| Abstract: |
| I will talk about a joint work with Yuya Okamoto concerning a fourth order overdetermined boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to
the homogeneous Dirichlet boundary condition.
It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball.
In this talk, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant.
We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the
deviation of a domain from a ball in terms of the perturbation of the boundary value. |
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