Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems
| Autor: | Bevan, J. |
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| Rok vydání: | 2014 |
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| Druh dokumentu: | Working Paper |
| Popis: | We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,\theta) = Rg(\theta)$, where $(R,\theta)$ are plane polar coordinates and $g: \mathbb{R}^{2} \to \mathbb{R}^{m}$, $m \geq 2$. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals $I(u) = \int_{B}W(x,\nabla u(x))\,dx$, where $D_{F}W(x,F)$ behaves like $\frac{1}{|x|}$ as $|x| \to 0$ and $W$ satisfies an ellipticity condition. Such solutions cannot exist when $|x|D_{F}W(x,F) \to 0$ as $|x| \to 0$, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality. We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals. Comment: This paper was submitted to the Journal of the London Mathematical Society on 8 November 2012 |
| Databáze: | arXiv |
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