Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results. (English) Zbl 1481.35006
Summary: Interpolation inequalities play an essential role in analysis with fundamental consequences in mathematical physics, nonlinear partial differential equations (PDEs), Markov processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.
MSC:
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35K55 | Nonlinear parabolic equations |
| 46B70 | Interpolation between normed linear spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 49J40 | Variational inequalities |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49K30 | Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
Keywords:
interpolation; Gagliardo-Nirenberg inequality; Caffarelli-Kohn-Nirenberg inequality; entropy methods; carré du champ; fast diffusion equation; Harnack principle; asymptotic behaviour; Hardy-Poincaré inequalities; spectral gap; intermediate asymptotics; self-similar Barenblatt solutions; rates of convergence; symmetry; symmetry breaking; bifurcation; entropy-entropy production inequalitiesReferences:
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