Yasuhito Miyamoto

The “hot spots” conjecture for a certain class of planar convex domains

We prove the “hot spots” conjecture of Rauch [“Five problems: An introduction to the qualitative theory of partial differential equations,” Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics (Springer, Berlin, 1975), Vol. 446, pp. 355–369] for a certain class of planar convex domains. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Ω attains its maximum (minimum) at points in ∂Ω. One class of domains is the planar convex domain Ω satisfying diam(Ω)2/|Ω|<1.378. When Ω is a disk, diam(Ω)2/|Ω|≈1.273. Hence, this condition indicates that Ω is a nearly circular planar convex domain. However, symmetries of the domain are not assumed. We give other sufficient conditions for domains for which the conjecture holds. We also give a new isoperimetric inequality.

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domains u t = D u Δu+ f(u, ξ) in Ω x R + and τξ t =|Ω | ∫∫ Ω g(u,ξ)dxdy in R + , ∂ v u =0 on ∂Ω xR + , where f and g satisfy the following: gξ 0, then the shape of u is like a boundary one-spike layer even if D u is not small.

Stability of a boundary spike layer for the Gierer-Meinhardt system

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point P of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at P , and hence the mean curvature at the boundary has a local maximum at P . Rigorous results for the stability of such a boundary spike-layer solution are given.

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

Abstract We study radial solutions of the semilinear elliptic equation Δ u + f ( u ) = 0 under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., Δ u + u p = 0 and Δ u + e u = 0 .