Inverse-monotone nonlinear differential operators of the second order

1978 ◽  
Vol 79 (3-4) ◽  
pp. 193-226 ◽  
Author(s):  
Johann Schröder
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Monotone Operators ◽  
Second Order ◽  
Differential Inequalities ◽  
Partial Differential ◽  
Nonlinear Differential Operators ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisThis paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

scholarly journals Two-scale convergence with respect to measures and homogenization of monotone operators

2005 ◽  
Vol 3 (2) ◽  
pp. 125-161 ◽  
Author(s):  
Dag Lukkassen ◽  
Peter Wall
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Monotone Operators ◽  
Partial Differential ◽  
Homogenization Problem ◽  
Partial Differential Operators ◽  
Basic Facts

In 1989 Nguetseng introduced two-scale convergence, which now is a frequently used tool in homogenization of partial differential operators. In this paper we discuss the notion of two-scale convergence with respect to measures. We make an exposition of the basic facts of this theory and develope it in various ways. In particular, we consider both variableLpspaces and variable Sobolev spaces. Moreover, we apply the results to a homogenization problem connected to a class of monotone operators.

On the uniformization of plane direction fields, and of second-order partial differential operators

1973 ◽  
Vol 73 (1) ◽  
pp. 87-109 ◽  
Author(s):  
Robert Finn
Keyword(s):  
Elliptic Equations ◽  
Differential Operators ◽  
Classification Problem ◽  
Second Order ◽  
Generalized Solutions ◽  
Partial Differential ◽  
Uniformization Theorem ◽  
Partial Differential Operators ◽  
Recent Developments ◽  
Direction Fields

One of the topics covered by most textbooks on partial differential equations is the classification problem for second-order equations in the plane. In a typical treatment, it is shown that under some smoothness conditions on the coefficients, hyperbolic and parabolic equations can be reduced locally to normal form (uniformized) by an elementary procedure. For elliptic equations, the procedure fails unless an extraneous hypothesis (analyticity of the coefficients) is introduced. It is then pointed out that a different and much deeper method (essentially the general uniformization theorem) is effective for the elliptic case and even yields a global result. It is striking, but not surprising in view of recent developments on generalized solutions, that the alternate (global) procedure for elliptic equations requires much less smoothness of the coefficients than is needed for a sensible local result in the other cases.

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