论文标题
$ c^{1,1} $开放集的非本地椭圆和抛物线方程的Sobolev规则性理论
Sobolev regularity theory for the non-local elliptic and parabolic equations on $C^{1,1}$ open sets
论文作者
论文摘要
我们研究椭圆方程的零外部问题$$δ^{α/2} u-λu= f，\ quad x \ in d \ in;; \ quad u | _ {d^c} = 0 $$以及抛物线方程$$ u_t =δ^{α/2} u+f，\ quad t> 0，\，\，x \ in d \ in; \ quad u（0，\ cdot）| _d = u_0，\，\，u | _ {[0，t] \ times d^c} = 0。 $$这里，$α\ in（0,2）$，$λ\ geq 0 $和$ d $是$ c^{1,1} $ open集。我们证明了加权Sobolev空间中解决方案的独特性和存在，并获得了全球Sobolev和Hölder估计解决方案及其任意阶衍生物的估计。我们使用由与边界距离的适当力量组成的权重系统测量解决方案及其任意导数的Sobolev和Hölder规律性。到边界距离的可接受力量的范围是锋利的。
We study the zero exterior problem for the elliptic equation $$ Δ^{α/2}u-λu=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=Δ^{α/2}u+f, \quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $α\in (0,2)$, $λ\geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and Hölder estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and Hölder regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.
