6 problems

Let $\Omega\subset\mathbb R^d$ be a domain of finite measure, and let $\lambda_k^{(m,t)}(\Omega)$ denote its $k$th buckling eigenvalue. Generalised Payne buckling inequality. For e…

Let $\Omega\subset\mathbb R^d$ be a domain of finite measure with inradius $\varrho$, and let $\lambda_{1}^{(m)}(\Omega)$ denote its first polyharmonic eigenvalue. Inradius asympto…

Let $\Omega$ be a domain of finite measure, let $\mathbb B$ denote the unit ball in $\mathbb R^d$, and let $\omega_d=|\mathbb B|$. For $m,t\in\mathbb N$, write…

Let $m$ and $\alpha$ be the parameters defining the weighted Sobolev quotient $\mathcal{S}(m,2,\alpha,\alpha,\infty)$, and suppose that this quotient admits a $0$-th nonsingular mi…

Let $m$ be the order of the polyharmonic operator, whose principal part is $(-\Delta)^m$, and let $s$ denote the Sobolev regularity order of its coefficients. Optimal-regularity co…

Let $Omega$ be a domain in the hemisphere $mathbb S^2_+$. For $p\geq 1$, let $R_1^{p,D}(z)$ denote the first Riesz mean of the Dirichlet eigenvalues of the $p$-harmonic operator on…