Displaying similar documents to “Unexpected solutions of first and second order partial differential equations.”

P-order necessary and sufficient conditions for optimality in singular calculus of variations

Agnieszka Prusińska, Alexey Tret'yakov (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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This paper is devoted to singular calculus of variations problems with constraint functional not regular at the solution point in the sense that the first derivative is not surjective. In the first part of the paper we pursue an approach based on the constructions of the p-regularity theory. For p-regular calculus of variations problem we formulate and prove necessary and sufficient conditions for optimality in singular case and illustrate our results by classical example of calculus...

On complete solutions and complete singular solutions of second order ordinary differential equations

Masatomo Takahashi (2007)

Colloquium Mathematicae

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A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular...