Generic nondegeneracy for solutions of the Allen-Cahn equation under a volume constraint in closed manifolds

Gustavo de Paula Ramos Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090, São Paulo, SP, Brazil. [email protected] http://www.ime.usp.br/ gpramos

Abstract.

Let $M^{n}$ be a connected closed smooth manifold with $n\geq 2$ . We adapt the techniques in [MP09] and [GM11] to prove the generic nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint in $M$ .

Keywords. nondegenerate critical points, Allen-Cahn equation, generic result.

2010 Mathematics Subject Classification. 58E05, 35J20.

1. Introduction and main result

Let $(M^{n},g)$ be a connected closed smooth Riemannian manifold, where $n\geq 2$ . Let $W\colon\mathbb{R}\to\mathbb{R}$ be a function of class $C^{2}$ . Fix $\nu,\epsilon>0$ . A pair $(u,\lambda)\in H_{g}(M)\times\mathbb{R}$ is a solution for the Van der Waals-Allen-Cahn Hilliard equation under volume constraint $\nu$ when

(

P_{W,\nu,\epsilon,g}

)

\begin{cases}-\epsilon^{2}\Delta_{g}u+W^{\prime}(u)=\lambda\\ \int_{M}u\mathop{}\!\mathrm{d}\mu_{g}=\nu\end{cases},

where $\mu_{g}$ is a measure induced by $g$ defined on the Borel subsets of $M$ and $H_{g}(M)$ is a convenient Sobolev space of functions defined in section 2.

In [BNAP20], the authors establish lower bounds on the number of solutions for $\eqref{eqn:nonlinear-problem}$ in function of topological invariants of $M$ for sufficiently small $\nu,\epsilon>0$ and under specific hypotheses on the potential function $W$ . In particular: if $\eqref{eqn:nonlinear-problem}$ only admits nondegenerate solutions, then Morse theory may be applied to prove that it admits at least $P_{M}(1)$ solutions, where $P_{M}(t)$ is the Poincaré polynomial of $M$ .

Our main result is that under suitable growth conditions for $W^{\prime}$ and $W^{\prime\prime}$ , this is indeed the case generically with respect to $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ , where $1\leq k<\infty$ and $\mathcal{M}^{k}$ is the space of Riemannian metrics of class $C^{k}$ on $M$ :

Theorem 1.1.

Fix $g_{0}\in\mathcal{M}^{k}$ . Suppose that (1) and (2) hold. Then

\mathcal{D}^{*}_{W,\nu}=\left\{(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}\mathrel{\mathop{\mathchar 58\relax}}\text{ any solution }(u,\lambda)\in H_{g_{0}}(M)\times\mathbb{R}\right.\\ \left.\text{ for }\eqref{eqn:nonlinear-problem}\text{ is nondegenerate}\right\}

is an open dense subset of $]0,\infty[\times\mathcal{M}^{k}$ .

This result is obtained by the application of an abstract transversality theorem through an appropriate adaptation of the techniques in [MP09] and [GM11] to the context of this article.

More precisely, we say that a solution $(u,\lambda)\in H_{g}(M)\times\mathbb{R}$ for $\eqref{eqn:nonlinear-problem}$ is nondegenerate when the only pair $(v,\Lambda)\in H_{g}(M)\times\mathbb{R}$ which solves the linearized problem

(

Q_{W,\epsilon,g,u}

)

\begin{cases}-\epsilon^{2}\Delta_{g}v+W^{\prime\prime}(u)v=\Lambda\\ \int_{M}v\mathop{}\!\mathrm{d}\mu_{g}=0\end{cases}

is the trivial one $(v,\Lambda)=(0,0)$ .

In fact, this notion coincides with the Morse theoretic notion of a nondegenerate critical point for the functional $J_{W,\epsilon,g}\colon H_{g}(M)\times\mathbb{R}\to\mathbb{R}$ given by

J_{W,\epsilon,g}(u,\lambda)=\int_{M}\frac{\epsilon^{2}}{2}g(\nabla u,\nabla u)+W(u)-\lambda u\mathop{}\!\mathrm{d}\mu_{g}-\lambda\nu.

