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Banach spaces have a relatively uninteresting topology, because they are contractible. This prevents the direct application of Morse-like min-max arguments to establish the existence of critical points. However, there is a trick that works $X$ is a Banach space and $F: X \to \mathbf{R}$ is an even functional, that is \begin{equation} F(-x) = F(x) \quad \text{ for all $x \in X$}. \end{equation} Specifically, one can quotient $X$ by the $\mathbf{Z}/2\mathbf{Z}$-action; the quotient $X / \pm 1$ is now an infinite-dimensional projective space, and its topology can be used to construct critical points.

Typically $X$ is a function space, say $W^{1,2}(M)$ for example, defined on a compact manifold without boundary.

Can this method be used when $X$ is a function space over a manifold $M$ with non-empty boundary, and $F: X \to \mathbf{R}$ is still even? (I am interested primarily in the case where the boundary data is non-zero.)

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The boundary case works without any major change (at least for a reasonable functional). For example, if you consider the Allen--Cahn energy $E_\epsilon$ and take $W^{1,2}(M)$ then critical points of $E_{\epsilon}$ will be solutions to Allen--Cahn with Neumann boundary conditions.

In the original papers (Guraco, Gaspar--Guaraco) this was not considered. However in their later paper they discuss the case of boundary, e.g. the remark after Theorem 2.1 (or just search the pdf for "boundary" for more comments).

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  • $\begingroup$ Thanks Otis, that's interesting! So there's no way of doing min-max with a Dirichlet boundary condition? Taking the Allen-Cahn functional as an example, say we have some boundary function $g: \partial M \to \mathbf{R}$ for which there exist multiple stable solutions. I guess one-parameter min-max should be OK, but are you saying there's no way of defining multi-parameter min-max among functions with $u = g$ on $\partial M$? $\endgroup$
    – Leo Moos
    Commented Jun 3, 2023 at 20:22
  • $\begingroup$ I think that you can also do Dirichlet boundary conditions no problem. Just use $W_0^{1,2}$ or more generally $E_\epsilon(\cdot + g)$ on $W_0^{1,2}$ for some extension of $g$. However, it's not clear to me what you get in this situation (note that the estimates $\omega_p \sim p^{1/n+1}$ is unlikely to hold in this setting unless you choose the boundary conditions in some clever way. $\endgroup$
    – Otis Chodosh
    Commented Jun 3, 2023 at 21:43
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    $\begingroup$ Hmm, the natural $\mathbf{Z}/2\mathbf{Z}$-action on functions $u \in W^{1,2}(M)$ with $u = g$ would be $u \mapsto -u + 2g$. But the functional wouldn't be invariant under this action, as $E(-u + 2g) = E(u - 2g) \neq E(u)$. What am I not seeing here? $\endgroup$
    – Leo Moos
    Commented Jun 3, 2023 at 21:54
  • $\begingroup$ Whoops I was too hasty. I guess only g=0 makes sense in this setting. It would be interesting to understand the growth rate of the corresponding p widths. $\endgroup$
    – Otis Chodosh
    Commented Jun 4, 2023 at 2:45

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