An Optimization Problem Related to the Best Sobolev Trace Constant in Thin Domains

Abstract

Let Ω ⊂ R N be a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W 1,p (Ω) → L q (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem S A = inf u p W 1,p (Ω) / u p L q (∂Ω) for functions that verify u| A = 0. It is known that there exists an optimal hole that minimizes the best constant S A among subsets of Ω of the prescribed volume.In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain.

Optimal boundary holes for the Sobolev trace constant

“…The ideas in this section follow closely the ones in [9] where the behavior of the best Sobolev trace constant for shrinking domains was analyzed and [13] where the interior set problem was studied.…”

“…So in this subsection we consider the case Ω 1 = (a, b) ⊂ R, an interval. In [13] the following Theorem regarding the limit problem for n = 1 is proved Theorem 5.6 ([13], Theorem 1.2). The optimal limit constant S(α) is attained only for a hole A * = (a, a + α(b − a)) or A * = (b − α(b − a), b), that is the best hole is an interval concentrated on one side of the interval (a, b).…”

“…Once the product case is studied, the extension to general geometries is analog to Theorem 1.1 in [9]. See also Theorem 1.3 in [13]. We omit the details.…”

“…Moreover, in the case p = 2 the interior regularity of optimal sets is studied in [14]. See [13], where some asymptotic behavior of optimal sets are studied (see also, Section 4). Further, in [8] and in [4] the so-called shape derivative for S(A) is computed with respect to regular deformations on the set A.…”

“…In [13] the following Theorem regarding the limit problem for n = 1 is proved , that is the best hole is an interval concentrated on one side of the interval (a, b). Moreover, the optimal limit constant is given by…”