Indeed, $J_{W,\epsilon,g}$ is a functional of class $C^{2}$ for which $(v,\Lambda)$ is a solution for $\eqref{eqn:linearized-problem}$ if, and only if, $\int_{M}v\mathop{}\!\mathrm{d}\mu_{g}=0$ and $(v,\Lambda)\in\ker\mathrm{Hess}(J_{W,\epsilon,g})_{(u,\lambda)}$ . Therefore, $(u,\lambda)$ is a nondegenerate solution for $\eqref{eqn:nonlinear-problem}$ precisely when $(u,\lambda)$ is a nondegenerate critical point of $J_{W,\epsilon,g}$ such that $\int_{M}u\mathop{}\!\mathrm{d}\mu_{g}=\nu$ .

For Differential Geometry, interest for the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint is justified by the results of [PR03], where Pacard and Ritoré showed that one can approach constant mean curvature hypersurfaces by the nodal sets of critical points for $J_{W,\epsilon,g,\lambda}$ as $\epsilon\to 0^{+}$ . If we consider critical points without the volume constraint, these sets approach a minimal hypersurface.

Acknowledgement

The author thanks Paolo Piccione for suggesting the topic and discussing drafts of this article.

2. Preliminaries

Basic constructions

Fix $1\leq k<\infty$ . Denote by $\mathcal{S}^{k}$ the Banach space of symmetric 2-covectors on $M$ of class $C^{k}$ . The space $\mathcal{M}^{k}$ of Riemannian metrics on $M$ of class $C^{k}$ is an open convex cone in $\mathcal{S}^{k}$ .

Consider any $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ . $(\epsilon,g)$ induces the following inner products on $C^{\infty}(M)$ :

\left\langle u,v\right\rangle_{g}=\int_{M}g\left(\nabla u,\nabla v\right)+uv\mathop{}\!\mathrm{d}\mu_{g};

E_{\epsilon,g}(u,v)\mathrel{\mathop{\mathchar 58\relax}}=\int_{M}\epsilon^{2}g\left(\nabla u,\nabla v\right)+uv\mathop{}\!\mathrm{d}\mu_{g}.

$H_{g}(M)$ , $H_{\epsilon,g}(M)$ are, respectively, the Hilbert spaces endowed with $\left\langle\cdot,\cdot\right\rangle_{g}$ , $E_{\epsilon,g}$ obtained as completions of $C^{\infty}(M)$ . Similarly: given $1\leq q<\infty$ , $L^{q}_{g}(M)$ is the Banach space obtained as completion of $C^{\infty}(M)$ with respect to the norm

\mathinner{\!\left\lVert u\right\rVert}_{q,g}\mathrel{\mathop{\mathchar 58\relax}}=\left(\int_{M}\mathinner{\!\left\lvert u\right\rvert}^{q}\mathop{}\!\mathrm{d}\mu_{g}\right)^{1/q}.

One may check that the norms induced by $\left\langle\cdot,\cdot\right\rangle_{g}$ , $E_{\epsilon,g}$ on $C^{\infty}(M)$ are equivalent. In particular, this implies $H_{g}(M)=H_{\epsilon,g}(M)$ as sets and that the canonical inclusion $H_{g}(M)\to H_{\epsilon,g}(M)$ is an isomorphism of Banach spaces. The same holds for the canonical inclusion $H_{g^{\prime}}(M)\to H_{g}(M)$ for any $g^{\prime}\in\mathcal{M}^{k}$ . For details, we refer the reader to [Heb00, Proposition 2.2].

Considered setting

Suppose that

(1)

\exists K_{1}>0\ \forall t\in\mathbb{R},\ \mathinner{\!\left\lvert W^{\prime}(t)\right\rvert}\leq K_{1}(1+\mathinner{\!\left\lvert t\right\rvert}^{p-1});

(2)

\exists K_{2}>0\ \forall t\in\mathbb{R},\ \mathinner{\!\left\lvert W^{\prime\prime}(t)\right\rvert}\leq K_{2}(1+\mathinner{\!\left\lvert t\right\rvert}^{p-2});

for a certain $p\in\left]2,p_{n}\right[$ , where $p_{n}=\infty$ for $n=2$ , $p_{n}=(2n)/(n-2)$ for $n\geq 3$ .

