Equivalent Norms of Solutions to Hyperbolic Poisson's Equations
Equivalent Norms of Solutions to Hyperbolic Poisson's Equations
复制标题
双曲泊松方程解的等效范数
DOI：
10.1007/s12220-020-00581-1
复制
发表时间：
2021
影响因子：
1.1
通讯作者：
Wang Xiantao
中科院分区：
文献类型：
--
作者：
Chen Jiaolong;Huang Manzi;Lee Seekeong;Wang Xiantao
我们假设，是具有边界条件的双曲泊松方程的解，其中是双曲拉普拉斯算子和。在Chen et al. （Calc Var Partial Differ Equ 57:32, 2018）中，本文的第一作者、第二作者和最后作者与Rasila一起研究了u的表达式，证明了，其中分别表示的泊松积分和的格林积分。在假设为常数的情况下，研究了其lipschitz型连续性。作为延续，在本文中，我们首先考虑解的存在性，并证明如果…如果映射u = P h [φ] - G h [ψ]∈C 2 （B n）rn)∩C (B¯n，R n)，则是上述狄利克雷问题的解。然后，利用快速模量，得到了与解相关的几个等价范数。这些证明主要是基于解与边界映射之间的lipschitz型连续性关系，这是一个独立的研究方向。作为一种应用，我们有Cho等人（Taiwan J Math 12:741-751, 2008）和Dyakonov （Acta Math 178:143-167, 1997）在解决方案设置中的主要结果的对应。
We assume that,is a solution to the hyperbolic Poisson equationinwith the boundary condition, whereis the hyperbolic Laplace operator and. In Chen et al. (Calc Var Partial Differ Equ 57:32, 2018), the first, the second, and the last author of this paper, together with Rasila, studied expressions ofu, and proved that, whereanddenote the Poisson integral ofand the Green integral ofwith respect to, respectively. With the assumptionis a constant), the Lipschitz-type continuity ofuwas also investigated. As a continuation, in this paper, we first consider the existence of the solutions, and demonstrate that if,,, and if the mapping u = P h [ ϕ ] - G h [ ψ ] ∈ C 2 ( B n , R n ) ∩ C ( B ¯ n , R n ) , thenuis a solution to the above Dirichlet problem. Then, by using fast majorants, we get several equivalent norms related to the solutions. The proofs are mainly based on the relationships of the Lipschitz-type continuity between the solutionsuand the boundary mappings, which are of independent interest. As an application, we have a counterpart of the main results in Cho et al. (Taiwan J Math 12:741–751, 2008) and Dyakonov (Acta Math 178:143–167, 1997) in the setting of the solutionsu.