| Abstract: |
| In this talk, we consider the initial and boundary value problem for the beam equation, which is known as one mathematical model for elastic materials. As the first step in modeling, we have introduced the stress function having the singular point. In our previous results, we have proved the existence and uniqueness of weak and strong solutions. Moreover, we obtained the lower bound of the strain. This estimate guarantees that the elastic curve does not shrink to the one point. This result indicates advantage of our model. We remark that in our model, an unknown function is vector-valued function and representing the position. For this reason, we need to treat the nonlinear strain function, and we adopt the singular stress function. For the construction of a model representing the characteristics of the elastic curve, we derive the initial and boundary value problem from the free energy including the curvature. For the mathematical difficulties, we approximate the equation by adding the sixth derivative term with respect to space variable. Based on the Galerkin method and the Aubin compact theorem, we can prove the existence of weak solutions. This is a joint work with Prof. Aiki from Japan Women`s University, Japan. |
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