插值不等式、非线性流、边界项、最优性和线性化
Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization
摘要：
研究了一类带权的快速扩散方程的熵方法中渐近边界项的计算.到目前为止，只考虑了椭圆型方程，我们的目标是证明，至少部分地，carr{é} du champ / Bakry-Emery / R{é}nyi熵方法到抛物型方程的扩展.这是有意义的，因为即使只考虑椭圆型方程，发展方程也是该方法的核心，但这也提出了关于存在权时解的正则性和增长性的难题。我们还研究了熵-熵生产不等式中的最优常数，信息-信息生产不等式中的最优常数，在假设权在x=0处不引入奇异边界项的条件下，讨论了Caffarelli-Kohn-Nirenberg不等式在发展方程作用下广义R{é}nyi熵幂的渐近增长率和对称破缺问题的最佳参数范围.即使在没有权重的情况下，这些考虑也是新的。例如，我们建立了carr{é} du champ和R{é}nyi熵方法的等价性，并解释了为什么熵方法在没有权重的情况下产生熵-熵产生和Gagliardo-Nirenberg不等式中的最佳常数，或者当权重存在时产生最佳对称范围。
英文摘要：
This paper is devoted to the computation of the asymptotic boundary terms in entropy methods applied to a fast diffusion equation with weights associated with Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, only elliptic equations have been considered and our goal is to justify, at least partially, an extension of the carr{é} du champ / Bakry-Emery / R{é}nyi entropy methods to parabolic equations. This makes sense because evolution equations are at the core of the heuristics of the method even when only elliptic equations are considered, but this also raises difficult questions on the regularity and on the growth of the solutions in presence of weights.We also investigate the relations between the optimal constant in the entropy - entropy production inequality, the optimal constant in the information - information production inequality, the asymptotic growth rate of generalized R{é}nyi entropy powers under the action of the evolution equation and the optimal range of parameters for symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities, under the assumption that the weights do not introduce singular boundary terms at x=0. These considerations are new even in the case without weights. For instance, we establish the equivalence of carr{é} du champ and R{é}nyi entropy methods and explain why entropy methods produce optimal constants in entropy - entropy production and Gagliardo-Nirenberg inequalities in absence of weights, or optimal symmetry ranges when weights are present.
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