Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients

Abstract

We study existence and uniqueness of distributional solutions w to the ordinary differential equation with discontinuous coefficients and right-hand side. For example, if a and w are non-smooth the product a · w″ has no obvious meaning. When interpreted on the most general level of the hierarchy of distributional products discussed by Oberguggenberger, M. [1992, Multiplication of distributions and applications to partial differential equations (Harlow: Longman Scientific & Technical)], it turns out that existence of a solution w forces it to be at least continuously differentiable. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of having a discontinuous displacement function as a solution. We also give conditions for unique solvability.

§Supported by Ministry of Science of Serbia, project 144016, and the Austrian Science Fund (FWF) START program Y237 on ‘Nonlinear distributional geometry’.

Keywords:

• Ordinary differential equations with discontinuous coefficients
• Distributional solutions
• Multiplication of distributions

Keywords:

• AMS 2000 Subject Classifications:
• 46F10
• 34A36

Acknowledgements

The authors are very grateful to Prof. Teodor Atanackovic for providing the mechanical model and valuable suggestions for mathematical investigations. We also thank Simon Haller for critical discussions on several details in the construction and regularity of the solution.

Notes

§Supported by Ministry of Science of Serbia, project 144016, and the Austrian Science Fund (FWF) START program Y237 on ‘Nonlinear distributional geometry’.