Multi-localized time-symmetric initial data for the Einstein vacuum equations

John Anderson; Justin Corvino; Federico Pasqualotto

doi:10.1515/crelle-2023-0088

Article

Multi-localized time-symmetric initial data for the Einstein vacuum equations

John Anderson , Justin Corvino and Federico Pasqualotto

Published/Copyright: January 31, 2024

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 808

Abstract

We construct a class of time-symmetric initial data sets for the Einstein vacuum equations modeling elementary configurations of multiple almost isolated systems. Each such initial data set consists of a collection of several localized sources of gravitational radiation, and lies in a family of data sets which is closed under scaling out the distances between the systems by arbitrarily large amounts. This class contains data sets which are not asymptotically flat, but to which nonetheless a finite ADM mass can be ascribed. The construction proceeds by a gluing scheme using the Brill–Lindquist metric as a template. Such initial data are motivated in part by a desire to understand the dynamical interaction of distant systems in the context of general relativity. As a by-product of the construction, we produce families of complete, scalar-flat initial data with trivial topology and infinitely many minimal spheres, as well as families of initial data with infinitely many Einstein–Rosen bridges.

Funding source: National Science Foundation

Award Identifier / Grant number: 2103266

Funding statement: John Anderson gratefully acknowledges that this work was partly supported by the National Science Foundation under Grant No. 2103266.

A On the proof of Proposition 2.1

Recall Ω = { x ∈ ℝ 3 : 1 < | x | < 2 } and the definition of the rotationally symmetric bump function ζ above, and we let g ̊ be the Euclidean metric with g ̊ i ⁢ j = δ i ⁢ j in the coordinates x.

We will prove the following extension of Proposition 2.1. To facilitate the statement, we extend some notation: let k 0 = 4 , and for m > 0 , let k m = 1 . We now consider g S m for m ≥ 0 , where we note g S 0 = g ̊ is the Euclidean metric. For m > 0 ,

f m ⁢ ( x ) = 1 - m 2 ⁢ | x | 1 + m 2 ⁢ | x |

spans the one-dimensional kernel of L g S m * , whereas

ker ⁡ ( L g ̊ * ) = span ⁡ { 1 , x 2 , x 2 , x 3 } .

So we let f 0 = 〈 1 , x 1 , x 2 , x 3 〉 , in order that we can express a general element of ker ⁡ ( L g ̊ * ) as b ⁢ f m , with b ∈ ℝ k m : in case m = 0 , b ⁢ f 0 is the dot product.

Remark A.1.

While we only require m ≥ 0 for our applications, for m < 0 , note that

( g S m ) i ⁢ j ⁢ ( x ) = ( 1 + m 2 ⁢ | x | ) 4 ⁢ δ i ⁢ j

is a scalar-flat metric on the region | x | > - m 2 , and

ker ⁡ ( L g S m * ) = span ⁡ { f m }

has dimension k m = 1 . The results in this section are formulated on Ω, and apply equally well to the case m > - 2 .

Proposition A.2.

Let 0 < α < 1 . Let Ω ′ be open and compactly contained in Ω. There is a constant C 0 > 0 such that for smooth metrics γ sufficiently near g S m in C 4 , α ⁢ ( Ω ¯ ) and with scalar curvature R ⁢ ( γ ) supported in Ω ′ ¯ , there exists b ⁢ ( γ ) ∈ R k m , along with a smooth symmetric tensor h = h ⁢ ( γ ) on R 3 which vanishes outside Ω, so that γ + h ⁢ ( γ ) is a metric on Ω ¯ with R ⁢ ( γ + h ⁢ ( γ ) ) = b ⁢ ( γ ) ⁢ ζ ⁢ f m . Moreover, ( h ⁢ ( γ ) , b ⁢ ( γ ) ) ∈ C 2 , α ⁢ ( Ω ¯ ) × R k m depends continuously on γ ∈ C 4 , α ⁢ ( Ω ¯ ) , with ∥ ( h ⁢ ( γ 1 ) , b ⁢ ( γ 1 ) ) - ( h ⁢ ( γ 2 ) , b ⁢ ( γ 2 ) ) ∥ C 2 , α ⁢ ( Ω ) × R k m ≤ C 0 ⁢ ∥ γ 1 - γ 2 ∥ C 4 , α ⁢ ( Ω ) and with ∥ h ⁢ ( γ ) ∥ C 2 , α ⁢ ( Ω ) ≤ C 0 ⁢ ∥ R ⁢ ( γ ) ∥ C 0 , α ⁢ ( Ω ) . Furthermore, for each k ∈ Z + there is a C k > 0 such that ∥ h ⁢ ( γ ) ∥ C k + 2 , α ⁢ ( Ω ) ≤ C k ⁢ ∥ R ⁢ ( γ ) ∥ C k , α ⁢ ( Ω ) .

In case γ is parity-symmetric, h ⁢ ( γ ) is parity-symmetric as well.

We briefly recall the strategy for proving this result, which is then carried out in the rest of the Appendix (recall Remark 2.2).

Our goal is to solve for a metric with vanishing scalar curvature modulo the kernel of the adjoint of the linearized scalar curvature operator around a given metric. This will be done by iteration, repeatedly solving the equation linearized around a fixed metric (see Section A.6). In order to do this, we must effectively solve the linearized equation. This is carried out in Section A.5 using Hilbert space methods. However, we want to solve the linearized equation in such a way that this procedure does not change the metric in the exterior of Ω. Having nontrivial compact solutions of this kind is not possible for elliptic equations in general, but in this case, we can construct such solutions by leveraging the underdetermined nature of the equations. From a technical viewpoint, there is an elliptic estimate which does not contain boundary terms (see Section A.1), and this estimate allows us to work in spaces which have weights that vanish, dual to spaces with weights that blow up, at all orders as we approach the boundary of Ω (see Sections A.2 through A.4). Working in these spaces naturally localizes the gluing to Ω, as desired.

A.1 Basic injectivity estimates

We prove two basic injectivity estimates for L g * for metrics g near g S m , on suitable spaces transverse to ker ⁡ ( L g S m * ) on Ω. For ε ∈ [ 0 , 1 ) , we let Ω ε = { x ∈ ℝ 3 : 1 + ε < | x | < 2 - ε } .

Lemma A.3.

Let S ⊂ L 2 ⁢ ( Ω , g S m ) be a subspace such that

L 2 ⁢ ( Ω , g S m ) = S ⊕ ker ⁡ ( L g S m * ) .

There is a C 2 ⁢ ( Ω ¯ ) -neighborhood U of g S m and a C > 0 so that for all g ∈ U and for all u ∈ H 2 ⁢ ( Ω , g ) ∩ S ,

(A.1) ∥ u ∥ H 2 ⁢ ( Ω , g ) ≤ C ⁢ ∥ L g * ⁢ u ∥ L 2 ⁢ ( Ω , g ) .

Proof.

As tr g ⁢ ( L g * ⁢ u ) = - ( n - 1 ) ⁢ Δ g ⁢ u - u ⁢ R ⁢ ( g ) , we see immediately that there is a C > 0 and a C 2 ⁢ ( Ω ¯ ) -neighborhood 𝒰 0 of g S m so that for all g ∈ 𝒰 0 , for all u ∈ H 2 ⁢ ( Ω , g ) , and for all ε ∈ [ 0 , 1 ) ,

∥ Hess g u ∥ L 2 ⁢ ( Ω ε , g ) ≤ C ( ∥ L g * u ∥ L 2 ⁢ ( Ω ε , g ) + ∥ u ∥ L 2 ⁢ ( Ω ε , g ) ) .

We can choose such a C ≥ 1 , and hence

(A.2) ∥ u ∥ H 2 ⁢ ( Ω ε , g ) ≤ C ⁢ ( ∥ L g * ⁢ u ∥ L 2 ⁢ ( Ω ε , g ) + ∥ u ∥ H 1 ⁢ ( Ω ε , g ) ) .

If estimate (A.1) fails, there is a sequence g i ∈ 𝒰 0 , g i → g S m in C 2 ⁢ ( Ω ¯ ) , and a sequence u i ∈ H 2 ⁢ ( Ω , g i ) ∩ S such that ∥ u i ∥ H 2 ⁢ ( Ω , g i ) = 1 but ∥ L g i * ⁢ u i ∥ L 2 ⁢ ( Ω , g i ) → 0 . Thus there are 0 < β 1 < 1 < β 2 such that β 1 ≤ ∥ u i ∥ H 2 ⁢ ( Ω , g S m ) ≤ β 2 ; moreover, ∥ L g S m * ⁢ u i ∥ L 2 ⁢ ( Ω , g S m ) → 0 . By the Rellich Lemma and (A.2) on Ω with g = g S m , applied to differences u i - u j , we have that u i converges to some u ∈ H 2 ⁢ ( Ω , g S m ) , with ∥ u ∥ H 2 ⁢ ( Ω , g S m ) ≥ β , and L g S m * ⁢ u = 0 . Moreover, u ∈ S as well, since S is closed. Thus we have a contradiction. ∎

Remark A.4.

