Variational aspects of phase transitions with prescribed mean curvature

Abstract

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the “prescribed mean curvature” function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asymptotics that are sharp to order $o(\varepsilon ^2)$ and $C^{2,\alpha }$ estimates on multiplicity-one phase transition layers.

Appendices

A Derivation of (66) and (79)

In what follows, (53) gets used repeatedly though implicitly when obtaining $O_{1,0,\alpha ,\varepsilon }$ bounds.

We project (65) onto $\Gamma$ by fixing $y \in B_{19}^\Gamma$, dotting with ${\overline{{\mathbb {H}}}}_\varepsilon '(y, z)$ and integrating over z. We start with the left hand side. We differentiate $\phi \perp {\overline{{\mathbb {H}}}}_\varepsilon '$ along y and use (54) to get:

$$\begin{aligned} \int _{\mathbf {R}}\varepsilon ^2 (\Delta _{g_z} \phi ) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz&= \int _{\mathbf {R}}\varepsilon ^2 (\Delta _\Gamma \phi ) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz + \int _{\mathbf {R}}\varepsilon ^2 (\Delta _{g_z} \phi - \Delta _\Gamma \phi ) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&= - \int _{\mathbf {R}}\varepsilon ^2 \phi (\Delta _\Gamma {\overline{{\mathbb {H}}}}_\varepsilon ') \, dz - \int _{\mathbf {R}}\varepsilon ^2 (\nabla _\Gamma \phi ) \cdot (\nabla _\Gamma {\overline{{\mathbb {H}}}}_\varepsilon ') \, dz \nonumber \\&\qquad + \int _{\mathbf {R}}\varepsilon ^2 (\Delta _{g_z} \phi - \Delta _\Gamma \phi ) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&= \int _{\mathbf {R}}\phi (\varepsilon (\Delta _\Gamma h) {\overline{{\mathbb {H}}}}_\varepsilon '' - |\nabla _\Gamma h|^2 {\overline{{\mathbb {H}}}}_\varepsilon ''') \, dz + \int _{\mathbf {R}}\varepsilon (\nabla _\Gamma \phi ) \cdot (\nabla _\Gamma h) {\overline{{\mathbb {H}}}}_\varepsilon '' \, dz \nonumber \\&\qquad + \int _{\mathbf {R}}\varepsilon ^2 (\Delta _{g_z} \phi - \Delta _\Gamma \phi ) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&= \varepsilon (\Delta _\Gamma h) \int _{\mathbf {R}}\phi {\overline{{\mathbb {H}}}}_\varepsilon '' \, dz + \varepsilon \cdot O_{1,0,\alpha ,\varepsilon }(\phi ) (O_{1,0,\alpha ,\varepsilon }(\nabla _\Gamma h))^2 \nonumber \\&\qquad + \varepsilon \cdot O_{1,0,\alpha ,\varepsilon }(\varepsilon \nabla _\Gamma \phi ) \cdot O_{1,0,\alpha ,\varepsilon }(\nabla _\Gamma h) \nonumber \\&\qquad + \varepsilon ^2 \cdot O_{1,0,\alpha ,\varepsilon }(\varepsilon ^2 \nabla ^2_\Gamma \phi , \varepsilon \nabla _\Gamma \phi ). \end{aligned}$$

(113)

Next, integrating by parts yields and using $\phi \perp {\overline{{\mathbb {H}}}}_\varepsilon '$ again:

$$\begin{aligned} \int _{\mathbf {R}}\varepsilon ^2 H_z (\partial _z \phi ) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz&= - \int _{\mathbf {R}}\varepsilon H_z \phi {\overline{{\mathbb {H}}}}_\varepsilon '' \, dz - \int _{\mathbf {R}}\varepsilon ^2 (\partial _z H_z) \phi {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&= - \int _{\mathbf {R}}\varepsilon (H_\Gamma \phi + O(1) z) {\overline{{\mathbb {H}}}}_\varepsilon '' \, dz - \int _{\mathbf {R}}\varepsilon ^2 (\partial _z H_z) \phi {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&= - \varepsilon H_\Gamma \int _{\mathbf {R}}\phi {\overline{{\mathbb {H}}}}_\varepsilon '' \, dz + \varepsilon ^3 \cdot O_{1,0,\alpha ,\varepsilon }(\phi ). \end{aligned}$$

