scholarly journals A Regularity Theorem for Certain 2nth-Order Nonlinear Differential Operator with a Parameter

1998 ◽  
Vol 222 (2) ◽  
pp. 460-483
Author(s):  
Liu Yingfan
Keyword(s):  
Differential Operator ◽  
Regularity Theorem ◽  
Nonlinear Differential Operator

scholarly journals Wave Front Sets with respect to the Iterates of an Operator with Constant Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
C. Boiti ◽  
D. Jornet ◽  
J. Juan-Huguet
Keyword(s):  
Differential Operator ◽  
Wave Front ◽  
Partial Differential Operator ◽  
Regularity Theorem ◽  
Wave Front Set ◽  
Open Set ◽  
Wave Front Sets ◽  
Constant Coefficients ◽  
New Type ◽  
Classical Distribution

We introduce the wave front setWF*P(u)with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distributionu∈𝒟′(Ω)in an open set Ω in the setting of ultradifferentiable classes of Braun, Meise, and Taylor. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.

scholarly journals Duality by reproducing kernels

2003 ◽  
Vol 2003 (6) ◽  
pp. 327-395 ◽  
Author(s):  
A. Shlapunov ◽  
N. Tarkhanov
Keyword(s):  
Differential Operator ◽  
Neumann Problem ◽  
Compact Manifold ◽  
Reproducing Kernels ◽  
Elliptic Differential Operator ◽  
Regularity Theorem ◽  
Neumann Problems ◽  
Smooth Compact Manifold ◽  
Convexity Properties ◽  
The Neumann Problem

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.

Rondo: Rank-order based nonlinear differential operator

1992 ◽  
Vol 25 (9) ◽  
pp. 1043-1059 ◽  
Author(s):  
Akira Asano ◽  
Kazuyoshi Itoh ◽  
Yoshiki Ichioka
Keyword(s):  
Differential Operator ◽  
Rank Order ◽  
Nonlinear Differential Operator

Existence of Multiple Solutions for Nonlinear Dirichlet Problems with a Nonhomogeneous Differential Operator

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Linear Differential Operator ◽  
Principal Eigenvalue ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Multiplicity Results ◽  
Nonlinear Elliptic ◽  
Nonlinear Differential Operator ◽  
Linear Differential

AbstractWe consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.

A Metod with a Controllable Exponential Convergence Rate for Nonlinear Differential Operator Equations

2009 ◽  
Vol 9 (1) ◽  
pp. 63-78 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
I. I. Lazurchak ◽  
V. L. Makarov ◽  
D. Sytnyk
Keyword(s):  
Differential Operator ◽  
Convergence Rate ◽  
Nonlinear Operator ◽  
Control Mechanism ◽  
Exponential Convergence ◽  
Operator Equations ◽  
Exponential Convergence Rate ◽  
Convergence Control ◽  
Nonlinear Differential Operator ◽  
Theoretical Results

AbstractWe propose a new analytical-numerical method with an embedded convergence control mechanism for solving nonlinear operator differential equations. The method provides the exponential convergence rate. A numerical example confirms the theoretical results.

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