The Green function \(G_0(x,y)\) for the biharmonic Dirichlet problem on a smooth domain \(\Omega \), that is \(\Delta ^{2}u=f\) in \(\Omega \) with \( u=u_{n}=0 \) on \(\partial \Omega \), can be written as the difference of a positive function, which bears the singularity at \(x=y\), and a rank-one positive function, both of which satisfy the boundary conditions. See Grunau et al. (Proc Am Math Soc 139:2151–2161, 2011). More precisely \(G_0(x,y)= H(x,y)- c\, d(x)^2 d(y)^2\) holds, where \(d(\cdot )\) is the distance to the boundary \(\partial \Omega \) and where H contains the singularity and is positive. We will extend the corresponding estimates to \( G_{\lambda }(x,y)\) for the differential operator \(\Delta ^{2}-\lambda \) with an optimal dependence on \(\lambda \). As a consequence, strict positivity of an eigenfunction with a simple eigenvalue \(\lambda _{i}\) implies a positivity preserving property for \(\left( \Delta ^{2}-\lambda \right) u=f\) in \(\Omega \) with \(u=u_{n}=0\) on \(\partial \Omega \) for \(\lambda \) in a left neighbourhood of \(\lambda _{i} \). This result can be viewed as a converse to the Krein–Rutman theorem.

中文翻译：

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与Krein–Rutman的双谐波逆向：本征函数正附近的最大原理

格林函数\（G_0（X，Y）\）用于在光滑域双调和Dirichlet问题\（\欧米茄\） ，即\（\德尔塔^ {2} U = F \）在\（\欧米茄\ ）与\（U = U_ {N} = 0 \）上\（\局部\欧米茄\） ，可写为一个正函数的差，其在承载所述奇点\（X = Y \） ，和一均满足边界条件的一阶正函数。参见Grunau等。（Proc Am Math Soc 139：2151–2161，2011年）。更精确地说\（G_0（x，y）= H（x，y）-c \，d（x）^ 2 d（y）^ 2 \）成立，其中\（d（\ cdot）\）是距离到边界\（\ partial \ Omega \）和H包含奇异且为正。我们将对差分算子\（\ Delta ^ {2}-\ lambda \）的对应估计值扩展到\（G _ {\ lambda}（x，y）\），并以对\（\ lambda \）的最佳依赖为前提。结果，具有简单特征值\（\ lambda _ {i} \）的本征函数的严格正性意味着\（\ left（\ Delta ^ {2}-\ lambda \ right）u = f \ ）在\（\欧米茄\）与\（U = U_ {N} = 0 \）上\（\局部\欧米茄\）为\（\拉姆达\）中的左侧附近\（\拉姆达_ {I} \）。该结果可以看作与Krein-Rutman定理相反。