(114)

Next, integrating by parts twice yields:

$$\begin{aligned} \int _{\mathbf {R}}\big [ \varepsilon ^2 (\partial _z^2 \phi ) - W''({\overline{{\mathbb {H}}}}_\varepsilon ) \phi \big ] {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz = \varepsilon ^3 \cdot O_{1,0,\alpha ,\varepsilon }(\phi ). \end{aligned}$$

(115)

We move on to the right hand side of (65). We have:

$$\begin{aligned} \int _{\mathbf {R}}\varepsilon {\mathfrak {h}}{\overline{{\mathbb {H}}}}_\varepsilon ' \, dz&= \int _{\mathbf {R}}\varepsilon ({\mathfrak {h}}(\cdot , 0) + (\partial _z {\mathfrak {h}})(\cdot , 0) z + O_{C^{1,\alpha }_\varepsilon }(1) z^2) {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&= 2 \varepsilon ^2 {\mathfrak {h}}(\cdot , 0) + 2 \varepsilon ^2 (\partial _z {\mathfrak {h}}(\cdot , 0)) h + O_{1,0,\alpha ,\varepsilon }(\varepsilon ^4). \end{aligned}$$

(116)

Next:

$$\begin{aligned} \int _{\mathbf {R}}\varepsilon (H_\Gamma - \Delta _\Gamma h) ({\overline{{\mathbb {H}}}}_\varepsilon ')^2 \, dz = \varepsilon ^2 {\mathfrak {e}}_0(H_\Gamma - \Delta _\Gamma h). \end{aligned}$$

(117)

Next:

$$\begin{aligned} \int _{\mathbf {R}}\varepsilon (|{{\,\mathrm{\mathrm {I\!I}}\,}}_\Gamma |^2 + {{\,\mathrm{Ric}\,}}({\varvec{n}}_\Gamma , {\varvec{n}}_\Gamma )) z {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz = 2 \varepsilon ^2 (|{{\,\mathrm{\mathrm {I\!I}}\,}}_\Gamma |^2 + {{\,\mathrm{Ric}\,}}({\varvec{n}}_\Gamma , {\varvec{n}}_\Gamma )) h. \end{aligned}$$

(118)

For now, we estimate:

$$\begin{aligned} \int _{\mathbf {R}}\tfrac{1}{2} W'''({\overline{{\mathbb {H}}}}_\varepsilon ) \phi ^2 {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz = \varepsilon \cdot (O_{1,0,\alpha ,\varepsilon }(\phi ))^2, \end{aligned}$$

(119)

though we will refine this estimate later once we get a more precise form of $\phi$. Finally:

$$\begin{aligned}&\int _{\mathbf {R}}\big [ (O_{1,0,\alpha ,\varepsilon }(\phi ))^3 + O_{1,0,\alpha ,\varepsilon }(\varepsilon \nabla _\Gamma ^2 h, \nabla _\Gamma h) z {\overline{{\mathbb {H}}}}_\varepsilon ' + (O_{1,0,\alpha ,\varepsilon }(\nabla _\Gamma h))^2 {\overline{{\mathbb {H}}}}_\varepsilon '' + O_{1,0,\alpha ,\varepsilon }(\varepsilon ^3) \big ] {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&\qquad = \varepsilon \cdot (O_{1,0,\alpha ,\varepsilon }(\phi ))^3 + \varepsilon ^2 \cdot O_{1,0,\alpha ,\varepsilon }(\varepsilon \nabla _\Gamma ^2 h, \nabla _\Gamma h) + \varepsilon \cdot (O_{1,0,\alpha ,\varepsilon }(\nabla _\Gamma h))|^2 + O_{1,0,\alpha ,\varepsilon }(\varepsilon ^4). \end{aligned}$$

(120)

At this point, (66) follows from combining (113), (114), (115), (116), (117), (118), (119), (120), and finally estimating h by $\phi$ as in [26, Lemma 9.6].

