| Abstract: |
| Let $\Omega$ be a bounded smooth domain in ${R}^{n}$ with two disjoint boundary components $C_1$ and $C_2$. The mixed Steklov Dirichlet problem is to find harmonic function $u$ in $\Omega$ such that $u=0$ on $C_1$ and outer normal derivative of $u$ is directly proportional to $u$ along $C_2$. This problem models the stationary heat distribution in $\Omega$ with the conditions that the temperature along $C_1$ is kept to zero and that the heat flux through $C_2$ is proportional to the temperature. In this talk, I will first discuss about behaviour of the first Steklov Dirichlet eigenvalue on doubly connected domains and then provide some isoperimetric bounds for higher Steklov Dirichlet eigenvalues. I will also talk about similar bounds for eigenvalues of higher Steklov Neumann eigenvalues. |
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