Spectral asymptotics for an elliptic operator in a locally periodic perforated domain

Subcritical perturbation of a locally periodic elliptic operator

“…These particular terms are In Theorem 2.1, we have established asymptotic series approximations for the eigenpairs . " j , u " j / to the spectral problem (1). The number of terms that are included in the series depend on the value of Ä.˛/, and most terms depend on the value of q.˛/.…”

“…In case (i), the eigenfunctions to localize in the scale ϵ α /4 in the vicinity of a fixed point, which is determined by the potential function c ( x , y ) and the magnitude of the oscillations is o (1) as ϵ tends to zero (e.g., ). In case (iii), which we call the critical case, the eigenfunctions localize in the scale at a fixed point which is determined by a ( x , y ) and c ( x , y ), and the magnitude of the oscillations is O (1) as ϵ tends to zero, which leads to a shift of order O (1) in the spectrum as demonstrated in .…”

“…In the present paper, we will construct the leading terms in the asymptotics of the eigenvalues " and an L 2 . / approximation to the eigenfunctions u " to (1).…”

“…In particular, the main contribution to the energy of the eigenfunctions is given by the second term in the asymptotic series approximation. The analysis of this case has not been included in the previous studies of the subcritical case [18,11], and our purpose here is to fill this gap between (i) and (iii).…”

Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

“…Boundary value and spectral problems in thin domains are usually treated using the analysis of resolvents ( [FS09]), the method of asymptotic expansions (see for example [CD79], [Pan05], [BF10], [MP10], [Naz01], [PS13]), two-scale convergence ( [EP96], [MMP00], [PP11], [PP15]), Γ-convergence ( [MS95], [AB01], [BFF00], [Gau+02], [BMT07], [BMT12]), compensated compactness agrument ( [GM03]), and the unfolding method ( [BG08], [AP11], [AVP17]). The presented list of works devoted to the homogenization in thin structures is far from being complete, but our primary focus is the case of thin domains with locally periodic rapidly varying thickness, and to our best knowledge the works closely related to our study are [MP10], [AP11], [FS09], [BF10], and [NPT16].…”