On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

For a certain class of second-order anisotropic elliptic equations with variable nonlinearity indices and L1 right-hand side we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable indices.

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Littlewood–Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-038-9 ◽

2016 ◽

Vol 59 (01) ◽

pp. 104-118 ◽

Cited By ~ 8

Author(s):

Ziyi He ◽

Dachun Yang ◽

Wen Yuan

Keyword(s):

Sobolev Spaces ◽

Second Order ◽

Area Function ◽

Lusin Area Function

Abstract In this paper, the authors characterize second-order Sobolev spaces W2,p(ℝn), with p ∊ [2,∞) and n ∊ N or p ∊ (1, 2) and n ∊ {1, 2, 3}, via the Lusin area function and the Littlewood–Paley g*λ -function in terms of ball means.

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Existence Result for a Second Order Nonlinear Degenerate Elliptic Equation in Weighted Orlicz-Sobolev Spaces

Lecture Notes in Pure and Applied Mathematics - Partial Differential Equations ◽

10.1201/9780203910108.ch9 ◽

2002 ◽

Author(s):

M Arias ◽

E Azroul

Keyword(s):

Elliptic Equation ◽

Sobolev Spaces ◽

Existence Result ◽

Degenerate Elliptic Equation ◽

Second Order ◽

Degenerate Elliptic ◽

Nonlinear Degenerate

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On the range of elliptic, second order, nonvariational operators in Sobolev spaces

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf02505888 ◽

2000 ◽

Vol 178 (1) ◽

pp. 67-80

Author(s):

P. Manselli

Keyword(s):

Sobolev Spaces ◽

Second Order

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Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

Mathematical Problems in Engineering ◽

10.1155/2011/171834 ◽

2011 ◽

Vol 2011 ◽

pp. 1-17 ◽

Cited By ~ 3

Author(s):

M. Safdar ◽

Asghar Qadir ◽

S. Ali

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Canonical Form ◽

Linear Equations ◽

Two Dimensions ◽

Second Order ◽

Equivalence Classes ◽

Order Ordinary Differential Equation ◽

Change Of Variables ◽

Symmetry Structure

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An “optimal (or simplest) canonical form” of linear systems had been established to obtain the symmetry structure, namely, with 5-, 6-, 7-, 8-, and 15-dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, we provide a “reduced optimal canonical form.” This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6-, 7-, and 15-dimensional algebras for these systems and illustrate our results with examples.

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Implicit Numerical Integration for Periodic Solutions of Autonomous Nonlinear Systems

Journal of Applied Mechanics ◽

10.1115/1.3162628 ◽

1982 ◽

Vol 49 (4) ◽

pp. 861-866 ◽

Cited By ~ 3

Author(s):

G. A. Thurston

Keyword(s):

Nonlinear Systems ◽

Critical Load ◽

Periodic Solutions ◽

Autonomous Systems ◽

Conservative System ◽

Constant Load ◽

Second Order ◽

Order System ◽

Change Of Variables ◽

System A

A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.

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