Fix $g_{0}\in\mathcal{M}^{k}$ . Consider any $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ . Due to the Kondrakov theorem, the canonical inclusion $i_{\epsilon,g}\colon H_{\epsilon,g}(M)\to L_{g}^{p}(M)$ is a compact operator. Set $p^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=p/(p-1)$ . We define $A_{\epsilon,g}$ as the adjoint of $i_{\epsilon,g}$ while considering the canonical Banach space isomorphisms $(L^{p}_{g}(M))^{\prime}\simeq L^{p^{\prime}}_{g}(M)$ and $H_{\epsilon,g}(M)\simeq(H_{\epsilon,g}(M))^{\prime}$ :

Definition 2.1.

$A_{\epsilon,g}=i_{\epsilon,g}^{*}\colon L_{g}^{p^{\prime}}(M)\to H_{\epsilon,g}(M).$

Remark 2.2.

$A_{\epsilon,g}$ is a compact self-adjoint operator and $E_{\epsilon,g}\left(A_{\epsilon,g}u,v\right)=\int_{M}uv\mathop{}\!\mathrm{d}\mu_{g}$ for any $u,v\in H_{g}(M)$ .

For details on lemmas 2.3 and 2.4, we refer the reader to[MP09, Lemmas 2.1, 2.3].

Lemma 2.3.

$E\colon]0,\infty[\times\mathcal{M}^{k}\to\mathrm{Bil}\left(H_{g_{0}}(M)\right)$ is a map of class $C^{1}$ , where $E(\epsilon,g)\mathrel{\mathop{\mathchar 58\relax}}=E_{\epsilon,g}$ . In particular,

\mathop{}\!\mathrm{d}E_{(\epsilon,g)}[\eta,h](u,v)=2\epsilon\eta\int_{M}g\left(\nabla u,\nabla v\right)\mathop{}\!\mathrm{d}\mu_{g}+\epsilon^{2}\int_{M}b_{g,h}\left(\nabla u,\nabla v\right)\mathop{}\!\mathrm{d}\mu_{g}+\\ +\frac{1}{2}\int_{M}\left(\mathrm{tr}_{g}h\right)uv\mathop{}\!\mathrm{d}\mu_{g},

where $b_{g,h}$ is a symmetric 2-covector on $M$ of class $C^{k}$ given locally by

\left(b_{g,h}\right)_{ij}=\left(\mathrm{tr}_{g}h\right)g^{ij}/2-g^{iq}h_{ql}g^{lj}.

Lemma 2.4.

$A\colon]0,\infty[\times\mathcal{M}^{k}\to B\left(L_{g_{0}}^{p^{\prime}}(M),H_{g_{0}}(M)\right)$ is a map of class $C^{1}$ , where $A(\epsilon,g)\mathrel{\mathop{\mathchar 58\relax}}=A_{\epsilon,g}$ . In particular,

\mathop{}\!\mathrm{d}E_{(\epsilon,g)}[\eta,h]\left(A_{\epsilon,g}u,v\right)+E_{\epsilon,g}\left(\mathop{}\!\mathrm{d}A_{(\epsilon,g)}[\eta,h]u,v\right)=\frac{1}{2}\int_{M}\left(\mathrm{tr}_{g}h\right)uv\mathop{}\!\mathrm{d}\mu_{g}.

$W^{\prime}\colon\mathbb{R}\to\mathbb{R}$ is a function of class $C^{1}$ with suitable growth conditions, so $H_{g_{0}}(M)\ni u\mapsto W^{\prime}(u)\in L_{g_{0}}^{p^{\prime}}(M)$ is a Nemytskii operator of class $C^{1}$ . For details on this argument, we recommend the reference [Kav93]. This implies:

Lemma 2.5.

The Nemytskii operator $B_{W}\colon H_{g_{0}}(M)\times\mathbb{R}\to L_{g_{0}}^{p^{\prime}}(M)$ given by $B_{W}(u,\lambda)=\lambda+u-W^{\prime}(u)$ is a map of class $C^{1}$ . In particular,

\mathop{}\!\mathrm{d}\left(B_{W}\right)_{(u,\lambda)}[v,\Lambda]=\Lambda+v-vW^{\prime\prime}(u).

In the next definition, we identify the space of constant real-valued functions on $M$ with $\mathbb{R}$ :

Definition 2.6.

Let $F_{W}\colon]0,\infty[\times\mathcal{M}^{k}\times(H_{g_{0}}(M)\setminus\mathbb{R})\times\mathbb{R}\to H_{g_{0}}(M)\times\mathbb{R}$ be given by

F_{W}\left(\epsilon,g,u,\lambda\right)=\left(u-A_{\epsilon,g}\circ B_{W}(u,\lambda),\int_{M}u\mathop{}\!\mathrm{d}\mu_{g}\right).

