逼近分数拉普拉斯算子的一族零级算子的一致等度连续性
Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian
摘要：
在本文中，我们考虑一个光滑的有界域$Ω\subset \R^N$和径向对称核$K_ε: \R^N \to \R_+$的参数族，使得对于每个$ε\in (0,1)$，它的$L^1-$范数是有限的，但它作为$ε\to 0$爆破。我们的目的是建立一个$ε$独立的连续模$Ω$，齐次Dirichlet问题的解决方案$u_ε$ \begin{equation*} \left \{ \begin{array}{rcll} - \I_ε[u] \&=\& f \& \mbox{in} \ Ω. \\ u \&=\& 0 \& \mbox{in} \ Ω^c, \end{array} \right . \end{equation*}其中$f \in C(\barΩ)$和操作$\I_ε$具有\begin{equation*} \I_ε[u](x) = \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_ε(z)dz \end{equation*}的形式，它接近分数拉普拉斯算子$ε\to 0$。连续性的模结合比较原理与$\I_ε$的平移不变性，构建合适的障碍，允许管理的解决方案$u_ε$可能对$\partial Ω$的不连续性。这一结果的扩展到完全非线性椭圆和抛物型算子进行了讨论。
英文摘要：
In this paper we consider a smooth bounded domain $Ω\subset \R^N$ and a parametric family of radially symmetric kernels $K_ε: \R^N \to \R_+$ such that, for each $ε\in (0,1)$, its $L^1-$norm is finite but it blows up as $ε\to 0$. Our aim is to establish an $ε$ independent modulus of continuity in $Ω$, for the solution $u_ε$ of the homogeneous Dirichlet problem \begin{equation*} \left \{ \begin{array}{rcll} - \I_ε[u] \&=\& f \& \mbox{in} \ Ω. \\ u \&=\& 0 \& \mbox{in} \ Ω^c, \end{array} \right . \end{equation*} where $f \in C(\barΩ)$ and the operator $\I_ε$ has the form \begin{equation*} \I_ε[u](x) = \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_ε(z)dz \end{equation*} and it approaches the fractional Laplacian as $ε\to 0$. The modulus of continuity is obtained combining the comparison principle with the translation invariance of $\I_ε$, constructing suitable barriers that allow to manage the discontinuities that the solution $u_ε$ may have on $\partial Ω$. Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed.
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