If m ~ is sufficiently close to m, then g S m ~ ∈ 𝒰 . We remark that for any u ∈ L 2 ⁢ ( Ω , g ) , u ∈ H 2 ⁢ ( Ω , g ) if and only if L g * ⁢ u ∈ L 2 ⁢ ( Ω , g ) . Furthermore, f ∈ H 2 ⁢ ( Ω , g ) if and only if f ∈ H 2 ⁢ ( Ω , g S m ) , and moreover one gets an equivalent estimate if one changes the metric used for the H 2 and L 2 norms.

Remark A.5.

For c ∈ ℝ 3 , let φ c ⁢ ( x ) = x - c , and c + Ω = { x : 1 < | x - c | < 2 } , so that φ c : ( c + Ω ) → Ω . Then g S m , c = φ c * ⁢ ( g S m ) , i.e.

( g S m , c ) i ⁢ j ⁢ ( x ) = ( 1 + m 2 ⁢ | x - c | ) 4 ⁢ δ i ⁢ j ,

and for m > 0 , the kernel of L g S m , c * is spanned by f m , c , where

f m , c ⁢ ( x ) = f m ∘ φ c ⁢ ( x ) = f m ⁢ ( x - c ) ,

and similarly for m = 0 . Furthermore, let S c and 𝒰 c be the pullbacks of S and 𝒰 under φ c ; note g S m , c ∈ 𝒰 c and L 2 ⁢ ( c + Ω , g S m , c ) = S c ⊕ span ⁡ ( f m , c ) . Then (A.1) holds for all g ∈ 𝒰 c and u ∈ H 2 ⁢ ( c + Ω , g ) ∩ S c .

Given 𝔪 = ( m k ) k ∈ ℤ + with m k > 0 and suitably chosen 𝔠 , if we now let 𝒰 ~ k be the corresponding neighborhoods for m = m ~ k on Ω, the calculations in Section 3 show that if for each k, m ~ k is close enough to m k , g BL 𝔪 , 𝔠 ∈ ⋂ k ∈ ℤ + 𝒰 ~ c k k . For the case of finitely many points, it suffices that 𝔪 ~ is close to 𝔪 and | c k - c ℓ | ≥ 5 for k ≠ ℓ .

Lemma A.6.

Let γ be a metric in C 0 ⁢ ( Ω ¯ ) . Let S γ be the L 2 ⁢ ( Ω , γ ) -orthogonal complement of ( ζ ⁢ ker ⁡ ( L g S m * ) ) . There is a C 2 ⁢ ( Ω ¯ ) -neighborhood U of g S m and a C > 0 so that for all ε ∈ [ 0 , 1 8 ] , for all g ∈ U and for all u ∈ H 2 ⁢ ( Ω , g ) ∩ S γ , we have

(A.3) ∥ u ∥ H 2 ⁢ ( Ω ε , g ) ≤ C ⁢ ∥ L g * ⁢ u ∥ L 2 ⁢ ( Ω ε , g ) .

Proof.

We first recall (cf. [27, Chapter 7]) that there is a constant D > 0 and for each ε ∈ [ 0 , 1 8 ] an extension operator E ε : H 2 ⁢ ( Ω ε , g ̊ ) → H 2 ⁢ ( Ω , g ̊ ) such that for all u ∈ H 2 ⁢ ( Ω ε , g ̊ ) ,

(A.4) ∥ E ε ⁢ ( u ) ∥ H 2 ⁢ ( Ω , g ̊ ) ≤ D ⁢ ∥ u ∥ H 2 ⁢ ( Ω ε , g ̊ ) .

We conclude that we can choose D > 0 and a neighborhood 𝒰 0 of g S m so that the extension operator satisfies the estimate (A.4) with g ̊ replaced by g ∈ 𝒰 0 .

The preceding lemma handles the case ε = 0 , So if the claim fails, there is a sequence g i → g S m in C 2 ⁢ ( Ω ¯ ) and ε i ∈ ( 0 , 1 8 ] as well as u i ∈ H 2 ⁢ ( Ω , g i ) ∩ S γ such that

∥ u i ∥ H 2 ⁢ ( Ω ε i , g i ) > i ⁢ ∥ L g i * ⁢ u i ∥ L 2 ⁢ ( Ω ε i , g i ) .

Let u ε i = E ε i ( ( u i ) ↾ Ω ε i ) . We can rescale to arrange 1 = ∥ u ε i ∥ H 2 ⁢ ( Ω , g i ) , and so it follows that ∥ L g i * ⁢ u i ∥ L 2 ⁢ ( Ω ε i , g i ) → 0 . There are constants 0 < β 1 < β 2 with β 1 ≤ ∥ u ε i ∥ H 2 ⁢ ( Ω , g S m ) ≤ β 2 . In particular, then ∥ u i ∥ H 2 ⁢ ( Ω ε i , g S m ) ≤ β 2 , and it follows that ∥ L g S m * ⁢ u i ∥ L 2 ⁢ ( Ω ε i , g S m ) → 0 .

On the other hand by the estimate (A.4) of the extension operator along with (A.2), we have

∥ u ε i ∥ H 2 ⁢ ( Ω , g S m ) ≤ D ⁢ ∥ u i ∥ H 2 ⁢ ( Ω ε i , g S m )

≤ D ⁢ C ⁢ ( ∥ L g S m * ⁢ u i ∥ L 2 ⁢ ( Ω ε i , g S m ) + ∥ u i ∥ H 1 ⁢ ( Ω ε i , g S m ) ) .

By applying Banach–Alaoglu, and Rellich’s Lemma, we then conclude that u ε i converges H 2 ⁢ ( Ω , g S m ) -weakly to some u ∈ H 2 ⁢ ( Ω , g S m ) , and strongly to u in H 1 ⁢ ( Ω , g S m ) . By the preceding estimate, u ε i converges to u strongly in H 2 ⁢ ( Ω , g S m ) , and so we conclude

∥ u ∥ H 2 ⁢ ( Ω , g S m ) ≥ β > 0 .

Furthermore, L g S m * ⁢ u = 0 , so u ∈ span ⁡ { f m } .

Now, by assumption, for all i, and for all f ∈ ker ⁡ ( L g S m * ) , ∫ Ω u i ⁢ ζ ⁢ f ⁢ 𝑑 μ γ = 0 . Thus since ζ vanishes outside Ω ε i , we conclude u ε i ∈ S γ . Since S γ is closed in L 2 ⁢ ( Ω , γ ) , we have by the L 2 -convergence that u ∈ S γ . Since we also have u ∈ ker ⁡ ( L g S m * ) , we conclude u = 0 , which is a contradiction. ∎

A.2 Weighted function spaces

We will use several weighted function spaces in the analysis, which we recall here.

We first define a weight function ρ on Ω. Let ρ ̊ ⁢ ( t ) = e - 1 t for 0 < t ≤ 1 16 , and ρ ̊ ⁢ ( t ) = 1 for t ≥ 1 8 , with ρ ̊ a smooth nondecreasing function supported on t ≥ 0 . For x ∈ Ω , let d ⁢ ( x ) be the Euclidean distance to the boundary ∂ ⁡ Ω , i.e. d ⁢ ( x ) = min ⁡ ( | x | - 1 , 2 - | x | ) . For x ∈ Ω , let ρ ⁢ ( x ) = ρ ̊ ⁢ ( d ⁢ ( x ) ) . If ∇ is the Euclidean connection, then for d ⁢ ( x ) < 1 16 ,

∇ ⁡ ρ ⁢ ( x ) = ( d ⁢ ( x ) ) - 2 ⁢ ρ ⁢ ( x ) ⁢ ∇ ⁡ d ⁢ ( x ) .

By induction, we have that for k ∈ ℤ + , there is C so that on Ω, | ∇ k ⁡ ρ | ≤ C ⁢ d - 2 ⁢ k ⁢ ρ .

