Polygonal heat conductors with a stationary hot spot

We consider a convex polygonal heat conductor whose inscribed circle touches every side of the conductor. Initially, the conductor has constant temperature and, at every time, the temperature of its boundary is kept at zero. The hot spot is the point at which temperature attains its maximum at each given time. It is proved that, if the hot spot is stationary, then the conductor must satisfy two geometric conditions. In particular, we prove that these geometric conditions yield some symmetries provided the conductor is either pentagonal or hexagonal. This research was partially supported by Grants-in-Aid for Scientific Research (B) (# 12440042) and (B) (# 15340047) of Japan Society for the Promotion of Science, and by a Grant of the Italian MURST.     

Zur Untersuchung des technischen Calciumcarbids

Ohne Zusammenfassung     

Critical points of solutions of degenerate elliptic equations in the plane

We study the minimizer u of a convex functional in the plane which is not Gâteaux-differentiable. Namely, we show that the set of critical points of any C ¹-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler–Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.     

Spatial critical points not moving along the heat flow II: The centrosymmetric case

We consider solutions of initial-boundary value problems for the heat equation on bounded domains in and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric with respect to the origin. Furthermore, we consider spatial zero points instead of spatial critical points, and prove some similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the wave equation. Received October 31, 1997; in final form February 3, 1998     

On Symmetric Invariants of Level Surfaces Near Regular Points

We consider the symmetric invariants of the level surfaces ofa smooth function away from its critical points, and prove forthem some formulae in divergence from. We then apply these formulaeto obtain an isoperimetric inequality for the surface area oflevel surfaces of p-capacity potentials.     

On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude — in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard.We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved.We offer an algorithm that restrains instability and produces effective approximate solutions.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem

We consider the Stekloff eigenvalue problem (1.1)–(1.2); Payne and Philippin conjectured that ifu is an eigenfunction which satisfies the overdetermined condition u=1 on , then should be a disk. In this paper we show that this conjecture holds if and only if the complex potentialF associated tou vanishes only at one point. Then we show how to construct non-symmetric domains in the case whereF vanishes at more than one point.