论文标题
加权的Gagliardo-Nirenberg通过最佳运输理论和应用
Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications
论文作者
论文摘要
我们证明，在$ \ Mathbb r^n $的开放式凸锥上证明了具有三个权重的Gagliardo-Nirenberg不平等的不平等现象 - 验证了联合凹条件。如果权重彼此等于彼此，则不平等现象变得尖锐，我们明确地计算了锋利的常数。对于一定范围的参数，我们可以表征极端函数类别；在这种情况下，我们还表明，主要三个加权的主要gliardo-nirenberg不平等的清晰度意味着重量必须等于某些恒定的乘法因子。我们的方法使用最佳的质量运输理论，并仔细分析了权重的关节凹陷条件。作为应用，我们建立了具有明显锋利常数的尖锐加权$ p $ -log-sobolev，faber-krahn和Isoperimetric不平等现象。
We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of $\mathbb R^n$. If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in this case, we also show that the sharpness in the main three-weighted Gagliardo-Nirenberg inequality implies that the weights must be equal up to some constant multiplicative factors. Our approach uses optimal mass transport theory and a careful analysis of the joint concavity condition of the weights. As applications we establish sharp weighted $p$-log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp constants.
