9 problems

Let $d$ be the dimension, let $cmathcal H(d)$ denote the Hankel subspace, and let $p^*=\max\{p,p/(p-1)\}$ be the conjugate exponent. Write $C(\mathcal H(d),p)$ for the optimal…

Let $\mu$ be a centered and isotropic probability measure on $\mathbb{R}^n$ with density $d\mu=e^{-\phi}dx$, meaning … Let $\gamma$ denote the standard Gaussian measure, and let…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, and let the fractional Korn inequalities under consideration be the fractional analogues of Korn's first and second i…

Let $d$ be the dimension and let $\phi$ satisfy the normalization, regularity, and weighted Poincaré assumptions denoted by … – … int{mathbb R^d} xi xj e^{-phi(x)},mathrm d x=delta…

Matching-asymptotics conjecture. The optimal constants $C_I$ and $C_{II}$ have the same asymptotics as $h$ goes to zero.

Korn scaling conjecture. One has

Consider a circular cylindrical shell with a shape imperfection, such as a small localized dent, and let the trivial branch denote the unbuckled solution branch. Let the fundamenta…

Consider the three cylindrical components of the gradient matrix in the Korn-like inequalities for a circular cylindrical shell, whose perfect symmetry produces three distinct scal…

Let a cylindrical shell be subject to imperfections that break the symmetry of the circular cylindrical shell, and consider the Korn-like inequalities for individual components of…