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H 1 -BMO duality on graphs

Emmanuel Russ — 2000

Colloquium Mathematicae

On graphs satisfying the doubling property and the Poincaré inequality, we prove that the space H m a x 1 is equal to H a t 1 , and therefore that its dual is BMO. We also prove the atomic decomposition for H m a x p for p ≤ 1 close enough to 1.

Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds

Emmanuel Russ;  Yannick Sire — 2011

Studia Mathematica

Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.

Symmetrization of functions and principal eigenvalues of elliptic operators

François Hamel;  Nikolai Nadirashvili;  Emmanuel Russ

Séminaire Laurent Schwartz — EDP et applications

In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of ℝ n . We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator - Δ + v · ∇ , for which the minimization problem is still well posed. Next, we deal with...

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