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arXiv 1906.01294math.AP
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高阶椭圆型方程和方程组的唯一性结果

Uniqueness results for higher order elliptic equations and systems

Daniele Cassani, Delia Schiera

摘要:

本文发展了Lane-Emden型多调和方程和系统的Gidas-Ni-Nirenberg技巧。至于我们所关心的Dirichlet边界条件,我们证明了唯一的解决方案,八阶方程,即涉及第四次迭代的拉普拉斯算子。然后,我们可以将结果推广到任何阶的任意多调和算子,只要满足一些自然边界条件,但不满足Dirichlet边界条件:障碍显然是一个新的现象,似乎是由于一些信息的损失,但远不清楚。当多调和算子是拉普拉斯算子的幂时,这是纳维边界条件的情况,作为副产品,解的唯一性在相当普遍的情况下成立。也建立了系统的新的存在性结果。

英文摘要:

In this paper we develop a Gidas-Ni-Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet's: the obstruction is apparently a new phenomenon and seems due to some loss of information though far from being clear. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier's boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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