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Register a new userA Local Faber-Krahn inequality and Applications to Schrodingers Equation
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Added by
Stefan Steinerberger
Publication date
2017
fields
Physics
and research's language is
English
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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.
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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 study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number) of the set [ left{ intint_{{u > 0}times{u>0}} frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 sigma}}d x d y : u in mathring H^sigma(mathbb{R}^n), int_{mathbb{R}^n} u^2 = 1, |{u > 0 }| leq 1right}. ] Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $mathbb{R}^n times mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.
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.
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.
A new technique was recently developed to approximate the solution of the Schroedinger equation. This approximation (dubbed ERS) is shown to yield a better accuracy than the WKB-approximation. Here, we review the ERS approximation and its application to one and three-dimensional systems. In particular, we treat bound state solutions. We further focus on random potentials in a quantum wire and discuss the solution in the context of Anderson localization.
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