TY - JOUR
T1 - Continuity of nonlinear eigenvalues in CD (K, ∞) spaces with respect to measured Gromov–Hausdorff convergence
AU - Ambrosio, Luigi
AU - Honda, Shouhei
AU - Portegies, Jacobus W.
N1 - Funding Information:
Acknowledgements The first author acknowledges the support of the MIUR PRIN 2015 grant. The second author acknowledges the support of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, the Grantin-Aid for Young Scientists (B) 16K17585 and the warm hospitality of SNS. The third author thanks Mark Peletier, Georg Prokert and Oliver Tse for helpful discussions and the SNS for its hospitality.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2018/4/1
Y1 - 2018/4/1
N2 - In this note we prove in the nonlinear setting of CD (K, ∞) spaces the stability of the Krasnoselskii spectrum of the Laplace operator -Δ under measured Gromov–Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of CD ∗(K, N) metric measure spaces with uniformly bounded diameter. Additionally, we show that every element λ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial u satisfying the eigenvalue equation -Δu=λu.
AB - In this note we prove in the nonlinear setting of CD (K, ∞) spaces the stability of the Krasnoselskii spectrum of the Laplace operator -Δ under measured Gromov–Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of CD ∗(K, N) metric measure spaces with uniformly bounded diameter. Additionally, we show that every element λ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial u satisfying the eigenvalue equation -Δu=λu.
KW - 49J35
KW - 49J52
KW - 49R05
KW - 58J35
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U2 - 10.1007/s00526-018-1315-0
DO - 10.1007/s00526-018-1315-0
M3 - Article
AN - SCOPUS:85041898253
SN - 0944-2669
VL - 57
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 2
M1 - 34
ER -