1. MULTIPLE SOLUTIONS FOR THE DIRICHLET PROBLEM FOR H-SYSTEMS WITH SMALL H.
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ISOBE, TAKESHI
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PAPER , *FIBERS , *WRITING materials & instruments , *STATIONERY , *ABSORBENT paper , *ALBUMEN paper - Abstract
Let ${\mathbb D}:=\{(x_1,x_2)\in {\mathbb R}^2:x_1^2+x_2^2 < 1\}$ be the 2-dimensional unit disk and $\gamma \in H^{1/2} (\partial {\mathbb D},{\mathbb R}^3) \cap L^\infty (\partial {\mathbb D})$. For suitably small H>0, we consider the Dirichlet problem for H-systems \[ \hskip3.4cm \left\{\begin{array}{rlll} \Delta u &=& 2H u_{x_1} \wedge u_{x_2} &\quad {\rm in}\ {\mathbb D}\\[5pt] u &=& \gamma &\quad {\rm on}\ \partial {\mathbb D}\,.\hskip3.45cm {\rm (H)} \end{array} \right. \] It is well-known that (H) admits at least two distinct solutions for non-constant γ and small H>0. In this paper, we introduce conditions on γ and develop methods of finding at least three distinct solutions to (H) under such conditions on γ. [ABSTRACT FROM AUTHOR]
- Published
- 2004
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