Strict monotonicity of the first q-eigenvalue of the fractional p-Laplace operator over annuli

Abstract

In this talk, we consider the first q-eigenvalue of the fractional (s,p)-Laplace operator over annular domains with homogeneous nonlocal Dirichlet boundary conditions for p>1, 0<s<1 and 1<q<p_s^*, where p_s^* is the critical fractional Sobolev exponent. We prove that the first q-eigenvalue is strictly decreasing as the inner ball moves towards the outer boundary of an annular domain. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for the first q-eigenvalue under polarization and also a version of the strong comparison principle involving a Sobolev function and its polarized function. This extends some monotonicity results obtained by Djitte-Fall-Weth [Calc. Var. Partial Differential Equations, 60:231, 2021] in the case of p=2 and q=1, 2 to the case of p>1 and 1<q<p^*_s. Our method bypasses the use of the shape derivative formula which is challenging and highly depends on the structure of the differential operator involved. Furthermore, we give some generalizations to differences of Steiner symmetric and foliated Schwarz symmetric domains.