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Register a new userA non-concentration estimate for partially rectangular billiards
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Hans Christianson
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We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $epsilon_0>0$, an $O(lambda^{-epsilon_0})$ quasimode must have $L^2$ mass in the wings bounded below by $lambda^{-2-delta}$ for any $delta>0$. The proof uses the authors recent work on 0-Gevrey smooth domains to approximate quasimodes on $C^{1,1}$ domains. There is an improvement for $C^{2,alpha}$ domains.
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In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in cite{M1}. There, the methods developed in Burq-Zworski cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ int_U |u|^2 geq c int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
Let $Omegasubset mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $alpha>0$, define the quantity [ Lambda(alpha)=inf_{uin W^{1,p}(Omega),, u otequiv 0} Big(int_Omega | abla u|^p,mathrm{d}x - alpha int_{partialOmega} |u|^p ,mathrm{d} sBig)Big/ int_Omega |u|^p,mathrm{d} x ] with $mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Omega$ with a non-linear $alpha$-dependent Robin boundary condition. We show the asymptotics $Lambda(alpha) =(1-p)alpha^{p/(p-1)}+o(alpha^{p/(p-1)})$ as $alpha$ tends to $+infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,lambda}$ domains.
We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.
We study a class of non-unitary so(2,d) representations (for even values of d), describing mixed-symmetry partially massless fields which constitute natural candidates for defining higher-spin singletons of higher order. It is shown that this class of so(2,d) modules obeys of natural generalisation of a couple of defining properties of unitary higher-spin singletons. In particular, we find out that upon restriction to the subalgebra so(2,d-1), these representations branch onto a sum of modules describing partially massless fields of various depths. Finally, their tensor product is worked out in the particular case of d=4, where the appearance of a variety of mixed-symmetry partially massless fields in this decomposition is observed.
On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground stat is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.
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