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- W3049709902 abstract "Let $Omegasubset mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $partial Omega$. For $sigma>0$, we consider the Robin boundary value problem [ -Delta f =lambda f, qquad frac{partial f}{partial n} + sigma f = 0 mbox{ on } partial Omega ] where $ frac{partial f}{partial n} $ is the derivative in the direction of the outward pointing normal to $partial Omega$. Let $0<lambda^sigma_0leq lambda^sigma_1leq dots $ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps [ d_n(sigma):=lambda_n^sigma-lambda_n^0 . ] For a wide class of planar domains we show that there is a limiting mean value, equal to $2{rm length}(partialOmega)/{rm area}(Omega)cdot sigma$ and in the smooth case, give an upper bound of $d_n(sigma)leq C(Omega ) n^{1/3}sigma $ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound." @default.
- W3049709902 created "2020-08-21" @default.
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- W3049709902 date "2020-08-17" @default.
- W3049709902 modified "2023-10-16" @default.
- W3049709902 title "Differences between Robin and Neumann eigenvalues" @default.
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