An isoperimetric inequality for twisted eigenvalues with one orthogonality constraint

We consider twisted eigenvalues $λ_{1}^{g}(Ω)$, defined as the minimum of the Rayleigh quotient of functions in $H^1_{0}(Ω)$ that are orthogonal to a given function $g\in L^2_\text{loc}(\mathbb R^d)$. We prove an isoperimetric inequality for $λ_1^g(Ω)$, which provides a uniform bound on twisted eigenvalues -- not only with respect to the domain $Ω$ (an open bounded set of $\mathbb R^d$) -- but also in relation to the orthogonality function $g$. Remarkably, the lower bound is uniquely attained when $Ω$ is the union of two disjoint balls of specific radii, and when the function $g$ in the orthogonality constraint is of bang-bang type, i.e., constant on each ball. As a consequence, we obtain a continuous 1-parameter family of optimal sets -- each being the union of two disjoint balls -- that interpolates between the optimal shapes of the first two Dirichlet eigenvalues of the Laplacian. This new isoperimetric inequality offers fresh perspectives on well established results, such as the Hong-Krahn-Szegő and the Freitas-Henrot inequalities. Notably, in these particular cases our proof avoids reliance on Bessel functions, suggesting potential extensions to nonlinear settings.