Functional Inequalities: Nonlinear Flows and Entropy Methods as a Tool for Obtaining Sharp and Constructive Results

Functional inequalities and applications to doubly nonlinear diffusion equations

“…In the range of parameters under consideration, inequality (32) does not hold for any function f ∈ C ∞ c ( N ), due to scaling arguments. This is very different from the case, 1 > m(p −1) > 1−(p −1)/N , where (32) is equivalent to a class of Gagliardo-Nirenberg-Sobolev inequalities, see [3,15,36,29]. However, when v(τ) is close enough to B, then (32) holds.…”

Monotonicity of the period and positive periodic solutions of a quasilinear equation

“…The study of (3) is motivated by rigidity and symmetry breaking results associated with interpolation inequalities on the unit sphere S d in one and higher dimensions, that is, d ≥ 1. If p = 2, a precise description of the threshold value of λ is known in the framework of Markov processes if q is not too large (see [3] for an overview with historical references that go back to [2]) and from [5,11,14,15,16,17,13] using entropy methods applied to nonlinear elliptic and parabolic equations; also see [12] for an overview and extensions to various related variational problems.…”

Monotonicity of the Period and Positive Periodic Solutions of a Quasilinear Equation

“…The study of (3) is motivated by rigidity and symmetry breaking results associated with interpolation inequalities on the unit sphere S d in one and higher dimensions, that is, d ≥ 1. If p = 2, a precise description of the threshold value of λ is known in the framework of Markov processes if q is not too large (see [3] for an overview with historical references that go back to [2]) and from [5,11,14,15,16,17,13] using entropy methods applied to nonlinear elliptic and parabolic equations; also see [12] for an overview and extensions to various related variational problems.…”