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In the present work we present a general framework which guarantees theexistence of optimal domains for isoperimetric problems within the class of$C^{1,1}$-regular domains satisfying a uniform ball condition as long as thedesired objective function satisfies certain properties. We then verify thatthe helicity isoperimetric problem studied by Cantarella, DeTurck, Gluck andTeytel in \cite{CDGT002} satisfies the conditions of our framework and henceestablish the existence of optimal domains within the given class of domains.We additionally use the same framework to prove the existence of optimaldomains among uniform $C^{1,1}$-domains for a first curl eigenvalue problemwhich has been studied recently for other classes of domains in \cite{EGPS23}.

Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition