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We consider the problem of placing a Dirichlet region made by n small balls of given radius in a given domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.
The present article is an overview of some mathematical results, which
provide elements of rigorous basis for some multiscale
computations in materials science. The emphasis is laid upon atomistic
to continuum limits for crystalline materials. Various mathematical
approaches are addressed. The
setting is stationary. The relation to existing techniques used in the engineering
literature is investigated.
Let (x,u,∇u) be a Lagrangian periodic of period 1 in
x1,...,xn,u. We shall study the non self intersecting
functions u: RnR minimizing ; non self intersecting means that, if u(x0 + k) + j = u(x0)
for some x0∈Rn and (k , j) ∈Zn × Z, then
u(x) = u(x + k) + jx. Moser has shown that each of these
functions is at finite distance from a plane
u = ρx and thus
has an average slope ρ; moreover, Senn has proven that it is
possible to define the average action of u, which is usually called since...
We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problems
whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method
for constrained optimization. At each iteration, primal variables are updated by solving
an unconstrained variational inequality problem, and then dual variables are updated through a closed formula.
A full convergence analysis is provided, allowing for inexact solution of the subproblems.
...
Si studiano problemi di autovalori per disequazioni variazionali semilineari ellittiche con un ostacolo puntuale sulla derivata prima della funzione incognita. Si mette in particolare in evidenza il ruolo della «ipotesi di non tangenza» tra il convesso, che viene definito dalla condizione di ostacolo, e la sfera dello spazio funzionale, su cui è naturale studiare un problema di autovalori. Tale condizione viene analizzata in alcuni casi concreti e si indicano alcune ipotesi che, garantendone la...
We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form . If is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace with , which is defined as the smallest degenerate elliptic function above .
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