Finally, let us assume we have a more refined ansatz for $\phi$, namely:

$$\begin{aligned} \phi = {\hat{\phi }} + \varepsilon {\mathfrak {h}}{\overline{{\mathbb {I}}}}_\varepsilon \end{aligned}$$

where ${\hat{\phi }} = O_{1,0,\alpha ,\varepsilon }(\varepsilon ^2)$. Then, we can replace (119) by

$$\begin{aligned}&\int _{\mathbf {R}}\tfrac{1}{2} W'''({\overline{{\mathbb {H}}}}_\varepsilon ) \phi ^2 {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz \nonumber \\&\qquad = \varepsilon ^2 \int _{\mathbf {R}}\tfrac{1}{2} W'''({\overline{{\mathbb {H}}}}_\varepsilon ) {\mathfrak {h}}^2 {\overline{{\mathbb {I}}}}_\varepsilon ^2 {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz + O_{1,0,\alpha ,\varepsilon }(\varepsilon ^4) \nonumber \\&\qquad = \varepsilon ^2 \int _{\mathbf {R}}\tfrac{1}{2} W'''({\overline{{\mathbb {H}}}}_\varepsilon ) ({\mathfrak {h}}(\cdot , 0) + O_{1,0,\alpha ,\varepsilon }(1) z)^2 {\overline{{\mathbb {I}}}}_\varepsilon ^2 {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz + O_{C^{0,\alpha }_\varepsilon }(\varepsilon ^4) \nonumber \\&\qquad = \varepsilon ^2 {\mathfrak {h}}(\cdot , 0)^2 \int _{\mathbf {R}}\tfrac{1}{2} W'''({\overline{{\mathbb {H}}}}_\varepsilon ) {\overline{{\mathbb {I}}}}_\varepsilon ^2 {\overline{{\mathbb {H}}}}_\varepsilon ' \, dz + O_{1,0,\alpha ,\varepsilon }(\varepsilon ^4) = O_{1,0,\alpha ,\varepsilon }(\varepsilon ^4), \end{aligned}$$

(121)

where in the last step we’ve used (68) and the fact that, by parity,

$$\begin{aligned} \int _{\mathbf {R}}W'''({\mathbb {H}}) {\mathbb {I}}^2 {\mathbb {H}}' \, dz = 0. \end{aligned}$$

Now, (79) follows from the same equations, with (121) replacing (119).

B Derivation of (68), (70), (78)

This section is meant to simplify and condense the exposition in [26, Sections 11-13] by exploiting the multiplicity-one setting. It is borrowed from collaborative notes written with O. Chodosh. In this appendix we will assume, without loss of generality, that $W''(\pm 1) = 2$.

Lemma 15

Consider $w \in C^{2}({\mathbf {R}}^{n})$ and $f \in C^{0}({\mathbf {R}}^{n-1})$ so that, for $(y, z) \in {\mathbf {R}}^{n-1} \times {\mathbf {R}}= {\mathbf {R}}^n$,

$$\begin{aligned} \Delta _{{\mathbf {R}}^{n-1}} w(y,z) + \partial _{z}^{2} w(y,z) - W''({\mathbb {H}}(z)) w(y,z) = f(y){\mathbb {H}}'(z). \end{aligned}$$

Then, there is some $c \in C^{2}({\mathbf {R}}^{n-1})$ so that $w=c(y){\mathbb {H}}'(z)$.