We let 0 < ϕ < d ≤ 1 be a smooth function on Ω, which near the boundary ∂ ⁡ Ω (say for d ⁢ ( x ) < 1 8 ) satisfies ϕ ⁢ ( x ) = ( d ⁢ ( x ) ) 2 . Note that for all x ∈ Ω , the Euclidean ball B ϕ ⁢ ( x ) ⁢ ( x ) ¯ ⊂ Ω . We note then that for each k ∈ ℤ + , there is a C so that | ϕ k ⁢ ρ - 1 ⁢ ∇ k ⁡ ρ | ≤ C . Moreover, there is a C > 1 so that for all x ∈ Ω and y ∈ B ϕ ⁢ ( x ) ⁢ ( x ) , both ϕ ⁢ ( x ) / ϕ ⁢ ( y ) and ρ ⁢ ( x ) / ρ ⁢ ( y ) lie in the interval [ C - 1 , C ] , since for y ∈ B ϕ ⁢ ( x ) ⁢ ( x ) , it follows that d ⁢ ( x ) - ϕ ⁢ ( x ) ≤ d ⁢ ( y ) ≤ d ⁢ ( x ) + ϕ ⁢ ( x ) .

Given k ∈ ℤ + and a metric g on Ω ¯ , we define the weighted Sobolev space H ρ k ⁢ ( Ω , g ) to be the space of functions (or sections of a tensor bundle) u so that | ∇ g j ⁡ u | g ∈ L 2 ⁢ ( Ω , ρ ⁢ d ⁢ μ g ) for all 0 ≤ j ≤ k , with

∥ u ∥ H ρ k ⁢ ( Ω , g ) 2 = ∑ j ≤ k ∫ Ω | ∇ g j ⁡ u | g 2 ⁢ ρ ⁢ 𝑑 μ g .

We extend this to k = 0 , and let L ρ 2 ⁢ ( Ω , g ) = H ρ 0 ⁢ ( Ω , g ) . Note that H ρ k ⁢ ( Ω , g ) has a natural Hilbert space structure. For economy of notation we will sometimes suppress the metric: H ρ k ⁢ ( Ω ) := H ρ k ⁢ ( Ω , g ) . We assume g is smooth, or at least C k ⁢ ( Ω ¯ ) , to define ∇ g j ⁡ u for a tensor field, with j ≤ k . We can interpret ∇ g j ⁡ u weakly, and we recall that by [22, Lemma 2.1], C ∞ ⁢ ( Ω ¯ ) is dense in H ρ k ⁢ ( Ω ) . We also note that we get an equivalent norm if we use the background Euclidean metric g ̊ .

For r , s ∈ ℝ , let φ = ϕ r ⁢ ρ s . For α ∈ ( 0 , 1 ] and k a nonnegative integer, define the weighted Hölder space C ϕ , φ k , α ⁢ ( Ω ) as the space of all u ∈ C loc k , α ⁢ ( Ω ) for which

∥ u ∥ C ϕ , φ k , α ⁢ ( Ω ) := sup x ∈ Ω ( ∑ j = 0 k φ ( x ) ϕ j ( x ) ∥ ∇ j u ∥ C 0 ⁢ ( B ϕ ⁢ ( x ) ⁢ ( x ) )

+ φ ( x ) ϕ k + α ( x ) [ ∇ k u ] 0 , α ; B ϕ ⁢ ( x ) ⁢ ( x ) )

is finite. Note that C ϕ , φ k , α ⁢ ( Ω ) is a Banach space. We would obtain an equivalent norm by using the connection ∇ g in place of the Euclidean connection, as in [14, Appendix A].

We will also make use of the following spaces:

ℬ 0 ⁢ ( Ω ) = C ϕ , ϕ 4 + 3 2 ⁢ ρ - 1 2 0 , α ⁢ ( Ω ) ∩ L ρ - 1 2 ⁢ ( Ω ) , ℬ 2 ⁢ ( Ω ) = C ϕ , ϕ 2 + 3 2 ⁢ ρ - 1 2 2 , α ⁢ ( Ω ) ∩ L ρ - 1 2 ⁢ ( Ω ) ,

while

ℬ 4 ⁢ ( Ω ) = C ϕ , ϕ 3 2 ⁢ ρ 1 2 4 , α ⁢ ( Ω ) ∩ H ρ 2 ⁢ ( Ω ) .

The norms on these spaces are defined by summing the relevant weighted Sobolev and Hölder norms, e.g.,

∥ u ∥ ℬ 0 ⁢ ( Ω ) := ∥ u ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ - 1 2 0 , α ⁢ ( Ω ) + ∥ u ∥ L ρ - 1 2 ⁢ ( Ω ) .

As remarked above, changing the metric gives an equivalent norm, so we often suppress the metric.

A.3 The basic weighted injectivity estimate

Recall that Ω = { x ∈ ℝ 3 : 1 < | x | < 2 } . We obtain a basic weighted coercivity estimate for L g * for g near g S m , in a suitable space transverse to ker ⁡ ( L g S m * ) on Ω.

Proposition A.7.

Let γ be a metric in C 0 ⁢ ( Ω ¯ ) . Let S γ be the L 2 ⁢ ( Ω , γ ) -orthogonal complement of ( ζ ⁢ ker ⁡ ( L g S m * ) ) . There is a C 2 ⁢ ( Ω ¯ ) -neighborhood U of g S m and a C > 0 so that for all g ∈ U and for all u ∈ H ρ 2 ⁢ ( Ω , g ) ∩ S γ , we have the following:

(A.5) ∥ u ∥ H ρ 2 ⁢ ( Ω , g ) ≤ C ⁢ ∥ L g * ⁢ u ∥ L ρ 2 ⁢ ( Ω , g ) .

Proof.

Let ε ̊ = 1 8 and let g ∈ 𝒰 , where 𝒰 is as in Lemma A.6 . If suffices by density to establish the estimate for u ∈ H 2 ⁢ ( Ω , g ) ∩ S γ . Using the monotonicity of ρ ̊ , we have by (A.3) that

∫ 0 ε ̊ ρ ̊ ′ ⁢ ( ε ) ⁢ ∥ u ∥ H 2 ⁢ ( Ω ε , g ) 2 ⁢ 𝑑 ε ≤ C 2 ⁢ ∫ 0 ε ̊ ρ ̊ ′ ⁢ ( ε ) ⁢ ∥ L g * ⁢ u ∥ L 2 ⁢ ( Ω ε , g ) 2 ⁢ 𝑑 ε .

Applying the co-area formula

∫ Ω ∖ Ω ε ̊ f ⁢ 𝑑 μ g = ∫ 0 ε ̊ ∫ { x : d ⁢ ( x ) = ε } f ⁢ | ∇ g ⁡ d | g - 1 ⁢ 𝑑 σ g ⁢ 𝑑 ε

and integrating by parts, we have

ρ ̊ ⁢ ( ε ̊ ) ⁢ ∥ u ∥ H 2 ⁢ ( Ω ε ̊ , g ) 2 + ∫ 0 ε ̊ ρ ̊ ⁢ ( ε ) ⁢ ∫ { x : d ⁢ ( x ) = ε } ∑ j ≤ 2 | ∇ g j ⁡ u | g 2 ⁢ | ∇ g ⁡ d | g - 1 ⁢ d ⁢ σ g ⁢ d ⁢ ε

≤ C 2 ⁢ ( ρ ̊ ⁢ ( ε ̊ ) ⁢ ∥ L g * ⁢ u ∥ L 2 ⁢ ( Ω ε ̊ , g ) 2 + ∫ 0 ε ̊ ρ ̊ ⁢ ( ε ) ⁢ ∫ { x : d ⁢ ( x ) = ε } | L g * ⁢ u | g 2 ⁢ | ∇ g ⁡ d | g - 1 ⁢ 𝑑 σ g ⁢ 𝑑 ε ) ,

i.e.

ρ ̊ ⁢ ( ε ̊ ) ⁢ ∥ u ∥ H 2 ⁢ ( Ω ε ̊ , g ) 2 + ∥ u ∥ H ρ 2 ⁢ ( Ω ∖ Ω ε ̊ , g ) 2 = ρ ̊ ⁢ ( ε ̊ ) ⁢ ∥ u ∥ H 2 ⁢ ( Ω ε ̊ , g ) 2 + ∫ Ω ∖ Ω ε ̊ ∑ j ≤ 2 | ∇ g j ⁡ u | g 2 ⁢ ρ ⁢ d ⁢ μ g

≤ C 2 ⁢ ( ρ ̊ ⁢ ( ε ̊ ) ⁢ ∥ L g * ⁢ u ∥ L 2 ⁢ ( Ω ε ̊ , g ) 2 + ∫ Ω ∖ Ω ε ̊ | L g * ⁢ u | g 2 ⁢ ρ ⁢ 𝑑 μ g )

≤ C 2 ⁢ ∥ L g * ⁢ u ∥ L ρ 2 ⁢ ( Ω , g ) 2 .

Since 0 < ρ ̊ ≤ 1 , ∥ u ∥ H ρ 2 ⁢ ( Ω ε ̊ , g ) 2 ≤ ∥ u ∥ H 2 ⁢ ( Ω ε ̊ , g ) 2 , and we can conclude

∥ u ∥ H ρ 2 ⁢ ( Ω , g ) 2 ≤ C 2 ⁢ ( ρ ̊ ⁢ ( ε ̊ ) ) - 1 ⁢ ∥ L g * ⁢ u ∥ L ρ 2 ⁢ ( Ω , g ) 2 . ∎

Remark A.8.

