论文标题
非对称稳定操作员：规律性理论和零件的集成
Non-symmetric stable operators: regularity theory and integration by parts
论文作者
论文摘要
我们研究$ lu = f $ in $ω\ subset \ mathbb r^n $的解决方案，是$ l $是任何，可能是非对称，稳定的lévy过程的生成器。 一方面，我们研究了$ω$，$ u = 0 $ in $ω^c $，$ c^{1，α} $ domains〜 $ω$的解决方案的规律性。 We show that solutions $u$ satisfy $u/d^γ\in C^{\varepsilon_\circ}\big(\overlineΩ\big)$, where $d$ is the distance to $\partialΩ$, and $γ=γ(L,ν)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ν$ to the boundary $ \partialΩ$。 另一方面，我们在此类操作员的一半空间中建立了零件身份的新集成。这些新的身份扩展了分数拉普拉斯的先前身份，但是非对称设置呈现出一些新的有趣功能。 最后，我们将零件身份在半空间中的零件身份概括为有限的$ c^{1，α} $域的情况。我们通过一个新的有效近似参数来实现这一目标，该参数利用了$ u/d^γ$的Hölder规律性。我们认为，即使在分数拉普拉斯主义者的情况下，这个新的近似论点也很有趣。
We study solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Lévy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $Ω^c$, in $C^{1,α}$ domains~$Ω$. We show that solutions $u$ satisfy $u/d^γ\in C^{\varepsilon_\circ}\big(\overlineΩ\big)$, where $d$ is the distance to $\partialΩ$, and $γ=γ(L,ν)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ν$ to the boundary $\partialΩ$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,α}$ domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of $u/d^γ$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.