Proof

We mimic [19, Lemma 3.7]. Write

$$\begin{aligned} w(y,z) = c(y) {\mathbb {H}}'(z) + {\bar{w}}(y,z) \end{aligned}$$

where $\int _{-\infty }^{\infty } w(z,y) {\mathbb {H}}'(z) dz = 0$ for all $y\in {\mathbb {R}}^{n-1}$. We thus find that

$$\begin{aligned} {\mathbb {H}}'(z) \Delta _{{\mathbf {R}}^{n-1}} c(y) + (\partial ^{2}_{z} {\bar{w}}(y,z) - W''({\mathbb {H}}(z)) {\bar{w}}(y,z) + \Delta _{{\mathbf {R}}^{n-1}} {\bar{w}}(y,z)) = f(y){\mathbb {H}}'(z). \end{aligned}$$

Multiplying by ${\mathbb {H}}'(z)$ and integrating, we find that $\Delta _{{\mathbf {R}}^{n-1}} c(y) = f(y)$, and so

$$\begin{aligned} \partial ^{2}_{z} {\bar{w}}(y,z) - W''({\mathbb {H}}(z)) {\bar{w}}(y,z) + \Delta _{{\mathbf {R}}^{n-1}} {\bar{w}}(y,z) = 0. \end{aligned}$$

At this point, the proof that ${\bar{w}} = 0$ is identical to [19, Lemma 3.7]. $\square$

Lemma 16

Fix $\sigma \in (0,1)$. Then, we can choose $L>0$ and $C>0$ depending on $\sigma$, and $K>0$ sufficiently large depending only on W with the following property. Suppose that

$$\begin{aligned} \varepsilon ^{2} (\Delta _{\Gamma } \psi + \partial ^{2}_{z}\psi ) - W''({\overline{{\mathbb {H}}}}_\varepsilon ) \psi = \varepsilon f_{1}(y) {\overline{{\mathbb {H}}}}_{\varepsilon }'(y,z) + f_{2}(y,z) + \varepsilon D_{i} f_{3}^{(i)}(y,z) \end{aligned}$$

(122)

on $B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon$. Then, for $\varepsilon >0$ sufficiently small, either

$$\begin{aligned} \Vert \psi \Vert _{C^{0}(B_{r}^\Gamma \times {\mathbf {I}}_{\varepsilon })} \le 2 {\mathbb {H}}'(0) {\mathfrak {e}}_0^{-1} \sup _{y\in B^{\Gamma }_{r+2L\varepsilon }} \left| \int _{-\varepsilon K}^{\varepsilon K} \psi (y,z) {\overline{{\mathbb {H}}}}'_{\varepsilon }(z-h(y)) \, dz \right| \end{aligned}$$

or

$$\begin{aligned} \Vert \psi \Vert _{C^{0}(B_{r}^{\Gamma } \times {\mathbf {I}}_\varepsilon )}&\le \sigma \left( \Vert \psi \Vert _{C^{1}_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert f_{1} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma })} \right) \\&\qquad + C \left( \Vert f_{2}\Vert _{C^{0}(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert \mathbf {f}_{3} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert \psi \Vert _{C^0(B_{r}^{\Gamma } \times {\mathbf {J}}_{\varepsilon ,L})} \right) , \end{aligned}$$

where ${\mathbf {J}}_{\varepsilon ,L}$ denotes the points of ${\mathbf {I}}_\varepsilon$ that are within $\varepsilon L$ of $\partial {\mathbf {I}}_\varepsilon$.

Proof

First, choose ${\tilde{\chi }} : B_{r+2L\varepsilon }^{\Gamma }\rightarrow [0,1]$ a cutoff function that is 1 on $B_{r}^{\Gamma }$ and has support in $B_{r+ L\varepsilon }^{\Gamma }$. We can arrange so that $\varepsilon L |\nabla _{\Gamma } {\tilde{\chi }}| + \varepsilon ^{2}L^{2} |\nabla ^{2}_{\Gamma } {\tilde{\chi }}|^{2} = O(1)$. Now, by replacing $\psi$ by ${\tilde{\chi }} \psi$ and absorbing the resulting error terms into $f_{2}$, it is clear that it suffices to prove that

$$\begin{aligned} \Vert \psi \Vert _{C^{0}(B_{r}^{\Gamma } \times {\mathbf {I}}_\varepsilon )}&\le \sigma \Vert f_{1} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} \nonumber \\&\qquad + C \left( \Vert f_{2}\Vert _{C^{0}(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert \mathbf {f}_{3} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert \psi \Vert _{C^0(B_r^\Gamma \times {\mathbf {J}}_{\varepsilon ,L})} \right) \end{aligned}$$

