Extremal functions for Adams' inequalities in dimension four
Xiaomeng Li · arXiv (Cornell University) · 2018
Let $Ω\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(Ω)$ be the usual Sobolev space. For any positive integer $\ell$, $λ_{\ell}(Ω)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{λ_1(Ω)}\oplus E_{λ_2(Ω)}\oplus\cdots\oplus E_{λ_{\ell}(Ω)}$, where $E_{λ_i(Ω)}$ is eigenfunction space associated with $λ_i(Ω)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(Ω)$. For $0\leqα