As remarked earlier, one could take a background metric such as g ̊ to define the norms and obtain an analogous estimate.

A.4 Weighted Schauder estimates

We record here a basic elliptic estimate in the weighted Hölder spaces. The proof of the relevant estimate is a fairly straightforward scaling argument using the interior Schauder estimates, and can be found in [20, Appendix A], following [14, Appendix B], where more general operators are treated, cf. [21, Appendix C]. We suppose ϕ and φ are defined as above.

Proposition A.9.

Let k be a nonnegative integer, 0 < α < 1 , and let g ∈ C k + 3 , α ⁢ ( Ω ¯ ) be a Riemannian metric. Suppose

P = Δ g 2 + ∑ | β | ≤ 3 b β ⁢ ∂ x β

on Ω, with b β ∈ C ϕ , ϕ 4 - | β | k , α ⁢ ( Ω ) . There is a constant C so that for all u ∈ C ϕ , φ k + 4 , α ⁢ ( Ω ) ,

(A.6) ∥ u ∥ C ϕ , φ k + 4 , α ⁢ ( Ω ) ≤ C ⁢ ( ∥ P ⁢ u ∥ C ϕ , ϕ 4 ⁢ φ k , α ⁢ ( Ω ) + ∥ u ∥ L ϕ - 3 ⁢ φ 2 2 ⁢ ( Ω ) ) .

Moreover, if u ∈ L ϕ - 3 ⁢ φ 2 2 ⁢ ( Ω ) and P ⁢ u ∈ C ϕ , ϕ 4 ⁢ φ k , α ⁢ ( Ω ) , then u ∈ C ϕ , φ k + 4 , α ⁢ ( Ω ) and the above estimate holds.

Remark A.10.

For g ∈ C k + 4 , α ⁢ ( Ω ¯ ) , the operator

P = 1 2 ⁢ ρ - 1 ⁢ L g ⁢ ρ ⁢ L g *

has the above form. The proposition also applies if we replace Δ g 2 with a fourth-order elliptic operator with coefficients in C k , α ⁢ ( Ω ¯ ) .

A.5 The linearized equation

We let ζ be as above, and let S g be the L 2 ⁢ ( Ω , g ) -orthogonal complement of ( ζ ⁢ ker ⁡ ( L g S m * ) ) , and let Π g be the corresponding orthogonal projection onto S g ; let

S ̊ = S g S m and Π ̊ = Π g S m .

We will establish local deformations for the nonlinear operator g ↦ Π ̊ ∘ R ⁢ ( g ) for g near g S m . The linearization of this operator is Π ̊ ∘ L g , for which we want to establish surjectivity in appropriate spaces. This will follow from the analogous result for the related operator Π g ∘ L g .

Let ψ ⊥ = ψ ⊥ g = ψ - Π g ⁢ ( ψ ) ∈ ( ζ ⁢ ker ⁡ ( L g S m * ) ) . Since ζ is supported in the set where ρ = 1 , we have that since Π g ⁢ ( ψ ) and ψ ⊥ are orthogonal in L 2 ⁢ ( Ω , g ) , they are also orthogonal in L ρ - 1 2 ⁢ ( Ω , g ) , hence

∥ ψ ∥ L ρ - 1 2 ⁢ ( Ω , g ) 2 = ∥ Π g ⁢ ( ψ ) ∥ L ρ - 1 2 ⁢ ( Ω , g ) 2 + ∥ ψ ⊥ ∥ L ρ - 1 2 ⁢ ( Ω , g ) 2 .

We also note that there is a constant C > 0 so that for all metrics g near g S m in C 0 ⁢ ( Ω ¯ ) , and all ψ ∈ ℬ 0 ⁢ ( Ω ) (using the fact that ψ ⊥ lies in a finite-dimensional space),

∥ Π g ⁢ ( ψ ) ∥ ℬ 0 ⁢ ( Ω , g ) ≤ ∥ ψ ∥ ℬ 0 ⁢ ( Ω , g ) + ∥ ψ ⊥ ∥ ℬ 0 ⁢ ( Ω , g )

≤ ∥ ψ ∥ ℬ 0 ⁢ ( Ω , g ) + C ⁢ ∥ ψ ⊥ ∥ L ρ - 1 2 ⁢ ( Ω , g )

≤ ∥ ψ ∥ ℬ 0 ⁢ ( Ω , g ) + C ⁢ ∥ ψ ∥ L ρ - 1 2 ⁢ ( Ω , g )

≤ ( 1 + C ) ⁢ ∥ ψ ∥ ℬ 0 ⁢ ( Ω , g ) .

We also note that applying Π g to ψ - Π ̊ ⁢ ( ψ ) we conclude

∥ Π g ⁢ ( ψ ) ∥ L ρ - 1 2 ⁢ ( Ω , g ) = ∥ Π g ⁢ ( Π ̊ ⁢ ( ψ ) ) ∥ L ρ - 1 2 ⁢ ( Ω , g )

≤ ∥ Π ̊ ⁢ ( ψ ) ∥ L ρ - 1 2 ⁢ ( Ω , g ) ≤ C ⁢ ∥ Π ̊ ⁢ ( ψ ) ∥ L ρ - 1 ⁢ ( Ω , g S m ) 2 ,

where C is uniform for g near g S m in C 0 ⁢ ( Ω ¯ ) .

If u ∈ C c ∞ ⁢ ( Ω ) , then in the L 2 ⁢ ( Ω , g ) -orthogonal decomposition u = u 0 + u 1 , where u 0 ∈ ( ζ ⁢ ker ⁡ ( L g S m * ) ) and u 1 ∈ S g , we have u 1 ∈ C c ∞ ⁢ ( Ω ) as well. To say that

Π g ⁢ ( L g ⁢ ( ρ ⁢ L g * ⁢ f ) ) = Π g ⁢ ( ψ )

weakly, i.e. L g ⁢ ( ρ ⁢ L g * ⁢ f ) - ψ ∈ ( ζ ⁢ ker ⁡ ( L g S m * ) ) weakly, can then be interpreted as

∫ Ω ( 〈 ρ ⁢ L g * ⁢ u , L g * ⁢ f 〉 g - ψ ⁢ u ) ⁢ 𝑑 μ g = 0

for all u ∈ C c ∞ ⁢ ( Ω ) ∩ S g .

Proposition A.11.

There is a C 2 ⁢ ( Ω ¯ ) -neighborhood U of g S m along with a constant C 1 > 0 so that for all g ∈ U and for ψ ∈ L ρ - 1 2 ⁢ ( Ω ) , there is a unique f ∈ H ρ 2 ⁢ ( Ω , g ) ∩ S g that weakly solves

(A.7) Π g ⁢ ( L g ⁢ ( ρ ⁢ L g * ⁢ f ) ) = Π g ⁢ ( ψ ) ,

along with the estimate

(A.8) ∥ f ∥ H ρ 2 ⁢ ( Ω , g ) ≤ C 1 ⁢ ∥ Π g ⁢ ( ψ ) ∥ L ρ - 1 2 ⁢ ( Ω , g ) .

For such f we have Π ̊ ⁢ ( L g ⁢ ( ρ ⁢ L g * ⁢ f ) ) = Π ̊ ⁢ ( ψ ) , with ∥ f ∥ H ρ 2 ⁢ ( Ω , g ) ≤ C 1 ⁢ ∥ Π ̊ ⁢ ( ψ ) ∥ L ρ - 1 2 ⁢ ( Ω , g ) .

Proof.

Given ψ ∈ L ρ - 1 2 ⁢ ( Ω , g ) , let 𝒢 ⁢ ( u ) be defined for u ∈ H ρ 2 ⁢ ( Ω , g ) ∩ S g by

𝒢 ⁢ ( u ) = ∫ Ω ( 1 2 ⁢ ρ ⁢ | L g * ⁢ u | g 2 - Π g ⁢ ( ψ ) ⁢ u ) ⁢ 𝑑 μ g = ∫ Ω ( 1 2 ⁢ ρ ⁢ | L g * ⁢ u | g 2 - ψ ⁢ u ) ⁢ 𝑑 μ g .

By (A.5), we have

∫ Ω ( 1 2 ⁢ | L g * ⁢ u | g 2 - Π g ⁢ ( ψ ) ⁢ u ) ⁢ 𝑑 μ g ≥ C ⁢ ∥ u ∥ H ρ 2 ⁢ ( Ω , g ) 2 - ∥ Π g ⁢ ( ψ ) ∥ L ρ - 1 2 ⁢ ( Ω , g ) ⋅ ∥ u ∥ L ρ 2 ⁢ ( Ω , g ) .

