3,413 research outputs found
An Epiperimetric Inequality for the Thin Obstacle problem
We prove an epiperimetric inequality for the thin obstacle problem, extending
the pioneering results by Weiss on the classical obstacle problem (Invent.
Math. 138 (1999), no. 1, 23-50). This inequality provides the means to study
the rate of converge of the rescaled solutions to their limits, as well as the
regularity properties of the free boundary
On the measure and the structure of the free boundary of the lower dimensional obstacle problem
We provide a thorough description of the free boundary for the lower
dimensional obstacle problem in up to sets of null
measure. In particular, we prove (i) local finiteness of
the -dimensional Hausdorff measure of the free boundary, (ii)
-rectifiability of the free boundary, (iii) classification
of the frequencies up to a set of dimension at most (n-2) and classification of
the blow-ups at almost every free boundary point
Phase field approximation of cohesive fracture models
We obtain a cohesive fracture model as a -limit of scalar damage
models in which the elastic coefficient is computed from the damage variable
through a function of the form , with diverging for close to the value describing undamaged
material. The resulting fracture energy can be determined by solving a
one-dimensional vectorial optimal profile problem. It is linear in the opening
at small values of and has a finite limit as . If the
function is allowed to depend on the index , for specific choices we
recover in the limit Dugdale's and Griffith's fracture models, and models with
surface energy density having a power-law growth at small openings
Existence of minimizers for the d stationary Griffith fracture model
We consider the variational formulation of the Griffith fracture model in two
spatial dimensions and prove existence of strong minimizers, that is
deformation fields which are continuously differentiable outside a closed jump
set and which minimize the relevant energy. To this aim, we show that
minimizers of the weak formulation of the problem, set in the function space
and for which existence is well-known, are actually strong minimizers
following the approach developed by De Giorgi, Carriero, and Leaci in the
corresponding scalar setting of the Mumford-Shah problem
Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result
In this note we prove an abstract version of a recent quantitative
stratifcation priciple introduced by Cheeger and Naber (Invent. Math., 191
(2013), no. 2, 321-339; Comm. Pure Appl. Math., 66 (2013), no. 6, 965-990).
Using this general regularity result paired with an -regularity
theorem we provide a new estimate of the Minkowski dimension of the set of
higher multiplicity points of a Dir-minimizing Q-valued function. The abstract
priciple is applicable to several other problems: we recover recent results in
the literature and we obtain also some improvements in more classical contexts.Comment: modified title; minor change
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