(123)

assuming that $\psi$ is supported in $B^{\Gamma }_{r+\frac{1}{2} L\varepsilon }\times {\mathbf {I}}_{\varepsilon }$ and satisfies (122) and

$$\begin{aligned} \sup _{y\in B^{\Gamma }_{r+2L\varepsilon }} \left| \int _{-\varepsilon K}^{\varepsilon K} \psi (y,z) {\overline{{\mathbb {H}}}}'_{\varepsilon }(z-h(y)) \, dz \right| < \frac{1}{2} {\mathbb {H}}'(0)^{-1} {\mathfrak {e}}_0\Vert \psi \Vert _{C^{0}(B^\Gamma _{r} \times {\mathbf {I}}_{\varepsilon })}. \end{aligned}$$

(124)

Assume, for contradiction, that (123) fails. Then, there are $C,L\rightarrow \infty$ as $\varepsilon \rightarrow 0$ so that

$$\begin{aligned} \Vert \psi \Vert _{C^{0}(B_{r}^{\Gamma } \times {\mathbf {I}}_\varepsilon )}&\ge \sigma \Vert f_{1} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} \\&\qquad + C \left( \Vert f_{2}\Vert _{C^{0}(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert \mathbf {f}_{3} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{r+2L\varepsilon }^{\Gamma } \times {\mathbf {I}}_\varepsilon )} + \Vert \psi \Vert _{C^0(B_r^\Gamma \times {\mathbf {J}}_{\varepsilon ,L})} \right) . \end{aligned}$$

Choose ${\bar{x}} = ({\bar{y}},{\bar{z}}) \in \overline{B_{r}^{\Gamma }\times {\mathbf {I}}_{\varepsilon }}$ attaining $\Vert \psi \Vert _{C^{0}(B_{r}^{\Gamma } \times {\mathbf {I}}_\varepsilon )}$. Set ${\tilde{z}} = \varepsilon ^{-1}{\bar{z}}$. We first assume that ${\tilde{z}} \rightarrow {\hat{z}}$ as $\varepsilon \rightarrow 0$. The case that ${\tilde{z}}$ is unbounded as $\varepsilon \rightarrow 0$ follows from a similar, but simpler argument, as we describe below. Dividing the equation by $\pm \Vert \psi \Vert _{C^{0}(B_{r}^{\Gamma } \times {\mathbf {I}}_\varepsilon )}$ and rescaling around ${\bar{x}}$ to scale $\varepsilon$ (labeling rescaled quantities with a tilde), we find that ${\tilde{\psi }}(0) = 1$, $\Vert {\tilde{\psi }}\Vert _{C^{0}(B_{L})} = 1$,

$$\begin{aligned} \Delta _{{\tilde{\Gamma }}} {\tilde{\psi }} + \partial ^{2}_{z}{\tilde{\psi }} - W''(\tilde{{\overline{{\mathbb {H}}}}}) {\tilde{\psi }} = {\tilde{f}}_{1}(y) \tilde{{\overline{{\mathbb {H}}}}}'(z-{\tilde{z}}-\varepsilon ^{-1}{\tilde{h}}(y)) + {\tilde{f}}_{2}(y,z) + D_{i} {\tilde{f}}_{3}^{(i)}(y,z), \end{aligned}$$

on $B_{L}$, and finally

$$\begin{aligned} \Vert {\tilde{f}}_{1} \Vert _{C^{0,\alpha }(B_{L})} \le \sigma ^{-1}, \text { and } \Vert {\tilde{f}}_{2}\Vert _{C^{0}(B_{L})} + \Vert \tilde{f}_{3} \Vert _{C^{0,\alpha }_{\varepsilon }(B_{L})} = o(1). \end{aligned}$$

Hence, ${\tilde{f}}_{2}\rightarrow 0$ in $C^{0}(B_{L})$ and ${\tilde{f}}_{3}^{(i)}\rightarrow 0$ in $C^{0,\alpha }(B_{L})$. Moreover, ${\tilde{f}}_{1}$ is bounded in $C^{0,\alpha }(B_{L})$. We can thus find ${\hat{f}}_{1}\in C^{0,\alpha }({\mathbf {R}}^{n-1})$ so that ${\tilde{f}}_{1}\rightarrow {\hat{f}}_{1}$ in $C^{0,\alpha '}_{\text {loc}}({\mathbf {R}}^{n-1})$ for $\alpha ' < \alpha$.

