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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Adams, R.A., Fournier, J.F.: Sobolev Spaces. Elsevier Pte Ltd., Singapore (2009)
Baffico, L., Conca, C., Rajesh, M.: Homogenization of a class of nonlinear eigenvalue problems. Proc. R. Soc. Edinburgh Sect. A 136, 7–22 (2006)
Berger, M., Gauduchon, P., Mazet, E.: Le Spectre dúne Variété Riemannienne, Lecture Notes in Math., 194. Springer, Berlin (1971)
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley, New York (1953)
Cuesta, M., Ramos Quoirin, H.: A weighted eigenvalue problem for the \(p\)-Laplacian plus a potential. NoDEA Nonlinear Differ. Equ. Appl. 16, 469–491 (2009)
Evans, L.: Partial Differential Equations. Am. Math. Soc, Providence, RI (2003)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2003)
Grigor’yan, A.: Heat kernels on weighted manifolds and applications. Contemp. Math. 398, 93–191 (2006)
Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. Amer. Math. Soc. and Int. Press (2009)
Ladyzhenskaya, O.A., Uralt́seva, N.N.: Linear and Quasilinear Elliptic Equations. Academic, New York (1968)
Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983)
Lou, Y., Yanagida, E.: Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics. Japan J. Ind. Appl. Math. 23, 275–292 (2006)
Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)
Meng, G., Zhang, M.: Continuity in weak topology: first order linear systems of ODE. Acta Math. Sin. Engl. Ser. 26, 1287–1298 (2010)
Salort, A.M.: Convergence rates in a weighted Fucik problem. Adv. Nonlinear Stud. 14, 385–402 (2014)
Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, New York (1979)
Yau, S.T., Schoen, R.: Lectures on Differential Geometry. Int. Press (1994)
Zhang, M.: Continuity in weak topology: Higher order linear systems of ODE. Sci. China Ser. A 51, 1036–1058 (2008)
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The first author was supported by the scientific starting foundation of Inner Mongolia University (Grant No. 21200-5175108). The second author was supported by the National Natural Science Foundation of China (Grant No. 11571125). The third author was supported by the National Natural Science Foundation of China (Grant No. 11231001, No. 11371213, and No. 11790273).
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Wen, Z., Yang, M. & Zhang, M. Complete Continuity of Eigen-Pairs of Weighted Dirichlet Eigenvalue Problem. Mediterr. J. Math. 15, 73 (2018). https://doi.org/10.1007/s00009-018-1118-8
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DOI: https://doi.org/10.1007/s00009-018-1118-8