First eigenvalue for the p -Laplace operator

By A. Matei. 2000

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Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, unknown): “In terms of the Dirichlet eigenvalues, the trace of the heat kernel, ZΩ(t) is given by (2.16) ZΩ(t) = ∞ ∑ n,m=1 e−λn,mt. and using (2.15), and the definition of Θ(x), we get, (2.17) ZΩ(t) = 1 4 [ Θ( π t a2 )− 1 ] [ Θ( π t b2 )− 1 ] .”
Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant (2003, background): “Cheeger’s original result [6] treated also manifolds without boundary, and an extension of this result from p = 2 to general p was done in [23], resulting in the same relation (6).”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, methodology): “[54] A.M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal.”
Sharp estimates on the first eigenvalue of the $$p$$p-Laplacian with negative Ricci lower bound (2012, background): “In [10], the author obtains lower and upper bounds on λ1,p as a function of Cheeger’s isoperimetric constant (see in particular [10, Theorems 4.”
Ricci curvature and $L^p$-convergence (2012, background): “It is worth pointing out that in [44, 46, 56], Matei, Naber-Valtorta, and Valtorta give the sharp lower bounds on λ1,p.”