Using remark 2.2, we can prove that the set of solutions $(u,\lambda)\in(H_{g_{0}}(M)\setminus\mathbb{R})\times\mathbb{R}$ for $\eqref{eqn:nonlinear-problem}$ is a level-set of $F_{W}$ :

Remark 2.7.

$(u,\lambda)\in H_{g_{0}}(M)\times\mathbb{R}$ is a solution for $\eqref{eqn:nonlinear-problem}$ if, and only if, $F_{W}(\epsilon,g,u,\lambda)=(0,\nu)$ .

Lemma 2.8.

$F_{W}\colon]0,\infty[\times\mathcal{M}^{k}\times(H_{g_{0}}(M)\setminus\mathbb{R})\times\mathbb{R}\to H_{g_{0}}(M)\times\mathbb{R}$ is a map of class $C^{1}$ . In particular,

\mathop{}\!\mathrm{d}\left(F_{W}\right)_{\left(\epsilon,g,u,\lambda\right)}[\eta,h,v,\Lambda]=\\ =\left(v-A_{\epsilon,g}\circ\mathop{}\!\mathrm{d}\left(B_{W}\right)_{(u,\lambda)}[v,\Lambda]-\mathop{}\!\mathrm{d}A_{(\epsilon,g)}[\eta,h]\circ B_{W}(u,\lambda),\right.\\ \left.,\int_{M}\frac{1}{2}\left(\mathrm{tr}_{g}h\right)u+v\mathop{}\!\mathrm{d}\mu_{g}\right)

3. Proof of main result

Consider the following abstract transversality theorem:

Theorem 3.1.

[HHL05, Theorem 5.4] Let $X,Y,Z$ be real Banach spaces and $U,V$ be respective open subsets of $X,Y$ . Let $F\colon V\times U\to Z$ be a map of class $C^{m}$ , where $m\geq 1$ . Let $z_{0}\in\mathrm{im}\ F$ . Suppose that

(1)

Given $y\in V$ , $F(y,\cdot)\colon x\mapsto F(x,y)$ is a Fredholm map of index $l<m$ , i.e., $\mathop{}\!\mathrm{d}F(y,\cdot)_{x}\colon X\to Z$ is a Fredholm operator of index $l$ for any $x\in U$ ;
(2)

$z_{0}$ is a regular value of $F$ , i.e., $\mathop{}\!\mathrm{d}F_{\left(y_{0},x_{0}\right)}\colon Y\times X\to Z$ is surjective for any $(y_{0},x_{0})\in F^{-1}(z_{0})$ ;
(3)

Let $\iota\colon F^{-1}(z_{0})\to Y\times X$ be the canonical embedding and $\pi_{Y}\colon Y\times X\to Y$ be the projection of the first coordinate. Then $\pi_{Y}\circ\iota\colon F^{-1}(z_{0})\to Y$ is $\sigma$ -proper, i.e., $F^{-1}(z_{0})=\bigcup_{s=1}^{\infty}C_{s}$ , where given $s=1,2,...$ , $C_{s}$ is a closed subset of $F^{-1}(z_{0})$ and $\left.\pi_{Y}\circ\iota\right|_{C_{s}}$ is proper.

Then the set $\{y\in V\mathrel{\mathop{\mathchar 58\relax}}z_{0}\text{ is a regular value of }F(y,\cdot)\}$ is an open dense subset of $V$ .

The first step to prove our main result is the lemma that follows, in which we restrict ourselves to nonconstant solutions:

Lemma 3.2.

Fix $g_{0}\in\mathcal{M}^{k}$ . Suppose that (1) and (2) hold. Then

\mathcal{D}_{W,\nu}=\left\{(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}\mathrel{\mathop{\mathchar 58\relax}}\text{ any solution }(u,\lambda)\in(H_{g_{0}}(M)\setminus\mathbb{R})\times\mathbb{R}\right.\\ \left.\text{ for }\eqref{eqn:nonlinear-problem}\text{ is nondegenerate}\right\}

is an open dense subset of $]0,\infty[\times\mathcal{M}^{k}$ .

