The viscosity method for min-max free boundary minimal surfaces

We adapt the viscosity method introduced by Rivi`ere to the free boundary case. Namely, given a compact oriented surface $Sigma$, possibly with boundary, a closed ambient Riemannian manifold $(mathcal{M}^m,g)$ and a closed embedded submanifold $mathcal{N}^nsubsetmathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Phi$ for the functional begin{align*} &E_sigma(Phi):=operatorname{area}(Phi)+sigmaoperatorname{length}(Phi|_{partialSigma})+sigma^4int_Sigma|{mathrm {I!I}}^Phi|^4,operatorname{vol}_Phi end{align*} on immersions $Phi:Sigmatomathcal{M}$ with the constraint $Phi(partialSigma)subseteqmathcal{N}$, as $sigmato 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $mathcal{F}$ of compact subsets of the space of smooth immersions $(Sigma,partialSigma)to(mathcal{M},mathcal{N})$, assuming $mathcal{F}$ to be stable under isotopies of this space we show that the min-max value begin{align*} &beta:=inf_{Ainmathcal{F}}max_{Phiin A}operatorname{area}(Phi) end{align*} is the sum of the areas of finitely many branched minimal immersions $Phi_{(i)}:Sigma_{(i)}tomathcal{M}$ with $partial_ uPhi_{(i)}perp Tmathcal{N}$ along $partialSigma_{(i)}$, whose (connected) domains $Sigma_{(i)}$ can be different from $Sigma$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_sigma$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.