Similarly, by $C^{1,\alpha }$-Schauder estimates we see that ${\tilde{\psi }}$ is uniformly bounded in $C^{1,\alpha }$ on compact subsets of ${\mathbf {R}}^{n}$. Thus, there is ${\hat{\psi }}\in C^{1,\alpha }_{\text {loc}}({\mathbf {R}}^{n})\cap L^{\infty }({\mathbf {R}}^{n})$ so that ${\tilde{\psi }} \rightarrow {\hat{\psi }}$ in $C^{1,\alpha '}_{\text {loc}}({\mathbf {R}}^{n})$. Integrating by parts against a test function, we see that ${\hat{\psi }}$ weakly solves

$$\begin{aligned} \Delta _{{\mathbf {R}}^{n-1}} {\hat{\psi }} + \partial ^{2}_{z} {\hat{\psi }} - W''({\mathbb {H}}(z-{\tilde{z}})) {\hat{\psi }} = {\hat{f}}_{1}(y) {\mathbb {H}}'(z-{\tilde{z}}). \end{aligned}$$

Schauder theory implies that ${\hat{\psi }} \in C^{2,\alpha }({\mathbf {R}}^{n})$. By Lemma 15, we have that ${\hat{\psi }} = c(y) {\mathbb {H}}'(z-{\hat{z}})$. Because ${\hat{\psi }}(0) = 1 = \Vert {\hat{\psi }}\Vert _{L^{\infty }({\mathbf {R}}^{n})}$, we see that ${\hat{z}} =0$ and $c(0) = {\mathbb {H}}'(0)^{-1}$. Thus, we see that

$$\begin{aligned} \int _{-K}^{K}{\hat{\psi }}(0,z) {\mathbb {H}}'(z) dz = {\mathbb {H}}'(0)^{-1} {\mathfrak {e}}_0+ O(e^{-\sqrt{2}K}) \end{aligned}$$

Returning to $\psi$, we thus find that

$$\begin{aligned} \sup _{y\in B^{\Gamma }_{r+2L\varepsilon }} \left| \int _{-\varepsilon K}^{\varepsilon K} \psi (y,z) {\overline{{\mathbb {H}}}}'_{\varepsilon }(z-h(y)) dz \right| \ge \left( {\mathbb {H}}'(0)^{-1} {\mathfrak {e}}_0+ O(e^{-\sqrt{2}K}) + o(1) \right) \Vert \psi \Vert _{C^{0}(B_{r}^\Gamma \times {\mathbf {I}}_{\varepsilon })} \end{aligned}$$

as $\varepsilon \rightarrow 0$. Taking K sufficiently large this contradicts (124) for $\varepsilon$ sufficiently small.

Finally, if the case that ${\tilde{z}}\rightarrow \infty$, then repeating the same rescaling as above (but using ${\mathbb {H}}(t)\rightarrow \pm 1$ as $t\rightarrow \pm \infty$), we find ${\hat{\psi }} \in C^{2,\alpha }_{\text {loc}}({\mathbf {R}}^{n})\cap L^{\infty }({\mathbf {R}}^{n})$, with ${\hat{\psi }}(0) = 1$ and so that

$$\begin{aligned} \Delta _{{\mathbf {R}}^{n}}{\hat{\psi }} - W''(\pm 1) {\hat{\psi }} = 0. \end{aligned}$$

Because ${\hat{\psi }}$ attains its maximum at 0, we see that ${\hat{\psi }} \equiv 0$, a contradiction. $\square$

