The optimal shape of a dendrite sealed at both ends

In this paper, we are interested in the geometric structures which appear in nature. We consider the example of a nerve fiber and we suppose that shapes in nature arise in order to optimize some criterion. Then, we try to solve the problem consisting in searching the shape of a nerve fiber for a given criterion. The first considered criterion represents the attenuation in space of the electrical message troughout the fiber and seems to be relevant. Our second criterion represents the attenuation in time of the electrical message and doesn’t provide a realistic shape. We prove that the associated optimization problem has no solution.