As ζ has compact support, it follows that S g is closed in H ρ 2 ⁢ ( Ω , g ) ∩ S g . A standard argument using Banach–Alaoglu and Riesz Representation for Hilbert spaces as in [17, pp. 150–152] provides a minimizer f ∈ H ρ 2 ⁢ ( Ω , g ) ∩ S g . Such a minimizer is unique by the strict convexity: if u 1 ≠ u 2 are two elements in H ρ 2 ⁢ ( Ω , g ) ∩ S g , then for 0 < s < 1 ,

𝒢 ⁢ ( s ⁢ u 1 + ( 1 - s ) ⁢ u 2 ) > s ⁢ 𝒢 ⁢ ( u 1 ) + ( 1 - s ) ⁢ 𝒢 ⁢ ( u 2 ) ,

by an easy estimate using the arithmetic-geometric mean inequality, and the injectivity of L g * on H ρ 2 ⁢ ( Ω , g ) ∩ S g . Moreover, since for the minimizer 𝒢 ⁢ ( f ) ≤ 0 , the desired estimate holds.

We now consider the Euler–Lagrange equations. For u ∈ S g ∩ C c ∞ ⁢ ( Ω ) ,

0 = d d ⁢ t | t = 0 ⁢ 𝒢 ⁢ ( f + t ⁢ u )

= ∫ Ω ( 〈 ρ ⁢ L g * ⁢ u , L g * ⁢ f 〉 g - Π g ⁢ ( ψ ) ⁢ u ) ⁢ 𝑑 μ g

= ∫ Ω ( 〈 ρ ⁢ L g * ⁢ u , L g * ⁢ f 〉 g - ψ ⁢ u ) ⁢ 𝑑 μ g . ∎

Remark A.12.

Let T be the parity map T ⁢ ( x ) = - x . If g and ψ are parity-symmetric (i.e. T * ⁢ g = g and ψ = ψ ∘ T on Ω), then the minimizer f inherits the symmetry, since the minimizer is unique and 𝒢 ⁢ ( f ) = 𝒢 ⁢ ( f ∘ T ) in this case.

Since the principal part of L g ⁢ L g * is 2 ⁢ Δ g 2 , elliptic regularity immediately gives us that f in the preceding is in H loc 4 ⁢ ( Ω ) , and for smooth g, if ψ is smooth, then f is smooth in Ω too. For suitable ψ we want to show that ρ ⁢ L g * ⁢ f extends smoothly by zero over the boundary of Ω. For that we have the following proposition. We suppress the metric notation in the norms, or for definiteness, we could just use the connection ∇ for g ̊ and the measure d ⁢ μ g ̊ in defining the norms (similarly for any fixed (sufficiently) smooth metric).

Proposition A.13.

Let 0 < α < 1 . There is a C 4 , α ⁢ ( Ω ¯ ) -neighborhood U of g S m along with a constant C 2 > 0 so that for all g ∈ U and for ψ ∈ B 0 ⁢ ( Ω ) , if f ∈ H ρ 2 ⁢ ( Ω ) ∩ S g weakly solves

(A.9) Π ̊ ⁢ ( L g ⁢ ( ρ ⁢ L g * ⁢ f ) ) = Π ̊ ⁢ ( ψ ) ,

then f ∈ B 4 ⁢ ( Ω ) and

(A.10) ∥ f ∥ ℬ 4 ⁢ ( Ω ) ≤ C 2 ⁢ ∥ Π ̊ ⁢ ( ψ ) ∥ ℬ 0 ⁢ ( Ω ) .

Moreover, if we let h = ρ ⁢ L g * ⁢ f , then

(A.11) ∥ h ∥ ℬ 2 ⁢ ( Ω ) ≤ C 2 ⁢ ∥ Π ̊ ⁢ ( ψ ) ∥ ℬ 0 ⁢ ( Ω ) .

Proof.

Let P = ρ - 1 ⁢ L g ⁢ ρ ⁢ L g * . Since ρ = 1 on the support of ζ, u ⊥ = ρ - 1 ⁢ ( ρ ⁢ u ) ⊥ . Then we see P ⁢ f = ρ - 1 ⁢ ( Π ̊ ⁢ ( ρ ⁢ P ⁢ f ) + ( ρ ⁢ P ⁢ f ) ⊥ ) = ρ - 1 ⁢ Π ̊ ⁢ ( ψ ) + ( P ⁢ f ) ⊥ . We apply (A.6) to obtain (we omit the domain Ω in the notation for the norms)

∥ f ∥ C ϕ , ϕ 3 2 ⁢ ρ 1 2 4 , α ≤ C ⁢ ( ∥ P ⁢ f ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α + ∥ f ∥ L ρ 2 )

≤ C ⁢ ( ∥ ρ - 1 ⁢ Π ̊ ⁢ ( ψ ) ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α + ∥ ( P ⁢ f ) ⊥ ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α + ∥ f ∥ L ρ 2 ) .

The last term on the right can be estimated by the preceding proposition, and for the first term we have

∥ ρ - 1 ⁢ Π ̊ ⁢ ( ψ ) ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α ≤ C ⁢ ∥ Π ̊ ⁢ ( ψ ) ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ - 1 2 0 , α .

Thus we only need to estimate

∥ ( P ⁢ f ) ⊥ ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α .

Since ( P ⁢ f ) ⊥ lies in a finite-dimensional space, there is a C > 0 so that for all f,

∥ ( P ⁢ f ) ⊥ ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α ≤ C ⁢ ∥ ( P ⁢ f ) ⊥ ∥ L 2 ⁢ ( spt ⁢ ( ζ ) ) .

On the other hand, by interpolation, for any ϵ > 0 , there is C ⁢ ( ϵ ) so that

∥ ( P ⁢ f ) ⊥ ∥ L 2 ⁢ ( spt ⁢ ( ζ ) ) ≤ ∥ f ∥ C 4 ⁢ ( spt ⁢ ( ζ ) )

≤ ϵ ⁢ ∥ f ∥ C 4 , α ⁢ ( spt ⁢ ( ζ ) ) + C ⁢ ( ϵ ) ⁢ ∥ f ∥ L 2 ⁢ ( spt ⁢ ( ζ ) )

≤ C ⁢ ( ϵ ⁢ ∥ f ∥ C ϕ , ϕ 3 2 ⁢ ρ 1 2 4 , α + C ⁢ ( ϵ ) ⁢ ∥ f ∥ L ρ 2 ) .

As for h = ρ ⁢ L g * ⁢ f , ∥ h ∥ L ρ - 1 2 ≤ C ⁢ ∥ f ∥ L ρ 2 , while for some C 0 ,

∥ h ∥ C ϕ , ϕ 2 + 3 2 ⁢ ρ - 1 2 2 , α = ∥ ρ ⁢ L g * ⁢ f ∥ C ϕ , ϕ 2 + 3 2 ⁢ ρ - 1 2 2 , α ≤ C 0 ⁢ ∥ f ∥ C ϕ , ϕ 3 2 ⁢ ρ 1 2 4 , α . ∎

A.6 Solving the nonlinear problem by iteration

We continue the proof of Proposition A.2. Given γ close to g S m , we want to solve for suitable h ⁢ ( γ ) with Π ̊ ⁢ ( R ⁢ ( γ + h ⁢ ( γ ) ) ) = 0 . In fact, we find h ⁢ ( γ ) in the form h ⁢ ( γ ) = ρ ⁢ L γ * ⁢ f , for suitable f. We do this iteratively, following the same framework as our previous works [17, 22, 20, 21], cf. [14].

Let ℒ γ = L γ ⁢ ρ ⁢ L γ * . We recursively define f j , h j and γ j as follows. Let γ 0 = γ and let f 0 solve Π ̊ ⁢ ( ℒ γ ⁢ f 0 ) = - Π ̊ ⁢ ( R ⁢ ( γ ) ) . Let h 0 = ρ ⁢ L γ * ⁢ f 0 . For γ close enough to g S m , R ⁢ ( γ ) is sufficiently small so that γ 1 = γ 0 + h 0 is a small perturbation of γ, and hence is a metric close to g S m . We next solve Π ̊ ⁢ ( ℒ γ ⁢ f 1 ) = - Π ̊ ⁢ ( R ⁢ ( γ 1 ) ) ; we let h 1 = ρ ⁢ L γ * ⁢ f 1 , and γ 2 = γ 1 + h 1 . We then justify h 1 is small, and γ 2 is a metric. We remark that we linearize at a fixed metric, to control some estimates required to close the argument; to illustrate, the operator L γ 1 involves derivatives of the tensor h 0 we generated, whereas we have fixed estimates on L γ . As such, the convergence to a limit may be slower than that of Newton’s method, since the improvement is sub-quadratic after the first iteration.

