论文标题
neumann eigenfunctions $ c^{\ infty} $有限的平面域的非集中和限制范围
Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\infty}$ bounded planar domains
论文作者
论文摘要
令$（ω，g）$为$ \ r^2 $中的分段平滑，有界的凸域，并考虑$ l^2 $ - 正态化的neumann eigenfunctions $ ϕ_λ $，带有eigenvalue $λ^2 $^2 $和$u_λ和$u_λ：= = = = ϕ_λ_λ | _ { $ ϕ_λ $）。我们的第一个主要结果（定理\ ref {t：non-con}）是一个小规模{\ em non-centration} estimate：我们证明，我们证明，对于{\ em any} $ x_0 \ in \ in \intllineΩ，$（包括边界角点）和[0,1,1） ϕ_h \ | _ {b（x_0，λ^{ - δ}）\capΩ} = o（λ^{ - δ/2}）。$$我们的后续结果涉及非集中估计的应用到$ l^2 $ l^2 $限制边界eigenfunctions tht undereSers cormanes cormaners ordarysers yourgansers yourgansers yourgansers的上限。特别是，在定理\ ref {dirichlet}中，我们证明，对于任何{\ em flat}边界边缘$γ$（可能包括转角点），边界限制$ u_h：= ϕ_H | _ {\ partialω} ε}），$$对于任何$ε>0。$指数$ 1/4 $都很尖锐，并且结果在$ O（λ^{1/3}）$ o（λ^{1/3}）$ universal $ l^2 $ - 范围内，由于tataru \ cite {ta}而绑定了Neumann eigenfunctions。 $ o（λ^{1/4}）$ bund也是沿着完全对象的超曲面的Burq-gerard-tzvetkov \ cite {bgt}的著名内部$ l^2 $限制边界的边界（包括角点）的扩展。
Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for $L^2$ restrictions of boundary eigenfunctions that are valid up to boundary corners. In particular, in Theorem \ref{dirichlet} we prove that for any {\em flat} boundary edge $Γ$ (possibly including corner points), the boundary restrictions $u_h:= ϕ_h |_{\partial Ω}$ satisfy the bounds $$ \|u_λ \|_{L^2(Γ)} = O_ε(λ^{1/4 + ε}),$$ for any $ε>0.$ The exponent $1/4$ is sharp and the result improves on the $O(λ^{1/3})$ universal $L^2$-restriction bound for Neumann eigenfunctions due to Tataru \cite{Ta}. The $O(λ^{1/4})$ -bound is also an extension to the boundary (including corner points) of well-known interior $L^2$ restriction bounds of Burq-Gerard-Tzvetkov \cite{BGT} along totally-geodesic hypersurfaces.
