Nonlinear boundary value problems of the calculus of variations

Felix Sadyrbaev

doi:10.3934/proc.2003.2003.760

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2003, Volume 2003: 760-770. Doi: 10.3934/proc.2003.2003.760 open access

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Nonlinear boundary value problems of the calculus of variations

Felix Sadyrbaev^1,

1.
Institute of Mathematics and Computer Science, University of Latvia, Rainis boul. 29, LV-1459 Riga, Lativa

Received: August 31, 2002

Revised: April 30, 2003

Published: March 2003

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Abstract

We consider nonlinear boundary value problems arising in the classical one-dimensional calculus of variations for scalar-valued unknown functions. Conditions for the existence of extremals (solutions of the Euler equation subject to related boundary conditions) are obtained and properties of extremals are discussed. The method of upper and lower solutions (functions) is our main tool. Several Bernstein - Nagumo type conditions are derived directly in terms of the Lagrangian. Both coercive and non-coercive (slow-growth) variational problems are considered.

Keywords:

Nonlinear boundary value problems,
extremals,
basic problem of the calculus of variations,
free end point problem,
regularity of solutions.

Mathematics Subject Classification: Primary: 34B15, 49N60; Secondary: 49K05.

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