Lemma A.14.

Let 0 < α < 1 and 0 < δ < 1 . There is a C > 0 and a C 4 , α ⁢ ( Ω ¯ ) -neighborhood U of g S m so that the recursion outlined above produces infinite sequences f j and h j = ρ ⁢ L γ * ⁢ f j and γ j with γ 0 = γ , and γ j + 1 = γ 0 + ∑ k = 0 j h k , where each γ j is a metric, with the following estimates:

(A.12) ∥ f j ∥ ℬ 4 ⁢ ( Ω ) ≤ C ⁢ ∥ Π ̊ ⁢ ( R ⁢ ( γ ) ) ∥ ℬ 0 ⁢ ( Ω ) 1 + j ⁢ δ ,

(A.13) ∥ h j ∥ ℬ 2 ⁢ ( Ω ) ≤ C ⁢ ∥ Π ̊ ⁢ ( R ⁢ ( γ ) ) ∥ ℬ 0 ⁢ ( Ω ) 1 + j ⁢ δ ,

(A.14) ∥ Π ̊ ⁢ ( R ⁢ ( γ j ) ) ∥ ℬ 0 ⁢ ( Ω ) ≤ ∥ Π ̊ ⁢ ( R ⁢ ( γ ) ) ∥ ℬ 0 ⁢ ( Ω ) 1 + j ⁢ δ .

Proof.

The proof follows along the same lines as proofs of the analogous results in earlier works, in particular [20, Lemma 3.5], cf. [21, Section 6.3], so we just indicate the ideas. The required estimate (A.12) for f 0 is given by (A.10), and likewise for h 0 we have (A.11). The required estimate (A.14) for Π ̊ ⁢ ( R ⁢ ( γ 1 ) ) follows fairly readily from estimating the quadratic Taylor remainder for the scalar curvature functional R ⁢ ( γ 1 ) = R ⁢ ( γ ) + L γ ⁢ ( h 0 ) + 𝒬 γ ⁢ ( h 0 ) (cf. [20, Section 3.5]). Note that for j = 1 we could allow δ = 1 , but in the end we use δ < 1 , as we get slower convergence due to using the linearization at γ = γ 0 only, as we illustrate now.

Feeding estimate (A.14) for Π ̊ ⁢ ( R ⁢ ( γ 1 ) ) into (A.10) gives the required estimate (A.12) of f 1 , and then the required estimate (A.13) of h 1 follows likewise. As for estimate (A.14) for j = 2 and j = 3 , we write

Π ̊ ⁢ ( R ⁢ ( γ 2 ) ) = Π ̊ ⁢ ( R ⁢ ( γ 1 ) + L γ ⁢ ( h 1 ) + ( L γ 1 - L γ ) ⁢ ( h 1 ) + 𝒬 γ 1 ⁢ ( h 1 ) )

= Π ̊ ⁢ ( ( L γ 1 - L γ ) ⁢ ( h 1 ) + 𝒬 γ 1 ⁢ ( h 1 ) ) ,

Π ̊ ⁢ ( R ⁢ ( γ 3 ) ) = Π ̊ ⁢ ( R ⁢ ( γ 2 ) + L γ ⁢ ( h 2 ) + ( L γ 2 - L γ 1 ) ⁢ ( h 2 ) + ( L γ 1 - L γ ) ⁢ ( h 2 ) + 𝒬 γ 2 ⁢ ( h 2 ) )

= Π ̊ ⁢ ( ( L γ 1 - L γ ) ⁢ ( h 1 ) + 𝒬 γ 1 ⁢ ( h 1 ) ) .

As γ 1 - γ = h 0 , and γ 2 - γ 1 = h 1 , ∥ Π ̊ ⁢ ( R ⁢ ( γ 2 ) ) ∥ ℬ 0 ⁢ ( Ω ) and ∥ Π ̊ ⁢ ( R ⁢ ( γ 3 ) ) ∥ ℬ 0 ⁢ ( Ω ) above are of quadratic order in ( h 0 , h 1 , h 2 ) . We also note that since we control h j in C 2 , α ⁢ ( Ω ¯ ) , there is a uniform estimate of ∥ 𝒬 γ j ⁢ ( h j ) ∥ ℬ 0 ⁢ ( Ω ) . From here it should be clear how to proceed, and we refer to [20, Lemma 3.5], cf. [21, Section 6.3], for more details. ∎

It follows from this lemma that the iteration converges and yields a solution

h = h ⁢ ( γ ) = ρ ⁢ L γ * ⁢ f

of Π ̊ ⁢ ( R ⁢ ( γ + h ⁢ ( γ ) ) ) = 0 , as desired. Moreover, we can conclude that the extension of h by zero across ∂ ⁡ Ω is in C 2 , α ⁢ ( ℝ 3 ) , and using Remark A.12 at each stage of the iteration, we have that if γ is parity-symmetric, so is h ⁢ ( γ ) . It remains to show that h is smooth, and that h ⁢ ( γ ) depends continuously on γ.

The operator u ↦ P ⁢ u := ρ - 1 ⁢ ( R ⁢ ( γ + ρ ⁢ L γ * ⁢ u ) - R ⁢ ( γ ) ) is a quasilinear fourth-order elliptic operator provided ρ ⁢ L γ * ⁢ u is sufficiently small, in which case the fourth-order part of the operator is close to that of ρ - 1 ⁢ L γ ⁢ ρ ⁢ L γ * (i.e. 2 ⁢ Δ γ 2 ). As such, Proposition A.6 holds for P (for ρ ⁢ L γ * ⁢ u sufficiently small), either as a corollary via a perturbation argument, or by Remark A.10. So in case γ is smooth, h = h ⁢ ( γ ) is smooth in Ω. To see that it extends smoothly by zero over the boundary ∂ ⁡ Ω , we consider the equation R ⁢ ( γ + h ⁢ ( γ ) ) - R ⁢ ( γ ) = b ⁢ ( γ ) ⁢ ζ ⁢ f m - R ⁢ ( γ ) ; as the right-hand side is supported outside a collar neighborhood of the boundary, we can apply Proposition A.6 for any nonnegative integer k, from which we can conclude smoothness, and the required higher-order estimates.

Remark A.15.

The support of h ⁢ ( γ ) is contained in Ω ¯ . We could of course have arranged the support to lie strictly inside Ω by running the construction on Ω ε in place of Ω, for some small ε > 0 .

A.7 Continuous dependence

Finally, we prove continuous dependence of h ⁢ ( γ ) and hence b ⁢ ( γ ) on γ ∈ C 4 , α ⁢ ( Ω ¯ ) .

Proof.

Suppose we have γ 1 and γ 2 near g S m , with associated h 1 = h ⁢ ( γ 1 ) = ρ ⁢ L γ 1 * ⁢ f 1 and h 2 = h ⁢ ( γ 2 ) = ρ ⁢ L γ 2 * ⁢ f 2 , so that Π ̊ ⁢ ( R ⁢ ( γ i + h i ) ) = 0 . Then writing

L γ 1 ⁢ ( h 2 - h 1 ) = L γ 2 ⁢ ( h 2 ) - L γ 1 ⁢ ( h 1 ) + ( L γ 1 - L γ 2 ) ⁢ ( h 2 )

and applying a Taylor expansion R ⁢ ( γ i + h i ) = R ⁢ ( γ i ) + L γ i ⁢ ( h i ) + 𝒬 γ i ⁢ ( h i ) , we see

L γ 1 ⁢ ( h 2 - h 1 ) = R ⁢ ( γ 2 + h 2 ) - R ⁢ ( γ 1 + h 1 ) + R ⁢ ( γ 1 ) - R ⁢ ( γ 2 )

+ 𝒬 γ 1 ⁢ ( h 1 ) - 𝒬 γ 2 ⁢ ( h 2 ) + ( L γ 1 - L γ 2 ) ⁢ ( h 2 )

= ( b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) ) ⁢ ζ ⁢ f m + R ⁢ ( γ 1 ) - R ⁢ ( γ 2 )

+ 𝒬 γ 1 ⁢ ( h 1 ) - 𝒬 γ 2 ⁢ ( h 2 ) + ( L γ 1 - L γ 2 ) ⁢ ( h 2 ) .

Based on the form of the Taylor remainder, there is a constant C > 0 so that

(A.15) ∥ 𝒬 γ 1 ⁢ ( h 1 ) - 𝒬 γ 2 ⁢ ( h 2 ) ∥ ℬ 0 ⁢ ( Ω ) ≤ C ⁢ ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) ⁢ ( ∥ γ 1 - γ 2 ∥ C 2 , α ⁢ ( Ω ) ⁢ ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) + ∥ h 1 - h 2 ∥ ℬ 2 ⁢ ( Ω ) ) .

