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K. Kurata

Publications and source records attributed to K. Kurata.

3 recordsLinked to original sources
arXiv2000

Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes

We consider the following eigenvalue optimization problem: Given a bounded domain $Ω\subset\R^n$ and numbers $α\geq 0$, $A\in [0,|Ω|]$, find a subset $D\subsetΩ$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Δ+ αχ_D$ is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than $Ω$; on the other hand, for convex $Ω$ reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.

math.AP↗
arXiv1999

The Free Boundary Problem in the Optimization of Composite Membranes

This is a continuation of the paper 'Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes' by S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Again, we consider the following eigenvalue optimization problem: Given a bounded domain $Ω\subset\R^n$ and numbers $α\geq 0$, $A\in [0,|Ω|]$, find a subset $D\subsetΩ$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Δ+ αχ_D$ is as small as possible. In this paper we focus on the study of the free boundary of optimal solutions on general domains.

math.AP↗