An isoperimetric inequality for twisted eigenvalues with one orthogonality constraint

We consider twisted eigenvalues \(\lambda _{1}^{g}(\Omega )\) ${\lambda }_1^g(\mathrm{\Omega })$ , defined as the minimum of the Rayleigh quotient of functions in \(H^1_{0}(\Omega )\) $H_0^1(\mathrm{\Omega })$ that are orthogonal to a given function \(g\in L^2_{\text {loc}}(\mathbb {R}^d)\) $g\in L_{\text{loc}}^2({\mathbb{R}}^d)$ . We prove an isoperimetric inequality for \(\lambda _1^g(\Omega )\) ${\lambda }_1^g(\mathrm{\Omega })$ , which provides a uniform bound on twisted eigenvalues – not only with respect to the set \(\Omega \) $\mathrm{\Omega }$ (an open bounded set of \(\mathbb {R}^d\) ${\mathbb{R}}^d$ ) – but also in relation to the orthogonality function g. Remarkably, the lower bound is uniquely attained when \(\Omega \) $\mathrm{\Omega }$ 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-Szego and the Freitas-Henrot inequalities. Notably, only for these two particular inequalities our proof avoids reliance on Bessel functions, suggesting potential extensions to nonlinear settings. However, extending the inequalities to the general case requires proof strategies that rely on properties of Bessel functions.