论文标题
sobolev不平等现象，凸锥上具有共同凹的重量
Sobolev inequalities with jointly concave weights on convex cones
论文作者
论文摘要
使用最佳的质量传输参数，我们证明了形式的加权sobolev不等式（\ int_e | u（x）|^q \，ω（x）\，dx \ right）^{1/q} \ leq k_0 u(x)|^p\,σ(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent.这里$ e \ subseteq \ mathbb r^n $是一个开放式凸锥，$ω，σ：e \ to（0，\ infty）$是两个均质权重，验证一般的凹入式结构状态。常数$ k_0 = k_0（n，p，q，ω，σ）> 0 $由显式公式给出。在权重的轻度规律性假设下，我们还证明$ k_0 $在（WSI）中是最佳的，并且仅当$ω$和$σ$等于乘法因子时。我们的陈述涵盖了一些先前已知的结果，包括单一重量和径向重量的病例。还提供了对PDE的进一步示例和申请。
Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form \[\left(\int_E |u(x)|^q\,ω(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,σ(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq \mathbb R^n$ is an open convex cone, and $ω,σ:E\to (0,\infty)$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0= K_0(n, p, q, ω, σ) >0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $ω$ and $σ$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.
