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Displaying similar documents to “On the domain dependence of solutions to the two-phase Stefan problem”

An application of eigenfunctions of p -Laplacians to domain separation

Herbert Gajewski (2001)

Mathematica Bohemica

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We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain Ω into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the p -Laplacians, p → 1 , under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients. ...

Weak lineal convexity

Christer O. Kiselman (2015)

Banach Center Publications

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A bounded open set with boundary of class C¹ which is locally weakly lineally convex is weakly lineally convex, but, as shown by Yuriĭ Zelinskiĭ, this is not true for unbounded domains. The purpose here is to construct explicit examples, Hartogs domains, showing this. Their boundary can have regularity C 1 , 1 or C ∞ . Obstructions to constructing smoothly bounded domains with certain homogeneity properties will be discussed.

Homogenization in polygonal domains

David Gérard-Varet, Nader Masmoudi (2011)

Journal of the European Mathematical Society

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We consider the homogenization of elliptic systems with ε -periodic coefficients. Classical two-scale approximation yields an O ( ε ) error inside the domain. We discuss here the existence of higher order corrections, in the case of general polygonal domains. The corrector depends in a non-trivial way on the boundary. Our analysis substantially extends previous results obtained for polygonal domains with sides of rational slopes.

Some pinching deformations of the Fatou function

Patricia Domínguez, Guillermo Sienra (2015)

Fundamenta Mathematicae

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We are interested in deformations of Baker domains by a pinching process in curves. In this paper we deform the Fatou function F ( z ) = z + 1 + e - z , depending on the curves selected, to any map of the form F p / q ( z ) = z + e - z + 2 π i p / q , p/q a rational number. This process deforms a function with a doubly parabolic Baker domain into a function with an infinite number of doubly parabolic periodic Baker domains if p = 0, otherwise to a function with wandering domains. Finally, we show that certain attracting domains can be deformed...

Isoperimetric estimates for the first eigenvalue of the p -Laplace operator and the Cheeger constant

Bernhard Kawohl, V. Fridman (2003)

Commentationes Mathematicae Universitatis Carolinae

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First we recall a Faber-Krahn type inequality and an estimate for λ p ( Ω ) in terms of the so-called Cheeger constant. Then we prove that the eigenvalue λ p ( Ω ) converges to the Cheeger constant h ( Ω ) as p → 1 . The associated eigenfunction u p converges to the characteristic function of the Cheeger set, i.e. a subset of Ω which minimizes the ratio | ∂ D | / | D | among all simply connected D ⊂ ⊂ Ω . As a byproduct we prove that for convex Ω the Cheeger set ω is also convex.