For bounded C^2 convex domains and p>1, the first weighted p-Laplace eigenfunction is log-concave and its first eigenvalue satisfies λ(Ω_t) ≤ (1-t)λ(Ω_0)+tλ(Ω_1).
Barles, Remarks on uniqueness results of the first eigenvalue of the p-Laplacian, Ann
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log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator
For bounded C^2 convex domains and p>1, the first weighted p-Laplace eigenfunction is log-concave and its first eigenvalue satisfies λ(Ω_t) ≤ (1-t)λ(Ω_0)+tλ(Ω_1).