Journal ArticleDOI
A symmetry problem in potential theory
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In this article, it was shown that the flow velocity of a viscous incompressible fluid moving in straight parallel streamlines through a straight pipe of given cross sectional form f2 is a function of x, y alone satisfying the Poisson differential equation.Abstract:
The proof of this result is given in Section 1 ; in Section 3 we give various generalizations to elliptic differential equations other than (1). Before turning to the detailed arguments it will be of interest to discuss the physical motivation for the problem itself. Consider a viscous incompressible fluid moving in straight parallel streamlines through a straight pipe of given cross sectional form f2. If we fix rectangular coordinates in space with the z axis directed along the pipe, it is well known that the flow velocity u is then a function of x, y alone satisfying the Poisson differential equation (for n = 2) A u = A i n f2read more
Citations
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Journal ArticleDOI
Symmetry and related properties via the maximum principle
TL;DR: In this paper, the authors show that positive solutions of second order elliptic equations are symmetric about the limiting plane, and that the solution is symmetric in bounded domains and in the entire space.
Book
The Porous Medium Equation: Mathematical Theory
TL;DR: The Porous Medium Equation (PME) as discussed by the authors is one of the classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood.
Journal ArticleDOI
The dirichlet problem for nonlinear second‐order elliptic equations I. Monge‐ampégre equation
TL;DR: On considere le probleme de Dirichlet as discussed by the authors for des equations elliptiques non lineaires for a fonction reelle u definie dans la fermeture d'un domaine borne Ω dans R n avec une frontiere ∂Ω C ∞
Journal ArticleDOI
The isoperimetric inequality
TL;DR: For a survey of generalizations of the isoperimetric inequality, see as mentioned in this paper, where the main focus is on geometric versions and generalisations of the inequality, with emphasis on recent contributions.
Journal ArticleDOI
Classification of solutions for an integral equation
TL;DR: In this paper, it was shown that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn.
References
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Book
Maximum principles in differential equations
TL;DR: The One-Dimensional Maximum Principle (MDP) as mentioned in this paper is a generalization of the one-dimensional maximum principle (OMP) for the construction of hyperbolic equations.