Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula

Abstract

The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. In these spaces, geodesic balls uniquely minimize the first eigenvalue of the Dirichlet Laplacian among all sets of a fixed volume. We prove that the gap between the first eigenvalue of a given set Ω and that of the ball quantitatively controls both the L 1 distance of this set from a ball and the L 2 distance between the corresponding eigenfunctions:where B denotes the nearest geodesic ball to Ω with |B| = |Ω| and u Ω denotes the first eigenfunction with suitable normalization. On Euclidean space, this extends a result of Brasco-De Phillipis-Velichkov; the eigenfunction control largely builds upon on new regularity results for minimizers of critically perturbed Alt-Cafarelli type functionals in our companion paper. On the round sphere and hyperbolic space, the present results are the first sharp quantitative results with respect to any distance; here the local portion of the analysis is based on new implicit spectral analysis techniques.Second, we apply these sharp quantitative Faber-Krahn inequalities in order to establish a quantitative form of the Alt-Caffarelli-Friedman (ACF) monotonicity formula. A powerful tool in the study of free boundary problems, the ACF monotonicity formula is nonincreasing with respect to its scaling parameter for any pair of admissible subharmonic functions, and is constant if and only if the pair comprises two linear functions truncated to complementary half planes. We show that the energy drop in the ACF monotonicity formula from one scale to the next controls how close a pair of admissible functions is from a pair of complementary half-plane solutions. In particular, when the square root of the energy drop summed over all scales is small, our result implies the existence of tangents (unique blowups) of these functions.

Degenerate stability of some Sobolev inequalities

“…Here S d denotes the optimal constant in the Sobolev inequality on R d and Q the set of its optimizers. Importantly, the right side in (1) involves the square of the distance to the set of optimizers, and simple examples show that this is best possible, in the sense that the inequality does not hold with a right side equal to a constant times ∇u 2−α 2 inf Q∈Q ∇(u − Q) α 2 for α < 2. In the last two decades there has been an abundance of stability results for various functional inequalities.…”

“…In the last two decades there has been an abundance of stability results for various functional inequalities. Examples include, for instance, isoperimetric inequalities [31,25,17,21], L p -Sobolev inequalities [11,27,37,28], fractional Sobolev inequalities [13], Gagliardo-Nirenberg inequalities [6], Brunn-Minkowski, concentration and rearrangement inequalities [24,23,26,15,30], eigenvalue inequalities [36,10,7,33,1], solutions to elliptic equations with critical exponents [12,22,18], Young's inequality [16], Hausdorff-Young inequality [14], etc. Many of these works use strategies inspired by the paper of Bianchi-Egnell and in essentially all works (exceptions being [28,23] and one version of a refined Hölder inequality in [10]) the remainder term is quadratic in the distance to the optimizers.…”

“…We get an upper bound on the Yamabe constant by taking a constant trial function. The resulting upper bound is strictly small than the Sobolev constant on the sphere or equivalently on R d , namely S d in (1). Consequently, Lions's theorem [35,Theorem 4.1] is applicable and yields relative compactness in H 1 (M) of minimizing sequences.…”

Quantitative stability for minimizing Yamabe metrics

“…In a slightly different direction, quantitative stability estimates for critical points have been addressed for the isoperimetric inequality on Euclidean space [23,38] and for the Sobolev inequality [22,28]. Quantitative stability estimates have wideranging applications to contexts including characterization of minimizers in variational problems [21], rates of convergence of PDE [16], regularity of interfaces in free boundary problems [2], and even data science [35]. Apart from [18], all of these results make crucial use of the explicit form of minimizers and critical points or of the symmetries of the ambient space.…”

“…Critical points of the volume-normalized Einstein-Hilbert action functional, R(g) = vol g (M ) −2/2 * M R g dvol g , are Einstein metrics, i.e. metrics g satisfying Ric g = λg for some λ ∈ R where Ric g is the Ricci curvature tensor of g. The Yamabe functional Q defined in (2) is the restriction of this functional (and thus the corresponding variational problem) to a given conformal class [g]. If Y (M, [g]) ≤ 0, then the Euler-Lagrange equation corresponding to the Yamabe functional (2) (see (9)) satisfies the maximum principle and thus there is a unique critical Yamabe metric.…”