233 research outputs found
Phase-field approximation for a class of cohesive fracture energies with an activation threshold
We study the -limit of Ambrosio-Tortorelli-type functionals
, whose dependence on the symmetrised gradient is
different in and in , for a
-elliptic symmetric operator , in terms of the
prefactor depending on the phase-field variable . This is intermediate
between an approximation for the Griffith brittle fracture energy and the one
for a cohesive energy by Focardi and Iurlano. In particular we prove that
functions with bounded -variation are
Existence and uniqueness for planar anisotropic and crystalline curvature flow
We prove short-time existence of \phi-regular solutions to the planar
anisotropic curvature flow, including the crystalline case, with an additional
forcing term possibly unbounded and discontinuous in time, such as for instance
a white noise. We also prove uniqueness of such solutions when the anisotropy
is smooth and elliptic. The main tools are the use of an implicit variational
scheme in order to define the evolution, and the approximation with flows
corresponding to regular anisotropies
The Stress-Intensity Factor for nonsmooth fractures in antiplane elasticity
Motivated by some questions arising in the study of quasistatic growth in
brittle fracture, we investigate the asymptotic behavior of the energy of the
solution of a Neumann problem near a crack in dimension 2. We consider non
smooth cracks that are merely closed and connected. At any point of density
1/2 in , we show that the blow-up limit of is the usual "cracktip"
function , with a well-defined coefficient (the "stress
intensity factor" or SIF). The method relies on Bonnet's monotonicity formula
\cite{b} together with -convergence techniques.Comment: (version 2 : r\'ef\'erences corrig\'ees
A Remark on the Anisotropic Outer Minkowski content
We study an anisotropic version of the outer Minkowski content of a closed
set in Rn. In particular, we show that it exists on the same class of sets for
which the classical outer Minkowski content coincides with the Hausdorff
measure, and we give its explicit form.Comment: We corrected an error in the orignal manuscript, on p. 14 (the
boundaries of the regularized sets are not necessarily C^{1,1}
Nonlocal curvature flows
This paper aims at building a unified framework to deal with a wide class of
local and nonlocal translation-invariant geometric flows. First, we introduce a
class of generalized curvatures, and prove the existence and uniqueness for the
level set formulation of the corresponding geometric flows.
We then introduce a class of generalized perimeters, whose first variation is
an admissible generalized curvature. Within this class, we implement a
minimizing movements scheme and we prove that it approximates the viscosity
solution of the corresponding level set PDE.
We also describe several examples and applications. Besides recovering and
presenting in a unified way existence, uniqueness, and approximation results
for several geometric motions already studied and scattered in the literature,
the theory developed in this paper allows us to establish also new results
The -limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension
In this paper we generalize to arbitrary dimensions a one-dimensional
equicoerciveness and -convergence result for a second derivative
perturbation of Perona-Malik type functionals. Our proof relies on a new
density result in the space of special functions of bounded variation with
vanishing diffuse gradient part. This provides a direction of investigation to
derive approximation for functionals with discontinuities penalized with a
"cohesive" energy, that is, whose cost depends on the actual opening of the
discontinuity
- …