A conjugate-set game induced from the conjugate point (Nonlinear Analysis and Convex Analysis)
Hidefumi Kawasaki · Kyoto University Research Information Repository (Kyoto University) · 2005
In some nonlinear diffusive phenomena, e.g., grain growth in annealing pure metal and segregation between biological species, the systems have three or more stable states, see the left picture of Fig.1.Sternberg and Zeimer(Ref. 1) established the existence of local minimizers to the problem of partitioning certain domain $\Omega\subseteq$ $R^{2}$ into three subdomains having least interfacial area.Further, Ikota and Yanagida investigated stability for stationary curves with one triple junction in Ref. 2and stability for stationary binary-tree type interfaces in Ref.3. In this paper, we consider astatic version of their diffusion problem, which is formulated as an unconstrained nonlinear programming problem.We consider second-0rder optimality conditions and discuss stability and instability for stationary curves with one or two triple junctions.The great difference between the previous researches and ours is that our main concern is not stability but instability.We give anew insight to this problem from the viewpoint of game theory.