Abstract

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Namely, we study the unique solvability of the mixed biharmonic problem with the Steklov and Steklov-type conditions on the boundary in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet integral with weight \(|x|^{a}\). For solving this biharmonic problem with application we use the variational principle, and depending on the value of the parameter \(a\), we obtained uniqueness (non-uniqueness) theorems or present exact formulas for the dimension of the space of solutions.

中文翻译：

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混合 Steklov 和 Steklov 型边界条件的外双调和问题

摘要

我们研究了无界域中广义解的性质以及无穷远椭圆边值问题解的渐近行为。也就是说，我们研究了混合双调和问题在紧集外部边界上具有 Steklov 和 Steklov 类型条件的唯一可解性，假设该问题的广义解具有权重\(|x |^{a}\)。为了解决这个应用的双调和问题，我们使用变分原理，并且根据参数\(a\) 的值，我们获得了唯一性（非唯一性）定理或提出了解空间维数的精确公式。