Boundary point lemmas and overdetermined problems

Boundary Regularity and Compactness for Overdetermined Problems

1 Let D be either the unit ball B1(0) or the half ball B + 1 (0), let f be a strictly positive and continuous function, and let u and Ω ⊂ D solve the following overdetermined problem: ∆u(x) = χ Ω (x)f(x) in D, 0 ∈ ∂Ω, u = |∇u| = 0 in Ω, where χ Ω denotes the characteristic function of Ω, Ωc denotes the set D \Ω, and the equation is satisfied in the sense of distributions. When D = B 1 (0), then...

Overdetermined boundary value problems for the ∞-Laplacian

We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞-Laplacian, the normalized or game-theoretic ∞-Laplacian and the limit of the p-Laplacian as p→∞ are considered and provide different answers. Mathematics Subject Classification (2000). 35R35, 4...

Overdetermined Boundary Value Problems, Quadrature Domains and Applications

We discuss an overdetermined problem in planar multiply connected domains Ω. This problem is solvable in Ω if and only if Ω is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution o...

Approximate radial symmetry for overdetermined boundary value problems

B-SPLINE METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEMS

In this work the collocation method based on quartic B-spline is developed and applied to two-point boundary value problem in ordinary diferential equations. The error analysis and convergence of presented method is discussed. The method illustrated by two test examples which verify that the presented method is applicable and considerable accurate.