scholarly journals Recent developments on Pseudo-Differential Operators (I)

2015 ◽  
Vol 46 (1) ◽  
pp. 1-30 ◽  
Author(s):  
D.-C. Chang ◽  
W. RUNGROTTHEERA ◽  
B.-W. SCHULZE
Keyword(s):  
Boundary Value Problems ◽  
Differential Calculus ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Higher Order ◽  
Symbolic Calculus ◽  
Elliptic Boundary Value ◽  
The Past ◽  
Pseudo Differential Operators ◽  
Recent Developments

In recent years the analysis of (pseudo-)differential operators on manifolds with second and higher order corners made considerable progress, and essential new structures have been developed. The main objective of this series of paper is to give a survey on the development of this theory in the past twenty years. We start with a brief background of the theory of pseudo-differential operators which including its symbolic calculus on $\R^n$. Next we introduce pseudo-differential calculus with operator-valued symbols. This allows us to discuss elliptic boundary value problems on smooth domains in $\R^n$ and elliptic problems on manifolds. This paper is based on the first part of lectures given by the authors while they visited the National Center for Theoretical Sciences in Hsinchu, Taiwan during May-July of 2014.

Admissible Regions for Higher-Order Finite Volume Method Grids

2017 ◽  
Vol 7 (2) ◽  
pp. 269-285
Author(s):  
Yuanyuan Zhang ◽  
Zhongying Chen
Keyword(s):  
Boundary Value Problems ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Elliptic Boundary ◽  
Higher Order ◽  
Elliptic Boundary Value Problems ◽  
Sufficient Condition ◽  
Elliptic Boundary Value ◽  
Admissible Region ◽  
Volume Method

AbstractAdmissible regions for higher-order finite volume method (FVM) grids are considered. A new Hermite quintic FVM and a new hybrid quintic FVM are constructed to solve elliptic boundary value problems, and the corresponding admissible regions are investigated. A sufficient condition for the uniform local-ellipticity of the new hybrid quintic FVM is obtained when its admissible region is known. In addition, the admissible regions for a large number of higher-order FVMs are provided. For the same class of FVM (Lagrange, Hermite or hybrid), the higher order FVM has a smaller admissible region such that stronger geometric restrictions are required to guarantee its uniform local-ellipticity.

scholarly journals Higher order elliptic boundary value problems in spaces with homogeneous norms

1981 ◽  
Vol 31 (1) ◽  
pp. 92-113 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
A Priori ◽  
Elliptic Boundary Value ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

scholarly journals Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guangcun Lu
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Bifurcation Theory ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Higher Order ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Potential Operators ◽  
Linear Elliptic Boundary

<p style='text-indent:20px;'>This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.</p>

scholarly journals GREEN'S FUNCTIONS AND REGULARIZED TRACES OF STURM–LIOUVILLE OPERATORS ON GRAPHS

2008 ◽  
Vol 51 (2) ◽  
pp. 315-335 ◽  
Author(s):  
Sonja Currie ◽  
Bruce A. Watson
Keyword(s):  
Boundary Value Problems ◽  
Green's Functions ◽  
Differential Operators ◽  
Green’S Functions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Inverse Spectral

AbstractAsymptotic approximations to the Green's functions of Sturm–Liouville boundary-value problems on graphs are obtained. These approximations are used to study the regularized traces of the differential operators associated with these boundary-value problems. Various inverse spectral problems for Sturm–Liouville boundary-value problems on graphs resembling those considered in Halberg and Kramer's ‘A generalization of the trace concept' (Duke Mathematics Journal27 (1960), 607–617), for Sturm–Liouville problems, and Pielichowski's ‘An inverse spectral problem for linear elliptic differential operators' (Universitatis Iagellonicae Acta Mathematica27 (1988), 239–246), for elliptic boundary-value problems, are solved.

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