We note how the first alternative of Lemma 16 can never apply to $\phi$, provided K is chosen sufficiently large. Indeed, it follows from (60) that

$$\begin{aligned} \left| \int _{-\varepsilon K}^{\varepsilon K} \phi (y, z) {\overline{{\mathbb {H}}}}'_\varepsilon (y, z) \, dz \right| = \left| \int _{{\mathbf {I}}_{\varepsilon } \setminus [-\varepsilon K, \varepsilon K]} \phi (y, z) {\overline{{\mathbb {H}}}}'_\varepsilon (y, z) \, dz \right| \le C e^{-\sqrt{2}K} \Vert \phi (y, \cdot ) \Vert _{C^0({\mathbf {I}}_\varepsilon )}.\nonumber \\ \end{aligned}$$

(125)

Therefore, for sufficiently large (but fixed) choices of K, the second alternative of Lemma 16 must always hold when $\psi = \phi$.

Let us use this fact to prove (68). We first note that (65) and (64) imply

$$\begin{aligned} \varepsilon ^2 \Delta \phi - W''({\overline{{\mathbb {H}}}}_\varepsilon ) \phi&= - \varepsilon (H_\Gamma - \Delta _\Gamma h) \overline{{\mathbb {H}}}_\varepsilon ' + O_{1,0,\alpha ,\varepsilon }(\varepsilon {\mathfrak {h}}, \varepsilon ^2) + (O_{1,0,\alpha ,\varepsilon }(\phi ))^2 \nonumber \\&\qquad + (O_{1,0,\alpha ,\varepsilon }(\varepsilon \nabla _\Gamma ^2 h, \nabla _\Gamma h))^2 \nonumber \\&= - \varepsilon (H_\Gamma - \Delta _\Gamma h) \overline{{\mathbb {H}}}_\varepsilon ' + O_{1,0,\alpha ,\varepsilon }(\varepsilon ) + (O_{1,0,\alpha ,\varepsilon }(\varepsilon ^2 \nabla ^2_\Gamma \phi , \varepsilon \nabla _\Gamma \phi , \phi ))^2, \end{aligned}$$

(126)

where the second equation follows from the first from our bounds on the prescribed function ${\mathfrak {h}}$ our ability to control the height adjustment h in terms of $\phi$ ( [26, Lemma 9.6]).

Fix $\sigma \in (0, 1)$. We apply Lemma 16 in $B_{19}^\Gamma \times {\mathbf {I}}_\varepsilon$ to get a $C^0$ estimate on $\phi$ in $B_{19 - 2 \varepsilon L}^\Gamma \times {\mathbf {I}}_\varepsilon$ (using (55) to treat $\varepsilon ^2 (\Delta - \Delta _\Gamma - \partial _z^2) \phi$ as a right hand side term), which can be enlarged to a $C^0$ estimate on $B_{19 - 2 \varepsilon L}^\Gamma \times (-1, 1)$ with at most an $O(\varepsilon )$ error using the decay of $\phi$ off $\Gamma$. Then use Schauder theory on (65), (66) and again [26, Lemma 9.6], and absorbing the terms that are quadratic in $\phi$ we get:

$$\begin{aligned} \Vert \phi \Vert _{C^{2,\alpha }_\varepsilon (B_{19 - 4 \varepsilon L}^\Gamma \times (-1,1))} + \Vert H_\Gamma - \Delta _\Gamma h \Vert _{C^{0,\alpha }_\varepsilon (B^\Gamma _{19-4\varepsilon L})} \nonumber \\ \le \sigma (\Vert \phi \Vert _{C^{2,\alpha }_\varepsilon (B_{19}^\Gamma \times (-1,1))} + \Vert H_\Gamma - \Delta _\Gamma h \Vert _{C^{0,\alpha }_\varepsilon (B^\Gamma _{19})}) + C' \varepsilon , \end{aligned}$$

(127)

for a fixed $C' > 0$. Iterating this procedure on $B_{19 - 4k \varepsilon L}^\Gamma \times {\mathbf {I}}_\varepsilon$ for $k = 1, \ldots , M |\log \varepsilon |$, where M depends on $\sigma \in (0, 1)$ but not $\varepsilon$, yields the $\phi$ estimate in (68) and thus also (69).

