scholarly journals On the non-vanishing property for real analytic solutions of the $p$-Laplace equation

2015 ◽  
Vol 144 (6) ◽  
pp. 2375-2382 ◽  
Author(s):  
Vladimir G. Tkachev
Keyword(s):  
Laplace Equation ◽  
Analytic Solutions ◽  
Real Analytic ◽  
Real Analytic Solutions

On the approximation of continuous functions by analytic solutions of universal functional-differential equations

2004 ◽  
Vol 41 (1) ◽  
pp. 1-15 ◽  
Author(s):  
C. Elsner
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Analytic Solutions ◽  
Continuous Functions ◽  
Compact Interval ◽  
The Real ◽  
Real Analytic ◽  
Functional Differential ◽  
Real Analytic Solutions

The existence of an algebraic functional-differential equation P (y′(x), y′(x + log 2), …, y′(x + 5 log 2)) = 0 is proved such that the real-analytic solutions are dense in the space of continuous functions on every compact interval. A similar result holds for an algebraic functional-differential equation P(y′(x − 4πi), y′(x − 2πi), …, y′(x + 4πi)) = 0 (with i2 = −1), which is explicitly given: There are real-analytic solutions on the real line such that every continuous function defined on a compact interval can be approximated by these solutions with arbitrary accuracy.

scholarly journals On the analyticity of the locus of singularity of real analytic solutions with minimal dimension

1986 ◽  
Vol 104 ◽  
pp. 63-84 ◽  
Author(s):  
Akira Kaneko
Keyword(s):  
Complex Analysis ◽  
Analytic Solution ◽  
Partial Differential Operator ◽  
Analytic Solutions ◽  
Partial Differential ◽  
Minimal Dimension ◽  
Real Analytic ◽  
The One ◽  
Real Analytic Solutions ◽  
Real Analytic Solution

Let P(x, D) be a linear partial differential operator with real analytic coefficients and let C ⊂ Rn be a germ of closed subset, say at the origin. We say that C is (the locus of) an irremovable singularity of a real analytic solution u of P(x, D)u = 0 if u is defined outside C on a neighborhood Ω of 0 but cannot be extended to the whole neighborhood Ω even as a hyperfunction solution of P(x, D)u = 0. This usage of the word “singularity” is the same as the one for the analytic functions in complex analysis, and is different of the usual usage of “singularities of solutions” in the theory of partial differential equations.

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