Existence of an optimal shape for the first eigenvalue of polyharmonic operators

We prove the existence of an open set minimizing the first eigenvalue of the Dirichlet polylaplacian of order m >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 1$$\end{document} under volume constraint. Moreover, the corresponding eigenfunction is shown to enjoy Cm-1,alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{m-1,\alpha }$$\end{document} H & ouml;lder regularity. This is performed for dimension 2 <= d <= 4m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le d\le 4m$$\end{document}. In particular, our analysis answers the question of the existence of an optimal shape for the clamped plate up to dimension 8.