Sturm--Liouville theory for the p-Laplacian

2003 ◽  
Vol 40 (4) ◽  
pp. 373-396 ◽  
Author(s):  
Paul Binding ◽  
Pável Drábek
Keyword(s):  
Variational Principles ◽  
Liouville Theory ◽  
One Dimensional ◽  
Type Transformation ◽  
Radial Case ◽  
Sturm Liouville ◽  
The One ◽  
And Variational Principles

A version of Sturm--Liouville theory is given for the one-dimensional p-Laplacian including the radial case. The treatment is modern but follows the strategy of Elbert's early work. Topics include a Prüfer-type transformation, eigenvalue existence, asymptotics and variational principles, and eigenfunction oscillation.

A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Schrödinger Equation ◽  
Liouville Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Classical Case ◽  
Liouville Theory ◽  
Weyl’S Theorem ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to second-order differential operators with singularities at the endpoints of the interval. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The chapter first considers the one-dimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account bound-state theorems and complex eigenvalues. It also describes Weyl's theorem, given the Sturm-Liouville equation, and looks at two cases: the limit point and limit circle. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.

A FRACTAL VARIATIONAL THEORY FOR ONE-DIMENSIONAL COMPRESSIBLE FLOW IN A MICROGRAVITY SPACE

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050024 ◽  
Author(s):  
JI-HUAN HE
Keyword(s):  
Compressible Flow ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Inverse Method ◽  
Variational Principles ◽  
Microgravity Condition ◽  
Multiplier Method ◽  
Variational Theory ◽  
One Dimensional ◽  
The One

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

scholarly journals GAUGING SL(2, R) AND SL(2, R)×U(1) BY THEIR NILPOTENT SUBGROUPS

1995 ◽  
Vol 10 (01) ◽  
pp. 115-131 ◽  
Author(s):  
M. ALIMOHAMMADI ◽  
F. ARDALAN ◽  
H. ARFAEI
Keyword(s):  
Black Hole ◽  
Black Hole Solution ◽  
Three Dimensional ◽  
Liouville Theory ◽  
Black String ◽  
String Solution ◽  
One Dimensional ◽  
First Case ◽  
Wznw Model ◽  
The One

We consider the gauging of SL(2, R) and SL(2, R)×U(1) WZNW model by their nilpotent subgroups. The resulting space-times of the corresponding sigma models are seen to collapse to the one-dimensional Liouville theory in the first case, and lead to an extremal three-dimensional black string in the second case. We show that these results are also obtained by boosting the Witten black hole solution and Horne and Horowitz black string solution respectively.

Finite Element Analysis of Thermal Buckling and Postbuckling of Composite Plates With Embedded SMA

2001 ◽  
Author(s):  
Zhong Weifang ◽  
Zou Jing ◽  
Wu Yongdong
Keyword(s):  
Variational Principles ◽  
Young Modulus ◽  
Composite Plates ◽  
Thermal Buckling ◽  
Recovery Stress ◽  
Element Analysis ◽  
Numerical Examples ◽  
One Dimensional ◽  
Thermal Postbuckling ◽  
And Variational Principles

Abstract Based on Von Karman strain-displacement relations and variational principles, the FEM formulations, which are used to analyze the thermal buckling, and thermal postbuckling of composite plates with embedded SMA are presented in this study. The recovery stress and young modulus are calculated by one-dimensional constitutive model. Some numerical examples are also presented. The results indicate that activated SMA can suppress the thermal buckling of the plates and reduce thermal postbuckling deflections.

scholarly journals A Survey on Extremal Problems of Eigenvalues

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ping Yan ◽  
Meirong Zhang
Keyword(s):  
Variational Method ◽  
Lebesgue Space ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Lower And Upper Bounds ◽  
Open Problems ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One ◽  
Neumann Eigenvalues

Given an integrable potentialq∈L1([0,1],ℝ), the Dirichlet and the Neumann eigenvaluesλnD(q)andλnN(q)of the Sturm-Liouville operator with the potentialqare defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when theL1metric forqis given;∥q∥L1=r. Note that theL1spheres andL1balls are nonsmooth, noncompact domains of the Lebesgue space(L1([0,1],ℝ),∥·∥L1). To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spacesLα([0,1],ℝ),1<α<∞will be used. Then theL1problems will be solved by passingα↓1. Corresponding extremal problems for eigenvalues of the one-dimensionalp-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.

SOME UNIQUENESS RESULTS FOR ELLIPTIC EQUATIONS WITHOUT CONDITION AT INFINITY

2003 ◽  
Vol 05 (05) ◽  
pp. 705-717 ◽  
Author(s):  
A. PORRETTA
Keyword(s):  
Elliptic Equations ◽  
Positive Answer ◽  
Dimensional Case ◽  
One Dimensional ◽  
Uniqueness Results ◽  
Radial Case ◽  
Whole Space ◽  
Dependent Absorption ◽  
The One

We prove in this work uniqueness results concerning a class of elliptic equations having gradient dependent absorption terms, whose simplest model is given in [Formula: see text] We deal with the question whether uniqueness for solutions in the whole space holds without any condition at infinity, giving a positive answer in the one dimensional case as well as in the radial case in ℝN.

scholarly journals A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Sertac Goktas
Keyword(s):  
Liouville Equation ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Mathematical Physics ◽  
Asymptotic Estimates ◽  
One Dimensional ◽  
Multiplicative Calculus ◽  
New Type ◽  
Sturm Liouville ◽  
The One

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

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