We will want to express everything in terms of f 1 and f 2 . Note that

(A.16) L γ 1 ⁢ ( ρ ⁢ L γ 1 * ⁢ ( f 2 - f 1 ) ) = L γ 1 ⁢ ( h 2 - h 1 ) + L γ 1 ⁢ ( ρ ⁢ ( L γ 1 * - L γ 2 * ) ⁢ ( f 2 ) ) = ( b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) ) ⁢ ζ ⁢ f m + R ⁢ ( γ 1 ) - R ⁢ ( γ 2 ) + 𝒬 γ 1 ⁢ ( h 1 ) - 𝒬 γ 2 ⁢ ( h 2 ) + ( L γ 1 - L γ 2 ) ⁢ ( h 2 ) + L γ 1 ⁢ ( ρ ⁢ ( L γ 1 * - L γ 2 * ) ⁢ ( f 2 ) ) = : ( b ( γ 2 ) - b ( γ 1 ) ) ζ f m + ℰ 1 + L γ 1 ( ρ ( L γ 1 * - L γ 2 * ) ( f 2 ) ) = : ( b ( γ 2 ) - b ( γ 1 ) ) ζ f m + ℰ 1 + ℰ 2 = : ( b ( γ 2 ) - b ( γ 1 ) ) ζ f m + ℰ .

We estimate each of these terms. By (A.15), and the fact that R ⁢ ( γ 1 ) - R ⁢ ( γ 2 ) is supported in Ω ′ ¯ , away from ∂ ⁡ Ω , we see for some C > 0 ,

(A.17) ∥ ℰ 1 ∥ ℬ 0 ⁢ ( Ω ) ≤ C ⁢ ( ∥ γ 1 - γ 2 ∥ C 2 , α ⁢ ( Ω 0 ) + ∥ h 1 - h 2 ∥ ℬ 2 ⁢ ( Ω ) ⁢ ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) )

and

(A.18) ∥ ℰ 2 ∥ C φ , φ 4 + 3 2 ⁢ ρ - 1 2 0 , α ⁢ ( Ω ) ≤ C ⁢ ∥ γ 1 - γ 2 ∥ C 4 , α ⁢ ( Ω ) ⁢ ∥ f 2 ∥ C φ , φ 3 2 ⁢ ρ 1 2 4 , α ⁢ ( Ω )

≤ C ⁢ ∥ γ 1 - γ 2 ∥ C 4 , α ⁢ ( Ω ) ⁢ ∥ f 2 ∥ ℬ 4 ⁢ ( Ω ) .

Later on we will also want to estimate ∫ Ω ( f 2 - f 1 ) ⁢ ℰ ⁢ 𝑑 μ γ 1 , for which we note that, using the density of C c ∞ ⁢ ( Ω ) in H ρ 2 ⁢ ( Ω ) ,

(A.19) | ∫ Ω ( f 2 - f 1 ) ⁢ ℰ 1 ⁢ 𝑑 μ γ 1 | ≤ ∥ f 2 - f 1 ∥ L ρ 2 ⁢ ( Ω , γ 1 ) ⁢ ∥ ℰ 1 ∥ L ρ - 1 2 ⁢ ( Ω , γ 1 )

≤ C ⁢ ∥ f 2 - f 1 ∥ L ρ 2 ⁢ ( Ω , γ 1 ) ⁢ ∥ ℰ 1 ∥ ℬ 0 ⁢ ( Ω ) ,

(A.20) | ∫ Ω ( f 2 - f 1 ) ⁢ ℰ 2 ⁢ 𝑑 μ γ 1 | = | ∫ Ω 〈 ρ ⁢ L γ 1 * ⁢ ( f 2 - f 1 ) , ( ( L γ 1 * - L γ 2 * ) ⁢ ( f 2 ) ) 〉 γ 1 ⁢ 𝑑 μ γ 1 |

≤ C ⁢ ∥ γ 1 - γ 2 ∥ C 2 ⁢ ( Ω ) ⁢ ∥ f 2 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ⁢ ∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) .

We next estimate ( b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) ) ⁢ ζ ⁢ f m = R ⁢ ( γ 2 + h 2 ) - R ⁢ ( γ 1 + h 1 ) . For simplicity of exposition, consider the case m > 0 . Applying the Taylor expansion we have

( b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) ) ⁢ ∫ Ω ζ ⁢ ( f m ) 2 ⁢ 𝑑 μ γ 1

= ∫ Ω f m ⁢ ( R ⁢ ( γ 2 ) - R ⁢ ( γ 1 ) ) ⁢ 𝑑 μ γ 1 + ∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 ) ⁢ 𝑑 μ γ 2

+ ∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 ) ⁢ ( d ⁢ μ γ 1 - d ⁢ μ γ 2 ) - ∫ Ω f m ⁢ L γ 1 ⁢ ( h 1 ) ⁢ 𝑑 μ γ 1

+ ∫ Ω f m ⁢ ( 𝒬 γ 2 ⁢ ( h 2 ) - 𝒬 γ 1 ⁢ ( h 1 ) ) ⁢ 𝑑 μ γ 1 .

Therefore by (A.15),

(A.21) | b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) | ≤ C ⁢ ( ∥ γ 1 - γ 2 ∥ C 2 , α ⁢ ( Ω ) + ∥ h 1 - h 2 ∥ ℬ 2 ⁢ ( Ω ) ⁢ ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) )

+ | ∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 ) ⁢ 𝑑 μ γ 2 - ∫ Ω f m ⁢ L γ 1 ⁢ ( h 1 ) ⁢ 𝑑 μ γ 1 | .

We just need to estimate the integrals in (A.21). This is easy, but the required expansion is cumbersome. Since L g S m * ⁢ f m = 0 , and h ⁢ ( γ i ) vanishes along with derivatives at ∂ ⁡ Ω , we obtain

∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 ) ⁢ 𝑑 μ γ 2 - ∫ Ω f m ⁢ L γ 1 ⁢ ( h 1 ) ⁢ 𝑑 μ γ 1

= ∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ g S m ) + ∫ Ω f m ⁢ ( L γ 2 - L g S m ) ⁢ ( h 2 ) ⁢ 𝑑 μ g S m

- ∫ Ω f m ⁢ L γ 1 ⁢ ( h 1 ) ⁢ 𝑑 μ γ 1

= ∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 - h 1 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ g S m ) + ∫ Ω f m ⁢ ( L γ 2 - L γ 1 ) ⁢ ( h 1 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ g S m )

+ ∫ Ω f m ⁢ L γ 1 ⁢ ( h 1 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ γ 1 ) + ∫ Ω f m ⁢ ( L g S m - L γ 1 ) ⁢ ( h 1 ) ⁢ 𝑑 μ g S m

+ ∫ Ω f m ⁢ ( L γ 2 - L g S m ) ⁢ ( h 2 ) ⁢ 𝑑 μ g S m

= ∫ Ω f m ⁢ L γ 2 ⁢ ( h 2 - h 1 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ g S m ) + ∫ Ω f m ⁢ ( L γ 2 - L γ 1 ) ⁢ ( h 1 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ g S m )

+ ∫ Ω f m ⁢ L γ 1 ⁢ ( h 1 ) ⁢ ( d ⁢ μ γ 2 - d ⁢ μ γ 1 ) + ∫ Ω f m ⁢ ( L γ 2 - L γ 1 ) ⁢ ( h 1 ) ⁢ 𝑑 μ g S m

+ ∫ Ω f m ⁢ ( L γ 2 - L g S m ) ⁢ ( h 2 - h 1 ) ⁢ 𝑑 μ g S m .

Together with (A.21) this allows us to conclude

(A.22) | b ( γ 2 ) - b ( γ 1 ) | ≤ C ∥ γ 1 - γ 2 ∥ C 2 , α ⁢ ( Ω ) + D ∥ h 1 - h 2 ∥ ℬ 2 ⁢ ( Ω ) ( ∥ γ 2 - g S m ∥ C 2 ⁢ ( Ω )

+ ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) ) .

We obtain the same estimate in the m = 0 case by integrating

( b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) ) ⁢ ζ ⁢ f 0 = R ⁢ ( γ 2 + h 2 ) - R ⁢ ( γ 1 + h 1 )

against each of the components of f 0 = 〈 1 , x 1 , x 2 , x 3 〉 , and estimating just as above.

