论文标题
Dirichlet到Neumann操作员的非对角线范围
Off-diagonal bounds for the Dirichlet-to-Neumann operator
论文作者
论文摘要
令$ω$为$ \ mathbb {r}^{n+1} $的有限域，$ n \ ge 1 $。我们假设$ω$的边界$γ$是Lipschitz。考虑Dirichlet到Neumann运算符$ n_0 $与系统尺寸$ m $的系统相关联，带有真正的对称和H \''的较旧的连续系数。我们证明了$ l^p（γ）\ to l^q（γ）$ off-diagonal形式的$$ \ | | 1_f e^{ - t n_0} 1_e f \ | _q \ simeSim（t \ wedge 1）^{\ frac {n} {q} {q} - \ frac {n} {p}} {p}}}} \ left（1 + \ frac {dist（1 + \ frac {dist） 1_e f \ | _p $$用于所有可测量的子集$ e $和$ f $ $γ$。如果$γ$是$ c^{1+κ} $对于某些$κ> 0 $和$ m = 1 $，我们从$ \ left（1 + \ frac {dist（e，f）} {t} {t} {t} \ right）^{ - 1} $可以替换为$ \ weft（1 + frac} {t} {t} {t} {t} { \ right）^{ - （1 + \ frac {n} {p} - \ frac {n} {q}）} $。此类界限也有效期为复杂的时间。对于$ n = 1 $，我们应用我们的偏外界，以证明与系统相关的dirichlet到neumann运算符会在所有$ p \ in（1，\ infty）$上生成$ l^p（γ）$的分析半群。此外，相应的进化问题具有$ l^q（l^p）$ - 最大规律性。
Let $Ω$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $Γ$ of $Ω$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$ with real symmetric and H\''older continuous coefficients. We prove $L^p(Γ)\to L^q(Γ)$ off-diagonal bounds of the form$$ \| 1_F e^{-t N_0} 1_E f \|_q \lesssim (t \wedge 1)^{\frac{n}{q}-\frac{n}{p}} \left( 1 + \frac{dist(E,F)}{t} \right)^{-1} \| 1_E f \|_p$$for all measurable subsets $E$ and $F$ of $Γ$. If $Γ$ is $C^{1+ κ}$ for some $κ> 0$ and $m=1$, we obtain a sharp estimate in the sense that $ \left( 1 + \frac{dist(E,F)}{t} \right)^{-1}$ can be replaced by$ \left( 1 + \frac{dist(E,F)}{t} \right)^{-(1 + \frac{n}{p} - \frac{n}{q})}$. Such bounds are also valid for complex time. For $n=1$, we apply our off-diagonal bounds to prove that the Dirichlet-to-Neumann operator associated with a system generates an analytic semigroup on $L^p(Γ)$ for all $p \in (1, \infty)$. In addition, the corresponding evolution problem has $L^q(L^p)$-maximal regularity.
