The Dirichlet elliptic problem involving regional fractional Laplacian
The Dirichlet elliptic problem involving regional fractional Laplacian
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涉及区域分数拉普拉斯的狄利克雷椭圆问题
DOI：
10.1063/1.5046685
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发表时间：
2018-07-01
影响因子：
1.3
通讯作者：
Chen, Huyuan
中科院分区：
文献类型：
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作者：
Chen, Huyuan
本文研究了R-N（N ≥ 2）中有界正则区域Omega中包含区域分数Laplacian（E）（-Delta）（Omega）（alpha）u = f的椭圆型方程的解，其中C-2边界偏导数Omega服从Dirichlet边界g，其中alpha是（1/2，算子（-Delta）（Omega）（alpha）表示区域分数拉普拉斯算子。在f是L-2（Ω）中的元素，f是L-1中的元素的条件下，证明了当g 0时，问题（E）存在唯一的弱解（Omega，rho（beta）dx），并且f是M的元素（Omega，rho（beta）），其中rho（x）= dist（x，偏导数Ω），beta = 2 alpha - 1，且M（Omega，rho（beta））是所有Radon测度nu满足integral（Omega）rho（beta）d vertical bar nu vertical bar < + infinity的空间。最后，我们给出了具有边界数据g的经典解的分部积分公式。出版社：AIP Publishing
In this paper, we study the solutions of elliptic equations involving regional fractional Laplacian (E) (-Delta)(Omega)(alpha)u = f in a bounded regular domain Omega in R-N (N >= 2) with C-2 boundary partial derivative Omega, subject to Dirichlet boundary g on partial derivative Omega, where alpha is an element of (1/2, 1) and the operator (-Delta)(Omega)(alpha) denotes the regional fractional Laplacian. We prove that when g 0, problem (E) admits a unique weak solution under the hypotheses that f is an element of L-2(Omega), f is an element of L-1(Omega, rho(beta) dx), and f is an element of M (Omega, rho(beta)), where rho(x) = dist(x, partial derivative Omega), beta = 2 alpha - 1, and M (Omega, rho(beta)) is a space of all Radon measures nu satisfying integral(Omega) rho(beta) d vertical bar nu vertical bar < + infinity. Finally, we provide an integration by parts formula for the classical solution of (E) with boundary data g. Published by AIP Publishing.