Relaxation for Optimal Design Problems with Non-standard Growth

Abstract

In this paper we investigate the possibility of obtaining a measure representation for functionals arising in the context of optimal design problems under non-standard growth conditions and perimeter penalization. Applications to modelling of strings are also provided. MSC (2010): 49J45, 74K10

Relaxation for an optimal design problem inBD(Ω)

“…We must now build a transition sequence between and , in such a way that an upper bound for the total variation of is obtained. In order to connect these functions without adding more interfaces, we argue as in [13] (see also [14]). For consider where is small enough so that in and Given , let .…”

“…It is sometimes convenient to replace this constraint by inserting, instead, a Lagrange multiplier in the modelling functional which, in the optimal design context, becomes Despite the fact that we have compactness for u in BD(Ω) for functionals of the form (1.3), it is well known that the problem of minimizing (1.3) with respect to (χ, u), adding suitable forces and/or boundary conditions, is ill-posed, in the sense that minimizing sequences χ n ∈ L ∞ (Ω; {0, 1}) tend to highly oscillate and develop microstructure, so that in the limit we may no longer obtain a characteristic function. To avoid this phenomenon, as in [2,34], we add a perimeter penalization along the interface between the two zones {χ = 0} and {χ = 1} (see [21] for the analogous analysis performed in BV , and [14,15,20] for the Sobolev settings, also in the presence of a gap in the growth exponents).…”

A Γ-convergence result for optimal design problems

“…In [14], the authors obtain the relaxation of an optimal design model involving fractured media which induces an analysis in the space of special bounded variation functions. In [3], in the context of linear growth and still with a perimeter penalization, the authors derive a lower semicontinuity result and a measure representation result for the relaxation of optimal design functionals.…”

Relaxation for an optimal design problem in $BD(Ω)$

“…Despite the fact that we have compactness for u in BD(Ω) for functionals of the form (1.3), it is well known that the problem of minimising (1.3) with respect to (χ, u), adding suitable forces and/or boundary conditions, is ill-posed, in the sense that minimising sequences χ n ∈ L ∞ (Ω; {0, 1}) tend to highly oscillate and develop microstructure, so that in the limit we may no longer obtain a characteristic function. To avoid this phenomenon, as in [2] and [35], we add a perimeter penalisation along the interface between the two zones {χ = 0} and {χ = 1} (see [21] for the analogous analysis performed in BV , and [20,14,15] for the Sobolev settings, also in the presence of a gap in the growth exponents).…”