomenon" by showing that nontrivial optimal sequences restricted between the undeformed and the everted configurations must have energy scaling of at least epsilon(4/3)

Gradient theory of phase-transitions in composite media

We study the behaviour of non-convex functionals singularly perturbed by a possibly oscillating inhomogeneous gradient term, in the spirit of the gradient theory of phase transitions. We show that a limit problem giving a sharp interface, as the perturbation vanishes, always exists, but may be inhomogeneous or anisotropic. We specialize this study when the perturbation oscillates periodically, highlighting three types of regimes, depending on the frequency of the oscillations. In the two extreme cases, a separation of scales effect is described