Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

Abstract

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in the largest natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in R 2 or R 3 we also validate the approximation of the solution of the Westervelt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.

Non-Lipschitz uniform domain shape optimization in linear acoustics

“…Theorem 10 follows similarly as [42, Theorem 3.2] by a variational convergence argument: Theorem 3 for the domains and Banach-Alaoglu and Proposition 3 for the measures µ Γ imply the existence of a subsequential limit (Ω * , µ * ) ∈ Ûad for a minimizing sequence (Ω m , µ m ) m ⊂ Ûad . The simultaneous validity of Poincaré inequalities with the same constant for all Ω m (which follows as in [20,Theorem 6] or, alternatively, by modification of the standard proof as in [22,Proposition 7.1] or [23,Section 5.8] together with the convergence the sense of characteristic functions) implies that the extensions Ext Ωm u m of the unique solutions u m on the Ω m are uniformly bounded in W 1,2 (D) and therefore have a subsequential weak limit u * . Using a variational convergence argument based on (32) and an application of Lemma 5 similarly as in the proof of Theorem 8 one can identify u * | Ω * as the unique weak solution on (Ω * , µ * ).…”

“…Theorem 7 (iii) and [3,Definition 7] motivate to define a class of domains suitable to discuss different types of boundary value problems (see also [20,49]).…”

“…Following the same arguments one can obtain versions of Theorem 8 if in (22) the space W 1,2 (Ω) is replaced by a suitable subspace of W 1,2 (Ω). For instance, one can consider V (Ω, Γ Dir ), defined as in (20).…”

“…To extend functions from closed sets to the whole space we can use harmonic extension operators, which in the case of an L 2 -framework are linear. These trace and extension results are formulated for a wider class of domains, which (roughly following [3,20,49]) we call W 1,2 -admissible domains, Definition 4. As a first application to boundary value problems we show the Mosco convergence of energy functionals associated with Robin problems on a sequence of suitably convergent domains, Theorem 8.…”

On the existence of optimal shapes in architecture

“…Remark 7 For simplicity, and because it is sufficient for many practical purposes, Γ Dir is assumed to be fixed in the above setup. It would actually be sufficient to keep the set Γ Dir ∩Γ n fixed, see [11,Theorem 10].…”