Its proof consists of a direct application of the abstract transversality theorem. Specifically, we consider $X=Z=H_{g_{0}}(M)\times\mathbb{R}$ , $Y=V=]0,\infty[\times\mathcal{S}^{k}$ , $U=(H_{g_{0}}(M)\setminus\mathbb{R})\times\mathbb{R}$ , $F=F_{W}$ and $z_{0}=(0,\nu)$ . We verify that its hypotheses hold in section 4.

After analysing the constant solutions for $\eqref{eqn:nonlinear-problem}$ , we refine lemma 3.2 to prove our main result:

Proof of Theorem 1.1.

Let $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ and $\mathcal{U}$ be a neighborhood of $(\epsilon,g)$ in $]0,\infty[\times\mathcal{M}^{k}$ .

$\mathcal{D}^{*}_{W,\nu}\cap\mathcal{U}$ is not empty. Indeed, let $(\overline{\epsilon},\overline{g})\in\mathcal{D}_{W,\nu}\cap\mathcal{U}$ . If $\left(P_{\nu,\overline{\epsilon},\overline{g},W}\right)$ does not admit constant solutions, then $(\overline{\epsilon},\overline{g})\in\mathcal{D}^{*}_{W,\nu}\cap\mathcal{U}$ . Otherwise, the volume constraint shows that the unique constant solution is $\nu/\mu_{\overline{g}}(M)$ . This is a degenerate solution if, and only if, $\eqref{eqn:linearized-problem}$ admits a nontrivial solution. This only happens when there exists $j=1,2,...$ such that

(3)

\overline{\epsilon}^{2}=-\frac{W^{\prime\prime}(\nu/\mu_{\overline{g}}(M))}{\alpha_{j}(\overline{g})},

where $\mathcal{E}_{\overline{g}}=\mathinner{\left\{\alpha_{j}\left(\overline{g}\right)\mathrel{\mathop{\mathchar 58\relax}}j=1,2,...\right\}}$ is the set of nonzero eigenvalues of $-\Delta_{\overline{g}}$ . $\mathcal{E}_{\overline{g}}$ is a discrete subset of $]0,\infty[$ , so there exists $\hat{\epsilon}>0$ such that $(\hat{\epsilon},\overline{g})\in\mathcal{D}_{W,\nu}\cap\mathcal{U}$ and (3) does not hold for any positive integer $j$ . This implies $(\hat{\epsilon},\overline{g})\in\mathcal{D}^{*}_{W,\nu}$ .

$\mathcal{D}^{*}_{W,\nu}$ is an open subset of $]0,\infty[\times\mathcal{M}^{k}$ . Indeed, let $(\hat{\epsilon},\hat{g})\in\mathcal{D}^{*}_{W,\nu}$ . If $(P_{W,\nu,\hat{\epsilon},\hat{g}})$ does not admit constant solutions, the result is a corollary of lemma 3.2. Otherwise, note that $\mathcal{M}^{k}\ni g\mapsto W^{\prime\prime}(\nu/\mu_{g}(M))\in\mathbb{R}$ and $\mathcal{M}^{k}\ni g\mapsto\alpha_{j}(g)\in\mathbb{R}$ are continuous maps for any positive integer $j$ , so $(\hat{\epsilon},\hat{g})$ admits a neighborhood $\mathcal{V}$ in $]0,\infty[\times\mathcal{M}^{k}$ in which the constant solutions are nondegenerate. To conclude, $\mathcal{V}\cap\mathcal{D}_{W,\nu}$ is a neighborhood of $(\epsilon,g)$ in $]0,\infty[\times\mathcal{M}^{k}$ for which the respective Allen-Cahn equation does not admit degenerate solutions. ∎

4. Technical steps

For a pair $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ , let $F_{W,\epsilon,g}\colon(H_{g_{0}}(M)\setminus\mathbb{R})\times\mathbb{R}\to H_{g_{0}}\times\mathbb{R}$ be given by $F_{W,\epsilon,g}(u,\lambda)=F_{W}(\epsilon,g,u,\lambda).$ We adopt similar notation when fixing other variables.

In lemma 4.2, we shall verify that the first hypothesis of theorem 3.1 holds. With that objective in mind, consider the following preliminary result:

Lemma 4.1.

