Isoperimetric inequalities for the fractional composite membrane problem

In this article, we investigate some isoperimetric-type inequalities related to the first eigenvalue of the fractional composite membrane problem. First, we establish an analogue of the renowned Faber–Krahn inequality for the fractional composite membrane problem. Next, we investigate an isoperimetric inequality for the first eigenvalue of the fractional composite membrane problem on the intersection of two domains - a problem that was first studied by Lieb (Invent Math 74(3):441–448, 1983) for the Laplacian. Similar results in the local case were previously obtained by Cupini–Vecchi (Commun Pure Appl Anal 18(5):2679–2691, 2019) for the composite membrane problem. Our findings provide further insights into the fractional setting, offering a new perspective on these classical inequalities.