On the existence of a solution in a domain identification problem

Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

“…Since P h1,h2 y h1,h2 belongs to H 1 0 (Ω h1,h2 ) ⊂ H 1 0 (B), the limitỹ(t) belongs to H 1 0 (B). Furthermore, since ∂Ω is Lipschitz by assumption and also ∂Ω h1,h2 by construction, it possess the so-called -cone property, introduced in [4]. This implies that the bounded sequence (Ω h1,h2 ) h1,h2 defined, for each h 1 , h 2 fixed, as the largest domain included in Ω and union of cell B j,k converges towards Ω with respect to the complementary-Hausdorff topology (see [20]).…”

“…The objects corresponding to this example are summarized on (3,1), (3,5), (7,3), (9, 0), (9, 1), (9, 2), (9, 4), (9, 5)}, (3,5), (4,5), (7,3), (8,3), (9,3), (1,6), (2,6), (5,6), (6,6), (7,6), (8,6)}.…”

An Optimum Design Problem in Magnetostatics

“…Let us recall some classical properties of the Hausdorff convergence, for the proofs we refer to [8,11], or [3]. …”

“…Moreover, we know (cf. [8] or [3]) that we are always able to choose the sequence of associated cone directions ξ Xn in order that they converge to ξ X . Therefore, for n large enough, the boundary of ω n in a neighbourhood of X n can be written as a graph of a function ψ n (L + 1)-Lipschitzian in some local coordinate system R X associated to the point X with vertical vector ξ X (see (15)).…”

“…This class can also be described as sets with the ε-cone property, and it has been introduced by Chenais in her important work, see [3,4]. In Section 1.2, we recall the definition of the ε-cone property.…”