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Nonlinear boundary value problems relative to harmonic functions
Nonlinear Analysis ( IF 1.4 ) Pub Date : 2020-08-28 , DOI: 10.1016/j.na.2020.112090
Y. Oussama Boukarabila , Laurent Véron

We study the problem of finding a function u verifying Δu=0 in Ω under the boundary condition un+g(u)=μ on Ω where ΩRN is a smooth domain, n is the normal unit outward vector to Ω, μ is a measure on Ω and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r)=|r|p1r, p>1, we give conditions in order an isolated singularity on Ω to be removable. We also give capacitary conditions on a measure μ in order the problem with g(r)=|r|p1r to be solvable for some μ. We also study the isolated singularities of functions satisfying Δu=0inΩ and un+g(u)=0 on Ω{0}.



中文翻译:

与谐波函数有关的非线性边值问题

我们研究寻找功能的问题 ü 验证中 -Δü=0Ω 在边界条件下 üñ+Gü=μΩ 哪里 Ω[Rñ 是一个平滑的领域, ñ 是法向单位向量 Ωμ 是对 ΩG连续的非递减函数。我们给充分条件G该问题可以通过任何措施解决。什么时候G[R=|[R|p-1个[Rp>1个,我们给出条件以使上的孤立奇点 Ω可移动。我们还给出了一定的容量条件μ 为了解决问题 G[R=|[R|p-1个[R 可以解决一些 μ。我们还研究了满足以下条件的函数的孤立奇异性-Δü=0Ωüñ+Gü=0Ω{0}

更新日期:2020-08-28
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