Half-eigenvalues of elliptic operators
2002 ◽
Vol 132
(6)
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pp. 1439-1451
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Keyword(s):
Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).
1979 ◽
Vol 31
(5)
◽
pp. 1107-1120
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2005 ◽
Vol 50
(3)
◽
pp. 323-329
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Keyword(s):
2007 ◽
Vol 76
(1)
◽
pp. 143-154
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1967 ◽
Vol 19
◽
pp. 667-672
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1970 ◽
Vol 13
(1)
◽
pp. 1-7
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2009 ◽
Vol 51
(3)
◽
pp. 513-524
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