Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula
282
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic 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 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.
rate research
Read More
For a domain $Omega subset mathbb{R}^n$ and a small number $frak{T} > 0$, let [ mathcal{E}_0(Omega) = lambda_1(Omega) + {frak{T}} {text{tor}}(Omega) = inf_{u, w in H^1_0(Omega)setminus {0}} frac{int | abla u|^2}{int u^2} + {frak{T}} int frac{1}{2} | abla w|^2 - w ] be a modification of the first Dirichlet eigenvalue of $Omega$. It is well-known that over all $Omega$ with a given volume, the only sets attaining the infimum of $mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $Omega$ with the same volume and barycenter as $B_R$ and whose boundaries are parametrized as small $C^2$ normal graphs over $partial B_R$ with bounded $C^2$ norm, [ int |u_{Omega} - u_{B_R}|^2 + |Omega triangle B_R|^2 leq C [mathcal{E}_0(Omega) - mathcal{E}_0(B_R)] ] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $Omega$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_{Omega}$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $Omega$. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.
We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $Delta + V$ on an arbitrary domain $Omega$ in $mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 in Omega$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $Omega$ with probability $ge 1/2$. For nice (e.g., convex) domains, $T(x_0) asymp d(x_0,partialOmega)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $asymp T(x_0)^{1/2}$ such that $$ | V |_{L^{frac{n}{2}, 1}(Omega cap B)} ge c_n > 0, $$ provided that $n ge 3$. In the case $n = 2$, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.
In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $mu_1(Omega)$ of the fully nonlinear eigenvalue problem [ label{eq} left{begin{array}{r c l l} -lambda_N(D^2 u) & = & mu u & text{in }Omega, u & = & 0 & text{on }partial Omega. end{array}right. ] Here $ lambda_N(D^2 u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $Omega subset mathbb{R}^N$, the inequality [ mu_1(Omega) leq frac{pi^2}{[text{diam}(Omega)]^2} = mu_1(B_{text{diam}(Omega)/2}),] where $text{diam}(Omega)$ is the diameter of $Omega$, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. Furthermore, we discuss the minimization of $mu_1(Omega)$ under different kinds of constraints.
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.
We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrodinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schrodinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
