scholarly journals Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

2021 ◽  
Vol 105 (0) ◽  
pp. 113-136
Author(s):  
C. Soize
Keyword(s):  
Random Fields ◽  
Elliptic Boundary ◽  
Positive Definite Matrix ◽  
Gaussian Random Fields ◽  
Positive Definite ◽  
Elliptic Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
3D Space ◽  
Non Gaussian

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

scholarly journals Random field representations for stochastic elliptic boundary value problems and statistical inverse problems

2014 ◽  
Vol 25 (3) ◽  
pp. 339-373 ◽  
Author(s):  
A. NOUY ◽  
C. SOIZE
Keyword(s):  
Random Fields ◽  
General Class ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Approximation Methods ◽  
Elliptic Boundary Value Problem ◽  
Identification Procedure ◽  
Elliptic Boundary Value ◽  
Statistical Inverse Problems ◽  
Non Gaussian

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of non-Gaussian positive-definite matrix-valued random fields, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analysed. A minimal parameterisation of discretised random fields belonging to this general class is proposed. Using this parameterisation of the general class, a complete identification procedure is proposed. New results of the mathematical and numerical analyses of the parameterised stochastic elliptic boundary value problem are presented. The numerical solution of this parametric stochastic problem provides an explicit approximation of the application that maps the parameterised general class of random fields to the corresponding set of random solutions. This approximation can be used during the identification procedure in order to avoid the solution of multiple forward stochastic problems. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a particular mathematical analysis is developed and dedicated approximation methods are introduced. In order to obtain an algorithm for constructing the approximation of a very high-dimensional map, complexity reduction methods are introduced and are based on the use of sparse or low-rank approximation methods that exploit the tensor structure of the solution which results from the parameterisation of the general class of random fields.

Global Positive Solutions of Semilinear Elliptic Equations

1983 ◽  
Vol 35 (5) ◽  
pp. 839-861 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Elliptic Boundary ◽  
Positive Definite ◽  
Elliptic Boundary Value Problem ◽  
Hölder Exponent ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Image Position

The semilinear elliptic boundary value problem1.1will be considered in an exterior domain Ω ⊂ Rn, n ≥ 2, with boundary ∂Ω ∊ C2 + α, 0 < α < 1, where1.2Di = ∂/∂xi, i = 1, …, n. The coefficients aij, bi in (1.2) are assumed to be real-valued functions defined in Ω ∪ ∂Ω such that each , , and (aij(x)) is uniformly positive definite in every bounded domain in Ω. The Hölder exponent α is understood to be fixed throughout, 0 < α < 1 . The regularity hypotheses on f and g are stated as H 1 near the beginning of Section 2.

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

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