论文标题
非均匀dirichlet问题的全球规律性应用于非区别的域上的热核和耐寒空间
Heat Kernels and Hardy Spaces on Non-Tangentially Accessible Domains with Applications to Global Regularity of Inhomogeneous Dirichlet Problems
论文作者
论文摘要
令$ n \ ge2 $和$ω$为$ \ mathbb {r}^n $的一个有限的非区域性访问域（简称NTA域）。假设$ l_d $是具有dirichlet边界条件的$ l^2（ω）$上的实价，有界，可测量系数的椭圆形操作员。本文的主要目的是三倍。首先，作者证明了加热内核$ \ {k_t^{l_d} \} _ {t> 0} $由$ l_d $生成的$是Hölder的连续。其次，对于（0,1] $中的任何$ p \，作者通过限制hardy space $ h^p（\ mathbb {r}^n）$的任何元素来介绍“几何”'hardy space $ h^p_r（ω）$ $ h^p_r（ω）= h^p（ω）= h^p_ {l_d}（ω）$带有等价的准核电，其中$ h^p（ω）$和$ h^p_ {l_d}（l_d}（ω）$分别表示与$ω$的硬质量和$ l_d $ $ l_d $ f.内核的Hölder连续性$ \ {k_t^{l_d} \} _ {t> 0} $。 $ p \ in（\ frac {n} {n+1}，1] $，对于不均匀的dirichlet问题，二阶差异的dirichlet问题形成有限的nta域上的椭圆方程，其中$ p_0 \ in（2，\ infty）$仅在$ n $，$ n $，$ c的$ c上，$ l _ coffix coftrix of cofectrix $ l _ cofectrix cofecix cofecix y。全球梯度估算的范围$ p \ in（1，p_0）$在lebesgue Spaces $ l^p（ω）$的规模上是锐利的，并且在没有任何其他假设上建立了上述结果，这两个系数矩阵的$ l_d $均为$ l_d $，and the域$ω$。
Let $n\ge2$ and $Ω$ be a bounded non-tangentially accessible domain (for short, NTA domain) of $\mathbb{R}^n$. Assume that $L_D$ is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on $L^2(Ω)$ with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels $\{K_t^{L_D}\}_{t>0}$ generated by $L_D$ are Hölder continuous. Second, for any $p\in(0,1]$, the authors introduce the `geometrical' Hardy space $H^p_r(Ω)$ by restricting any element of the Hardy space $H^p(\mathbb{R}^n)$ to $Ω$, and show that, when $p\in(\frac{n}{n+δ_0},1]$, $H^p_r(Ω)=H^p(Ω)=H^p_{L_D}(Ω)$ with equivalent quasi-norms, where $H^p(Ω)$ and $H^p_{L_D}(Ω)$ respectively denote the Hardy space on $Ω$ and the Hardy space associated with $L_D$, and $δ_0\in(0,1]$ is the critical index of the Hölder continuity for the kernels $\{K_t^{L_D}\}_{t>0}$. Third, as applications, the authors obtain the global gradient estimates in both $L^p(Ω)$, with $p\in(1,p_0)$, and $H^p_z(Ω)$, with $p\in(\frac{n}{n+1},1]$, for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where $p_0\in(2,\infty)$ is a constant depending only on $n$, $Ω$, and the coefficient matrix of $L_D$. It is worth pointing out that the range $p\in(1,p_0)$ for the global gradient estimate in the scale of Lebesgue spaces $L^p(Ω)$ is sharp and the above results are established without any additional assumptions on both the coefficient matrix of $L_D$, and the domain $Ω$.
