论文标题
用于准线性方程的强大权重的空间：Maz'ya型的特征和足够的条件，以使最小化的存在
The space of Hardy-weights for quasilinear equations: Maz'ya-type characterization and sufficient conditions for existence of minimizers
论文作者
论文摘要
令$ p \ in（1，\ infty）$和$ω\ subset \ mathbb {r}^n $为域。令$ a：=（a_ {ij}）\ in l^{\ infty} _ {\ text {loc}}（ω; \ m mathbb {r}^{n \ times n}）$是符号符号和本地均匀的正阳性矩阵。设置$ |ξ| _a^2：= \ displayStyle \ sum_ {i，j = 1}^n a_ {ij}（x）ξ_iξ_j$，$ξ\ in \ mathbb {r}^n $，让$ v $在某些当地的Morrey空间中具有潜在的潜力。我们假设能量功能 $ q_ {p，a，v}（ϕ）：= \ displayStyle \int_Ω[| \ nabla ϕ | _a^p + v | ϕ |^p] {\ rm dx} $ $在$ W^{1，p}（p}（ω）\ cap c_c（ω）中是$ W^{1，P}（ω）中的$ w^^{\ rm dx} $。 我们介绍了$ q_ {p，a，v} $的广义概念，并表征了功能性$ q_ {p，a，v} $的所有强大量的空间，从而扩展了马兹亚（Maz'ya）对$ p $ laplacian的Hardy-Weights的众所周知的表征。此外，我们为潜在的$ V $和HARTY权威$ g $提供了各种足够的条件，以便在适当的Beppo-Levi空间中实现相应变分问题的最佳常数。
Let $p \in (1,\infty)$ and $Ω\subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(Ω; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|ξ|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) ξ_i ξ_j$, $ξ\in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(ϕ):=\displaystyle \int_Ω [|\nabla ϕ|_A^p + V|ϕ|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(Ω)\cap C_c(Ω)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all Hardy-weights for the functional $Q_{p,A,V}$, extending Maz'ya's well known characterization of the space of Hardy-weights for the $p$-Laplacian. In addition, we provide various sufficient conditions on the potential $V$ and the Hardy-weight $g$ such that the best constant of the corresponding variational problem is attained in an appropriate Beppo-Levi space.