Let $g\in\mathcal{M}^{k}$ , $C_{g}\colon H_{g_{0}}(M)\to\mathbb{R}$ be given by $C_{g}(v)=\int_{M}v\mathop{}\!\mathrm{d}\mu_{g}$ and $T_{g}\colon H_{g_{0}}(M)\times\mathbb{R}\to H_{g_{0}}(M)\times\mathbb{R}$ be given by $T_{g}(v,\Lambda)=\left(v,C_{g}(v)\right)$ . Then $T_{g}$ is a Fredholm operator of index $0$ .

Proof.

$C_{g}$ is a linear functional, so $\mathrm{codim}\ \ker C_{g}=1$ in $H_{g_{0}}(M)$ . This implies

\mathrm{codim}\ T_{g}(\ker C_{g}\times\mathbb{R})=2

in $H_{g_{0}}(M)\times\mathbb{R}$ .

T_{g}(\ker C_{g}\times\mathbb{R})\cap T_{g}(\mathbb{R}(1,0))=0,

\mathrm{codim}\ [T_{g}(\ker C_{g}\times\mathbb{R})+T_{g}(\mathbb{R}(1,0))]=\mathrm{codim}\ T_{g}(\ker C_{g}\times\mathbb{R})-1=1.

H_{g_{0}}(M)\times\mathbb{R}=(\ker C_{g}\times\mathbb{R})\oplus(\mathbb{R}(1,0)),

so $\mathrm{codim}\ \mathrm{im}\ T_{g}=1.$ $\ker T_{g}=\mathinner{\left\{0\right\}}\times\mathbb{R}$ , so $T_{g}$ is a Fredholm operator of index $0$ . ∎

Lemma 4.2.

Given $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ , $F_{W,\epsilon,g}$ is a Fredholm map of index $0$ .

Proof.

Fix $(u,\lambda)\in H_{g_{0}}(M)\times\mathbb{R}$ and let $K_{W,\epsilon,g,u,\lambda}\colon H_{g_{0}}(M)\times\mathbb{R}\to H_{g_{0}}(M)\times\mathbb{R}$ be given by

K_{W,\epsilon,g,u,\lambda}(v,\Lambda)=\left(A_{\epsilon,g}\circ\mathop{}\!\mathrm{d}\left(B_{W}\right)_{(u,\lambda)}[v,\Lambda],0\right).

$\mathop{}\!\mathrm{d}\left(F_{W,\epsilon,g}\right)_{(u,\lambda)}=T_{g}-K_{W,\epsilon,g,u,\lambda}$ , where $T_{g}$ was defined in lemma 4.1. Therefore, it suffices to prove that $K_{W,\epsilon,g,u,\lambda}$ is a compact operator to conclude that $\mathop{}\!\mathrm{d}\left(F_{W,\epsilon,g}\right)_{(u,\lambda)}$ is a Fredholm operator with index $0$ . This is indeed the case, because $A_{\epsilon,g}$ is a compact operator. ∎

Let us examine the second hypothesis of the abstract transversality theorem. Let $(\epsilon,g,u,\lambda)\in F_{W}^{-1}(0,\nu)$ . To conclude that $\mathop{}\!\mathrm{d}\left(F_{W}\right)_{(\epsilon,g,u,\lambda)}$ is surjective, it suffices to show that

(4)

\left[\mathrm{im}\ \mathop{}\!\mathrm{d}\left(F_{W,\epsilon,g}\right)_{(u,\lambda)}\right]^{\perp}\subset\mathrm{im}\ \mathop{}\!\mathrm{d}\left(F_{W,\epsilon,u,\lambda}\right)_{g},

which we shall prove in lemma 4.4.

The following defines an inner product on $H_{g_{0}}(M)\times\mathbb{R}$ :

\left\langle\left(u_{1},t_{1}\right),\left(u_{2},t_{2}\right)\right\rangle_{\epsilon,g}^{\prime}=E_{\epsilon,g}\left(u_{1},u_{2}\right)+t_{1}t_{2}.

This allows us to establish the characterization:

Remark 4.3.

Let $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ and $(u,\lambda),\ (v,\Lambda)\in H_{g_{0}}(M)\times\mathbb{R}$ . Then

(v,\Lambda)\in\left[\mathrm{im}\ \mathop{}\!\mathrm{d}\left(F_{W,\epsilon,g}\right)_{(u,\lambda)}\right]^{\perp}\text{ if, and only if, }(v,-\Lambda)\text{ is a solution for }\eqref{eqn:linearized-problem}.

