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Register a new userSharp quantitative stability for isoperimetric inequalities with homogeneous weights
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We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromovs proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
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We prove that a plane domain which is almost isoperimetric (with respect to the $L^1$ metric) is close to a square whose sides are parallel to the coordinates axis. Closeness is measured either by $L^infty$ Haussdorf distance or Fraenkel asymmetry. In the first case, we determine the extremal domains.
Let $A_infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $mathsf M^+:L^p(w)to L^{p,infty}(w)$ for some $p>1$, where $mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $win A_infty ^+$ if and only if there exist numerical constants $gammain(0,1)$ and $c>0$ such that $$ w({x in mathbb{R} : , mathsf M ^+mathbf 1_E (x)>gamma})leq c w(E) $$ for all measurable sets $Esubset mathbb R$. Furthermore, letting $$ mathsf C_w ^+(alpha):= sup_{0<w(E)<+infty} frac{1}{w(E)} w({xinmathbb R:,mathsf M^+mathbf 1_E (x)>alpha}) $$ we show that for all $win A_infty ^+$ we have the asymptotic estimate $mathsf C_w ^+ (alpha)-1lesssim (1-alpha)^frac{1}{c[w]_{A_infty ^+}}$ for $alpha$ sufficiently close to $1$ and $c>0$ a numerical constant, and that this estimate is best possible. We also show that the reverse Holder inequality for one-sided Muckenhoupt weights, previously proved by Martin-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of $A_infty ^+$. Our methods also allow us to show that a weight $win A_infty ^+$ satisfies $win A_p ^+$ for all $p>e^{c[w]_{A_infty ^+}}$.
We consider the punctured plane with volume density $|x|^alpha$ and perimeter density $|x|^beta$. We show that centred balls are uniquely isoperimetric for indices $(alpha,beta)$ which satisfy the conditions $alpha-beta+1>0$, $alphaleq 2beta$ and $alpha(beta+1)leqbeta^2$ except in the case $alpha=beta=0$ which corresponds to the classical isoperimetric inequality.
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 for any open set $Omega$, [ lambda_1(Omega) - lambda_1(B) gtrsim |Omega Delta B|^2 + int |u_{Omega} - u_B|^2, ] where $B$ denotes the nearest geodesic ball to $Omega$ with $|B|=|Omega|$ and $u_Omega$ 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.
We present reverse Holder inequalities for Muckenhoupt weights in $mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_infty$ weights with Fujii-Wilson constant $(w)_{A_infty}to 1^+$. That is, the local integrability exponent in the reverse Holder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Holder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Holder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Holder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.
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