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.
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
UR - http://www.scopus.com/inward/record.url?scp=85041898253&partnerID=8YFLogxK
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 -