We use this characterization to prove inclusion (4):

Lemma 4.4.

Let $(\epsilon,g,u,\lambda)\in F_{W}^{-1}(0,\nu)$ . Let $(v,-\Lambda)\in H_{g_{0}}(M)\times\mathbb{R}$ be a solution for $\eqref{eqn:linearized-problem}$ . If

\left\langle\mathop{}\!\mathrm{d}\left(F_{W,\epsilon,u,\lambda}\right)_{g}[h],(v,\Lambda)\right\rangle_{\epsilon,g}^{\prime}=0

for all $h\in\mathcal{S}^{k}$ , then $(v,\Lambda)=(0,0)$ .

Proof.

Due to lemmas 2.3, 2.4 and 2.8, the equation on the statement is rewritten

(5)

\int_{M}\epsilon^{2}b_{g,h}\left(\nabla u,\nabla v\right)+\frac{\left(\mathrm{tr}_{g}h\right)}{2}\left[\left(W^{\prime}(u)-\lambda\right)v+\Lambda u\right]\mathop{}\!\mathrm{d}\mu_{g}=0,

where we recall that $b_{g,h}$ is a symmetric 2-covector on $M$ of class $C^{k}$ given locally by

\left(b_{g,h}\right)_{ij}=\left(\mathrm{tr}_{g}h\right)g^{ij}/2-g^{iq}h_{ql}g^{lj}.

An argument with normal coordinates centered at arbitrary $x\in M$ which considers specific perturbations of $g$ proves that

(6)

g\left(\nabla u,\nabla v\right)=b_{g,h}\left(\nabla u,\nabla v\right)=0\in L^{1}_{g_{0}}(M)

for any $h\in\mathcal{S}^{k}$ . For details, see [GM11, Lemma 12].

Taking $h=\varphi g$ for arbitrary $\varphi\in C^{\infty}(M)$ shows that (5) and (6) imply

(7)

\left(W^{\prime}(u)-\lambda\right)v+\Lambda u=0\in L^{1}_{g_{0}}(M).

On one hand: integrating the equation above (7) yields

(8)

\int_{M}\left(W^{\prime}(u)-\lambda\right)v\mathop{}\!\mathrm{d}\mu_{g}=-\Lambda\nu.

On the other hand: taking into account (6) and the fact that $u$ is a weak solution for $-\epsilon^{2}\Delta_{g}u+W^{\prime}(u)=\lambda$ ,

\int_{M}W^{\prime}(u)v\mathop{}\!\mathrm{d}\mu_{g}=\int_{M}\lambda v\mathop{}\!\mathrm{d}\mu_{g}.

$\nu\neq 0$ , so the last equation and (8) imply $\Lambda=0$ . Due to (7), $\Lambda=0$ implies

\left(W^{\prime}(u)-\lambda\right)v=0\in L^{1}_{g_{0}}(M).

If $\lambda-W^{\prime}(u)\equiv 0$ , then $u$ is a weak solution for $-\epsilon^{2}\Delta_{g}u=0$ – which only happens with a constant $u$ . We do not consider constant solutions, so $\lambda-W^{\prime}(u)$ does not vanish identically. Due to proposition 4.5, $u,v$ are functions of class $C^{1}$ . Therefore, $\lambda-W^{\prime}(u)$ is a continuous function which does not vanish identically.

In particular, $v$ vanishes in a nonempty open subset of $M$ . In this context, we can use strong unique continuation ([PRS08, Theorem A.5]) in problem $\eqref{eqn:linearized-problem}$ to conclude that $v=0\in H_{g_{0}}(M)$ . ∎

Proposition 4.5.

Fix $(\epsilon,g)\in]0,\infty[\times\mathcal{M}^{k}$ and $\alpha\in]0,1[$ . If $(u,\lambda)\in H_{g_{0}}(M)\times\mathbb{R}$ is a solution for $\eqref{eqn:nonlinear-problem}$ , then $u\in C^{1,\alpha}(M)$ . If it also holds that $(v,\Lambda)\in H_{g_{0}}(M)\times\mathbb{R}$ is a solution for $\eqref{eqn:linearized-problem}$ , then $v\in C^{1,\alpha}(M)$ .

Proof.

