Generalized solutions for the Euler-Bernoulli model with distributional forces

TL;DR

This paper proves existence, uniqueness, and regularity of generalized solutions for an Euler-Bernoulli beam model with discontinuous and distributional forces, using advanced functional analysis techniques.

Contribution

It introduces a novel approach to handle distributional coefficients in the Euler-Bernoulli equation, extending classical solutions to a generalized framework.

Findings

Existence and uniqueness of generalized solutions established.

Regularity properties of solutions analyzed.

Applicable to beams with discontinuous and distributional forces.

Abstract

We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves discontinuous, and even distributional coefficients and right-hand side. The general problem is solved by application of functional analytic techniques to obtain estimates for the solutions to regularized problems. Finally, we prove coherence properties and provide a regularity analysis of the generalized solution.