Stability for the Sobolev inequality in cones
Filomena Pacella, Giulio Ciraolo, Camilla Chiara Polvara
TL;DR
This work proves a quantitative, BE-type stability estimate for the Sobolev inequality on Lipschitz cones $\Sigma_D$, tying the deficit $\Delta_D(\varphi)=\|\nabla\varphi\|_2^2-S_D^2\|\varphi\|_{2^*}^2$ to the squared distance from $\varphi$ to a finite minimizer manifold. The analysis hinges on a spectral study of the linearized operator $\mathcal{L}_V=V^{2-2^*}\Delta$ around minimizers: for generic nondegenerate local minimizers, the first two eigenvalues satisfy $\mu_1=S_V^{2^*}$ with eigenfunction $V$ and $\mu_2=(2^*-1)S_V^{2^*}$ with eigenfunction $\partial_sV_s|_{s=1}$, and the third eigenvalue controls the quadratic deficit via a sharp expansion. When the minimizer is the bubble $U$, a separation of variables reduces the problem to a radial-angular eigenvalue analysis involving $\lambda_j(D)$, giving explicit formulas for $\mu_{k,\lambda_j}$ and establishing nondegeneracy for many cones provided $\lambda_1(D)>N-1$. The global result follows by compactness, yielding a finite-family minimizer stability and, in the bubble regime, sharp constants and symmetry-based strictness in several cones, with implications for the optimality of local estimates and for the structure of minimizers on cones.
Abstract
We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate, which is the case of most cones. When the minimizers are the classical bubbles we have more precise results. Finally, we show that local estimates are not enough to get the optimal constant for the quantitative Sobolev inequality.