Regularity is a local problem, so we fix a coordinate system $\rho\colon\Omega\to\mathbb{R}^{n}$ where the $g^{ij}$ s are bounded and $\Omega$ is the coordinate open subset of $M$ . Let $\tilde{u}\colon\rho(\Omega)\to\mathbb{R}$ be the local expression of $u$ . Note that $\tilde{u}$ is a weak solution for

-\epsilon^{2}\partial_{i}\left(g^{ij}\partial_{j}\tilde{u}\right)+\epsilon^{2}b^{i}\left(\partial_{i}\tilde{u}\right)+W^{\prime}\left(\tilde{u}\right)=\lambda,

where $b^{i}=\partial_{j}\left(g^{ij}\right)+g^{ij}\Gamma_{kj}^{k}$ for any $i=1,...,n$ .

A slight adaptation of [Str10, Lemma B.3] shows that $\tilde{u}\in L^{q}\left(\rho(\Omega)\right)$ for every $q<\infty$ . Arguing as in [Jos13, Theorem 12.2.2], one may show that given $q>1$ , $W^{\prime}\left(\tilde{u}\right)\in L^{q}\left(\rho(\Omega)\right)$ implies $u\in H^{2,q}\left(\rho(\Omega)\right)$ . To conclude, we use the Sobolev Embedding Theorem. ∎

The third hypothesis is proved analogously as[GM11, Lemma 11]:

Lemma 4.6.

$\pi_{Y}\circ\iota\colon F_{W}^{-1}(0,\nu)\to Y$ is $\sigma$ -proper, where $\pi_{Y}$ and $\iota$ are defined in theorem 3.1.

Proof.

Given $s=1,2,...$ ; let

C_{s}=\left([1/s,s]\times\overline{\mathscr{B}_{s}}\times\overline{I(0,s)\setminus B(\mathbb{R},1/s)}\times[-s,s]\right)\cap F_{W}^{-1}(0,\nu),

where $\mathscr{B}_{s},I(0,s)$ are respective open balls in $\mathcal{S}^{k},H_{g_{0}}(M)$ centered at $0$ with radius $s$ and

B(\mathbb{R},1/s)=\mathinner{\left\{u\in H_{g_{0}}(M)\mathrel{\mathop{\mathchar 58\relax}}\inf_{v\in\mathbb{R}}\mathinner{\!\left\lVert u+v\right\rVert}_{H_{g_{0}}}<1/s\right\}}.

Fix a positive integer $s$ . Let us prove that $\left.\pi_{Y}\circ\iota\right|_{C_{s}}$ is a proper map. Let $\mathinner{\left\{\left(\epsilon_{n},g_{n},u_{n},\lambda_{n}\right)\right\}}_{n}\subset C_{s}$ be a sequence such that $\lim_{n}g_{n}=g\in\mathcal{M}^{k}$ , $\lim_{n}\epsilon_{n}=\epsilon\in[1/s,s]$ and given $n$ , $\left(u_{n},\lambda_{n}\right)$ is a solution for $(P_{W,\nu,\epsilon_{n},g_{n}})$ .

We claim that $\left(\left(u_{n},\lambda_{n}\right)\right)_{n}$ has a convergent subsequence. Due to the Kondrakov theorem, the canonical inclusion $i_{\epsilon,g,t}\colon H_{\epsilon,g_{0}}(M)\to L_{g_{0}}^{t}(M)$ is a compact operator for any $t\in\left]2,p_{n}\right[$ , so $\left(u_{n}\right)_{n}$ converges in $L_{g_{0}}^{t}(M)$ up to subsequence to a certain $u\in L_{g_{0}}^{t}(M)$ . $\left(\lambda_{n}\right)_{n}$ is bounded, so it converges up to a subsequence to a certain $\lambda\in[-s,s]$ . Arguing as in lemma 4.2, we see that $\lim_{n}A_{\epsilon,g}\circ B_{W}\left(u_{n},\lambda_{n}\right)=A_{\epsilon,g}\circ B_{W}\left(u,\lambda\right)$ . We can use the Mean Value Inequality and lemma 2.4 to prove that, in fact, $\lim_{n}A_{\epsilon_{n},g_{n}}\circ B_{W}\left(u_{n},\lambda_{n}\right)=A_{\epsilon,g}\circ B_{W}\left(u,\lambda\right)$ . ∎

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