We move on to verifying (70). Differentiating (65) in the directions parallel to $\Gamma$ (i.e., in $y_i$ in Fermi coordinates) we see similarly to (126) that:

$$\begin{aligned} \varepsilon ^2 \Delta (\varepsilon \partial _{y_i} \phi ) - W''({\overline{{\mathbb {H}}}}_\varepsilon ) (\varepsilon \partial _{y_i} \phi ) = - \varepsilon (\varepsilon \partial _{y_i} (H_\Gamma - \Delta _\Gamma h)) \overline{{\mathbb {H}}}_\varepsilon ' + {\mathcal {R}}\end{aligned}$$

(128)

where the error term can be estimated (using (68)) by:

$$\begin{aligned}\Vert {\mathcal {R}}\Vert _{C^{0,\alpha }_\varepsilon } \le C \varepsilon ^2 + C (\varepsilon ^2 \Vert \nabla ^2_\Gamma \varepsilon \partial _{y_i} \phi \Vert _{C^{0,\alpha }_\varepsilon } + \varepsilon \Vert \nabla _\Gamma \varepsilon \partial _{y_i} \phi \Vert _{C^{0,\alpha }_\varepsilon } + \Vert \varepsilon \partial _{y_i} \phi \Vert _{C^{0,\alpha }_\varepsilon })^2 \end{aligned}$$

Next, one differentiates (60) in the horizontal directions to show, similarly as in (125) but also estimating the error term $\langle \phi , \partial _{y_i} {\overline{{\mathbb {H}}}}_\varepsilon ' \rangle _{L^2}$, that

$$\begin{aligned} \left| \int _{-\varepsilon K}^{\varepsilon K} \varepsilon (\partial _{y_i} \phi )(y, z) {\overline{{\mathbb {H}}}}_\varepsilon '(y, z) \, dz \right| \le C e^{-\sqrt{2} K} \Vert \varepsilon (\partial _{y_i} \phi )(y, \cdot ) \Vert _{C^0({\mathbf {I}}_\varepsilon )} + C \varepsilon ^3. \end{aligned}$$

(129)

Lemma 16’s first alternative can only hold for $\psi = \varepsilon \partial _{y_i} \phi$, then, in case $\Vert \varepsilon \partial _{y_i} \phi \Vert = O(\varepsilon ^3)$ (which is smaller than the worse upper bound we wish to prove, and thus does not break the applicability of our previous strategy). Arguing as above, using (128) instead of (126) yields (70).

Finally, we establish (78). Recall that, by (69) and (77), ${\hat{\phi }} = \phi - \varepsilon {\mathfrak {h}}{\mathbb {I}}$ satisfies:

$$\begin{aligned} \varepsilon ^2 \Delta {\hat{\phi }} - W''({\overline{{\mathbb {H}}}}_\varepsilon ) {\hat{\phi }} = O_{C^{0,\alpha }_\varepsilon }(\varepsilon ^2), \end{aligned}$$

(130)

The function ${\hat{\phi }}$ satisfies an estimate similar to (129), namely:

$$\begin{aligned} \left| \int _{-\varepsilon K}^{\varepsilon K} {\hat{\phi }}(y, z) {\overline{{\mathbb {H}}}}_\varepsilon '(y, z) \, dz \right| \le C e^{-\sqrt{2} K} \Vert {\hat{\phi }}(y, \cdot ) \Vert _{C^0({\mathbf {I}}_\varepsilon )} + C \varepsilon ^3. \end{aligned}$$

(131)

Thus, as before, Lemma 16’s first alternative can only hold for $\psi = {\hat{\phi }}$, then, in case $\Vert {\hat{\phi }} \Vert = O(\varepsilon ^3)$ (which is smaller than the worse upper bound we wish to prove, and thus does not break the applicability of our previous strategy). The rest of the argument goes through as before, applying (130) and (131) instead of (128) and (129).