/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Convex bodies: the brunn–minkowski theory

Advanced Mathematical Methods for Scientists and Engineers

We consider the problem of determining the scalar coefficient γ in the elliptic equation div(γ grad u) = 0 in ω when, for every Dirichlet datum u = ∅ on ∂ω , the Neumann datum γ(∂/ ∂ n)u = ∧.γ∅ is known. We prove a continuous dependence result

Stable determination of conductivity by boundary measurements

The Stekloff eigenvalue problem (1.1) has a countable number of eigenvalues $\{ p_n \} n = 1,2, \ldots $, each of finite multiplicity. In this paper the authors give an upper estimate, in terms of the integer n, of the multiplicity of $p_n $, and the number of critical points and of nodal domains of the eigenfunctions corresponding to $p_n $.In view of a possible application to inverse conductivity problems, the result for the general case of elliptic equations with discontinuous coefficients in divergence form is proven by replacing the classical concept of critical point with the more suitable notion of geometric critical point.

/pdf/elliptic-equations-in-divergence-form-geometric-critical-3b5vm9edlk.pdf

Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions

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/pdf/the-index-of-isolated-critical-points-and-solutions-of-4e2cjul2w5.pdf

The index of isolated critical points and solutions of elliptic equations in the plane

We consider a bounded heat conductor that satisfies the exterior sphere condition. Suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. We show that if the conductor contains a proper sub-domain, satisfying the interior cone condition and having constant boundary temperature at each given time, then the conductor must be a ball.

/pdf/matzoh-ball-soup-heat-conductors-with-a-stationary-3cpjxfdbvs.pdf

Matzoh ball soup: Heat conductors with a stationary isothermic surface

In this work, we consider a wave propagation problem in a two-dimensional (2-D) waveguide. The problem arises in the study of light in optical fibers. We construct a transform theory as a framework for studying this problem. An explicit representation for the solution to problems involving light sources is derived. We derive a decay rate for the nonguided part of the solution in the direction of the core. The approach is also amenable to computations, and we demonstrate this in numerical examples for the case of slab waveguides.

/pdf/wave-propagation-in-a-2-d-optical-waveguide-39t8kkgove.pdf

Wave propagation in a 2-D optical waveguide

We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the former is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Amp\`ere equations; the latter relies on Alexandrov's reflection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations.

/pdf/the-location-of-the-hot-spot-in-a-grounded-convex-conductor-jco0ru416m.pdf

Rolando Magnanini

Papers

Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions

The index of isolated critical points and solutions of elliptic equations in the plane

Matzoh ball soup: Heat conductors with a stationary isothermic surface

Wave propagation in a 2-D optical waveguide

The location of the hot spot in a grounded convex conductor