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Structure of the spectrum and estimates for the eigenvalues of nonlinear homogeneous operators
V. R. Kardashov
Abstract:
In this paper conditions are given for the spectrum in an eigenvalue problem of the form $$\lambda A(u)=B(u)$$ to be discrete, where $A$ and $B$ are operators that are odd-homogeneous of degree $p-1$ $(p\geqslant2)$, acting from a reflexive Banach space into the dual. It is proved that the eigenvalues vary monotonically as $A$ and $B$ vary in the normed linear space of homogeneous operators of degree $p-1$. Explicit formulas for the eigenvalues and functions are obtained for the case where $A$ and $B$ are the gradients of the norms in the spaces $W_p^1[\Omega_0]$ and $L_p[\Omega_0]$ ($\Omega_0$ is a parallelepiped in $E^m$). Using these formulas the author obtains estimates for the eigenvalues in homogeneous and asymptotically homogeneous problems with variable coefficients in the space $\overset{\circ}{W_p^1}[\Omega]$, where $\Omega$ is an arbitrary bounded domain in $E^m$.
Bibliography: 12 titles.
Received: 15.06.1981
Citation:
V. R. Kardashov, “Structure of the spectrum and estimates for the eigenvalues of nonlinear homogeneous operators”, Mat. Sb. (N.S.), 120(162):3 (1983), 354–370; Math. USSR-Sb., 48:2 (1984), 349–363
Linking options:
https://www.mathnet.ru/eng/sm2135https://doi.org/10.1070/SM1984v048n02ABEH002679 https://www.mathnet.ru/eng/sm/v162/i3/p354
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Abstract page: | 282 | Russian version PDF: | 98 | English version PDF: | 18 | References: | 33 |
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