Complete Continuity of Eigen-Pairs of Weighted Dirichlet Eigenvalue Problem |
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Authors: | Email author" target="_blank">Zhiyuan?WenEmail author Meihua?Yang Meirong?Zhang |
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Institution: | 1.School of Mathematical Sciences,Inner Mongolia University,Hohhot,China;2.School of Mathematics and Statistics,Huazhong University of Science and Technology,Wuhan,China;3.Department of Mathematical Sciences,Tsinghua University,Beijing,China;4.Zhou Pei-Yuan Center for Applied Mathematics,Tsinghua University,Beijing,China |
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Abstract: | In this paper, we will study the dependence of eigen-pairs \((\lambda _k(\rho ), \varphi _k(x,\rho ))\) of weighted Dirichlet eigenvalue problem on weights \(\rho \). It will be shown that \(\lambda _k(\rho )\) and \(\varphi _k(x,\rho )\) are completely continuous (CC) in \(\rho \). Precisely, when \(\rho _n\) is weakly convergent to \(\rho \) in some Lebesgue space, \(\lambda _k(\rho _n)\) is convergent to \(\lambda _k(\rho )\). As for the convergence of eigenfunctions, since eigenvalues may have multiple eigenfunctions, it will be shown that the distance from \(\varphi _k(x,\rho _n)\) to the eigen space \(V_k(\rho )\) of \(\lambda _k(\rho )\) is tending to zero. As applications, the CC dependence of solutions of linear inhomogeneous equations and the CC dependence of the heat kernels on coefficients will be given. |
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