Next, we estimate ∥ f 2 - f 2 ∥ L ρ 2 ⁢ ( Ω , γ 1 ) . To this end, multiply (A.16) by ( f 2 - f 1 ) and integrate, using that f i ∈ S γ i along with the density of C c ∞ ⁢ ( Ω ) in H ρ 2 ⁢ ( Ω ) , to obtain

(A.23) ∥ L γ 1 * ⁢ ( f 2 - f 1 ) ∥ L ρ 2 ⁢ ( Ω , γ 1 ) 2 = ∫ Ω ( f 2 - f 1 ) ⁢ ( b ⁢ ( γ 2 ) - b ⁢ ( γ 1 ) ) ⁢ ζ ⁢ f m ⁢ 𝑑 μ γ 1 + ∫ Ω ( f 2 - f 1 ) ⁢ ℰ ⁢ 𝑑 μ γ 1 .

We use this to estimate f 2 - f 1 . Observe that if L γ 1 * has trivial kernel, then by the same arguments in the proof of Proposition A.7, there is a constant C 1 so that for all f ∈ H ρ 2 ⁢ ( Ω , γ 1 ) ,

∥ f ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ C 1 ⁢ ∥ L γ 1 * ⁢ f ∥ L ρ 2 ⁢ ( Ω , γ 1 ) ,

which gives us an estimate by setting f = f 2 - f 1 . In general, f 2 = Π γ 1 ⁢ ( f 2 ) + f ~ 2 , where for m > 0 ,

(A.24) f ~ 2 = 〈 f 2 , ζ ⁢ f m 〉 L 2 ⁢ ( Ω , γ 1 ) 〈 ζ ⁢ f m , ζ ⁢ f m 〉 L 2 ⁢ ( Ω , γ 1 ) ⁢ ζ ⁢ f m = ∫ Ω f 2 ⁢ ζ ⁢ f m ⁢ ( d ⁢ μ γ 1 - d ⁢ μ γ 2 ) 〈 ζ ⁢ f m , ζ ⁢ f m 〉 L 2 ⁢ ( Ω , γ 1 ) ⁢ ζ ⁢ f m ,

and analogously for m = 0 . Hence f ~ 2 = a ⁢ ζ ⁢ f m , where ∥ a ∥ ≤ C ⁢ ∥ γ 2 - γ 1 ∥ C 0 ⁢ ( Ω ) . Using this and (A.5), we have

∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ ∥ Π γ 1 ⁢ ( f 2 ) - f 1 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) + ∥ f ~ 2 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ C ⁢ ∥ L γ 1 * ⁢ [ Π γ 1 ⁢ ( f 2 ) - f 1 ] ∥ L ρ 2 ⁢ ( Ω , γ 1 ) + ∥ f ~ 2 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ C ⁢ ∥ L γ 1 * ⁢ ( f 2 - f 1 ) ∥ L ρ 2 ⁢ ( Ω , γ 1 ) + C ⁢ ∥ L γ 1 * ⁢ f ~ 2 ∥ L ρ 2 ⁢ ( Ω , γ 1 ) + ∥ f ~ 2 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ C ⁢ ∥ L γ 1 * ⁢ ( f 2 - f 1 ) ∥ L ρ 2 ⁢ ( Ω , γ 1 ) + C ′ ⁢ ∥ f ~ 2 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ C 1 ⁢ ( ∥ L γ 1 * ⁢ ( f 2 - f 1 ) ∥ L ρ 2 ⁢ ( Ω , γ 1 ) + ∥ γ 2 - γ 1 ∥ C 0 ⁢ ( Ω ) ) .

We thus conclude, using (A.23) together with (A.17), (A.19), (A.20) and (A.22),

∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) 2 ≤ 2 ⁢ ( C 1 ) 2 ⁢ ( ∥ L γ 1 * ⁢ ( f 2 - f 1 ) ∥ L ρ 2 ⁢ ( Ω , γ 1 ) 2 + ∥ γ 2 - γ 1 ∥ C 0 ⁢ ( Ω ) 2 )

≤ C 2 ∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ( ∥ γ 2 - γ 1 ∥ C 2 , α ⁢ ( Ω )

+ ∥ h 2 - h 2 ∥ ℬ 2 ⁢ ( Ω ) ( ∥ γ 2 - g S m ∥ C 2 ⁢ ( Ω ) + ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) ) )

+ 2 ⁢ C 1 2 ⁢ ∥ γ 2 - γ 1 ∥ C 0 ⁢ ( Ω ) 2 .

Noting for nonnegative β and δ, w 2 ≤ β ⁢ w + δ 2 implies w ≤ β + δ , we conclude

(A.25) ∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω , γ 1 ) ≤ C ( ∥ γ 2 - γ 1 ∥ C 2 , α ⁢ ( Ω ) + ∥ h 2 - h 2 ∥ ℬ 2 ⁢ ( Ω ) ( ∥ γ 2 - g S m ∥ C 2 ⁢ ( Ω ) + ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) ) ) .

We have the following, using (A.17), (A.18), (A.22), (A.25) and the weighted Schauder estimate (A.6) (where the constant “C” can change from line to line):

∥ h 2 - h 1 ∥ ℬ 2 ⁢ ( Ω ) = ∥ ρ ⁢ L γ 2 * ⁢ f 2 - ρ ⁢ L γ 1 * ⁢ f 1 ∥ ℬ 2 ⁢ ( Ω )

≤ ∥ ρ ⁢ L γ 1 * ⁢ ( f 2 - f 1 ) ∥ ℬ 2 ⁢ ( Ω ) + ∥ ρ ⁢ ( L γ 2 * - L γ 1 * ) ⁢ ( f 2 ) ∥ ℬ 2 ⁢ ( Ω )

≤ C ⁢ ∥ f 2 - f 1 ∥ ℬ 4 ⁢ ( Ω ) + ∥ ρ ⁢ ( L γ 2 * - L γ 1 * ) ⁢ ( f 2 ) ∥ ℬ 2 ⁢ ( Ω )

≤ C ⁢ ( ∥ f 2 - f 1 ∥ ℬ 4 ⁢ ( Ω ) + ∥ γ 2 - γ 1 ∥ C 4 , α ⁢ ( Ω ) ⁢ ∥ f 2 ∥ ℬ 4 ⁢ ( Ω ) )

≤ C ( ∥ ρ - 1 L γ 1 ( ρ L γ 1 * ( f 2 - f 1 ) ) ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α ⁢ ( Ω ) + ∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω )

+ ∥ γ 2 - γ 1 ∥ C 4 , α ⁢ ( Ω ) ∥ f 2 ∥ ℬ 4 ⁢ ( Ω ) )

≤ C ( ∥ ρ - 1 ( b ( γ 2 ) - b ( γ 1 ) ) ζ f m ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α ⁢ ( Ω ) + ∥ ρ - 1 ℰ ∥ C ϕ , ϕ 4 + 3 2 ⁢ ρ 1 2 0 , α ⁢ ( Ω )

+ ∥ f 2 - f 1 ∥ H ρ 2 ⁢ ( Ω ) + ∥ γ 2 - γ 1 ∥ C 4 , α ⁢ ( Ω ) ∥ f 2 ∥ ℬ 4 ⁢ ( Ω ) )

≤ C ( ∥ γ 2 - γ 1 ∥ C 4 , α ⁢ ( Ω )

+ ∥ h 2 - h 1 ∥ ℬ 2 ⁢ ( Ω ) ( ∥ γ 2 - g S m ∥ C 2 ⁢ ( Ω ) + ∥ ( h 1 , h 2 ) ∥ ℬ 2 ⁢ ( Ω ) ) ) .

For γ 1 and γ 2 close enough to g S m , we have ∥ h i ∥ ℬ 2 ⁢ ( Ω ) ≤ C ⁢ ∥ R ⁢ ( γ i ) ∥ ℬ 0 ⁢ ( Ω ) is sufficiently small so that we can absorb the second term on the right-hand side, to conclude

∥ h 2 - h 1 ∥ ℬ 2 ⁢ ( Ω ) ≤ C ′ ⁢ ∥ γ 2 - γ 1 ∥ C 4 , α ⁢ ( Ω ) ,

as desired. ∎

Remark A.16.

We observe that if L γ 1 * had nontrivial kernel, then as R ⁢ ( γ 1 ) is constant ([24, Theorem 1], cf. [17, Proposition 2.3] or [20, Proposition 2.1]), it must vanish. In that case, h 1 = 0 , and then R ⁢ ( γ ) → 0 in ℬ 0 ⁢ ( Ω ) for γ → γ 1 in C 2 , α ⁢ ( Ω ¯ ) and with R ⁢ ( γ ) supported in a fixed subset Ω ′ ¯ ⊂ Ω . Via Lemma A.14, we can then conclude h ⁢ ( γ ) → 0 = h 1 in ℬ 2 ⁢ ( Ω ) .

Acknowledgements

The authors thank an anonymous referee for several useful comments and suggestions.

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Received: 2023-02-25

Revised: 2023-09-14

Published Online: 2024-01-31

Published in Print: 2024-03-01

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