7 problems

Consider a composite material whose two phases have unequal volume fractions and are separated by a surface minimizing the total surface area. The upper bound of effective transpor…

Let a perfectly conducting inclusion in three dimensions have a fixed volume, and let its average polarizability be obtained by averaging over all orientations of the applied field…

Let a perfectly conducting inclusion in three dimensions have a fixed volume, and define its average polarizability by averaging over all orientations of the applied field. Pólya–S…

Conjecture. All eigenvalues are real, the set of eigenvalues is countable or finite, and the corresponding eigenfunctions $omega_p(z)$ satisfy the inequality stated in the source.

Let $\Omega$ be a ball, and let $B^*$ be a subdomain of $\Omega$ with the prescribed volume that minimizes the first eigenvalue of the mixture of two conducting materials. A concen…

Let $\mathcal T$ be a real finite-dimensional representation of $SO(3)$ of the form … where $W_{0}$, $W_{1}$, and $W_{2}$ are respectively the weight $0$, natural, and symmetric tr…

Let $S\subset\Omega$ be an open set with closure contained in $\Omega$. Consider continuously graded composites with inclusions having $C^{1,\alpha}$ boundaries, and let…