Calderón Projectors on Phi-Manifolds with Boundarypankrashkin/...Prologue ClassicalCalderón...

Prologue Classical Calderón φ-Manifolds with Boundary An Example φ-Calderón Epilogue

Calderón Projectors on φ-Manifolds with BoundaryKarsten Fritzsch1

(ongoing joint work with Daniel Grieser2 and Elmar Schrohe1)

Aspect’19Université Paris-Sud, Orsay

30 Sept – 4 Oct 2019

1Leibniz Universität Hannover 2Carl-von-Ossietzky Universität Oldenburg


Summary

Calderón projectors are useful tools in spectral geometry there is a rather “explicit” construction of these (Seeley ’66)

the fibred cusp geometry is applicable to interesting examples there is a good ΨDO-calculus adapted to this geometry(Mazzeo-Melrose ’98)

slightly generalising the setup by including non-singular boundaryvalue–boundary faces and adapting the classical construction, weconstruct Calderón projectors for elliptic fibred cusp DOs

Karsten Fritzsch (LUH): Calderón Projectors on φ-Manifolds 1 / 17


Outline

i. The (Classical) Calderón Projector

ii. φ-Manifolds with Boundary

iii. An Example

iv. Calderón Projectors on φ-Manifolds with Boundary



Calderón Projectors

X a smooth manifold with boundary, ν a vector field transversal to ∂X,P ∈ Diffm(X) elliptic.

Calderón projector C for P : any projection

C : C∞(∂X)m −→ γ kerP =γu∣∣Pu = 0

,

whereγu =

(u|∂X , ∂νu|∂X , . . . , ∂

m−1ν u|∂X

)∈ C∞(∂X)m

is the Cauchy data of u ∈ C∞(X).

In 1966, Seeley [See66, See69] gave a general construction using the thenrather new theory of ΨDOs (see also [Gru96, Hör85]).



Construction

i. Let X = 2X, choose a metric g and let S1 =(

0 P∗

P 0).

ii. Use self-adjointness of S1 to extend it to an elliptic ΨDO S2 on Xand let P1 =

(0 S∗

2S2 0

).

iii. Add projection Π1 : L2(X) −→ ker0P1 onto

ker0P1 = kerP1 ∩ C∞(X) =u ∈ kerP1

∣∣ suppu ⊂ X.

Then, P2 = P1 + Π1 satisfies ker0P2 = 0.

iv. Choose dual basis ωj of kerP2 so that suppωj ⊂ X−.

v. Add projection Π2 onto spanωj: P = P2 + Π2.

Then: P ∈ Ψm(X) elliptic and invertible, hence P−1 ∈ Ψ−m(X), withP u = (P + Πker0P )u for u ∈ C∞(X).



Construction

vi. Using the transmission condition:

P (u0) = (Pu)0 + γ∗Pγu ,

where u0 is extension by 0, the Green’s matrix P is determined by Pand ν, and γ∗U =

∑Ul ⊗ δ(l)

∂X , for U ∈ C∞(∂X)m is δ-extensionfrom the boundary .

vii. Finally, letC = γP−1γ∗P

and C be (1, 1)-component of C.

Then: C = (Ckl) is a Calderón projector for P on X, where Ckl is a ΨDOof order k − l.



φ-Manifolds with Boundary

A φ-manifold with boundary is given by the following data: X: a compact manifold with corners B: a closed manifold without boundary (the base) F : a compact manifold with non-empty boundary (the typical fibre) a decomposition

∂X = ∂sX ∪ ∂BVX

of ∂X into (collective) boundary hypersurfaces a commutative diagram of fibrations

F ∂sX

B

∂F ∂s,BVX

φ

φBV



φ-Manifolds with Boundary

In adapted local coordinates (x, y, z) near ∂s,BVX:

x : bdf for ∂sXy : base coordinatesz : fibre coordinatesz1 : bdf for ∂BVX

φ : (0, y, z) 7→ y

φBV : (0, y, 0, z′) 7→ y



φ-Differential Operators

a φ-DO P ∈ Diffmφ (X) is of the form

P =∑

k+|α|+|β|≤m

ak,α,β(x, y, z)(x2Dx)k(xDy)αDβz ,

its φ-principal symbol is given by

φσm(P ) =∑

k+|α|+|β|=m

ak,α,β(x, y, z)τkηαζβ ,

and its normal family is

N(P )(τ, y, η) =∑

k+|α|+|β|≤m

ak,α,β(0, y, z)τkηαDβz ∈ Diffm(Fy) .



The φ-Calculus

P φ-elliptic :⇐⇒ φσm(P ) invertible for (τ, η, ζ) 6= 0P fully elliptic :⇐⇒ P φ-elliptic, N(P ) invertible for all (τ, y, η)

Theorem (Mazzeo-Melrose [MM98]):Fully elliptic elements T ∈ Ψm

φ (X) have parametrices in Ψ−mφ (X) withremainders in x∞Ψ−∞φ (X). They define Fredholm operators

T : xαHs+mφ (X) −→ xαHs

φ(X) for all α, s ∈ R,

their null spaces are contained in x∞C∞(X) and there is a complementto their ranges in x∞C∞(X).

If T ∈ Ψmφ (X) is fully elliptic and invertible, then T−1 ∈ Ψ−mφ (X).



An Example: Touching Domains

(M, g) a smooth closed Riemannian manifold, Ω ⊂M the complementof two touching balls B± ⊂M .

Is there a sensible notion of Dirichlet-to-Neumann map N for ∆g on Ω?And if so, what happens at p0?




If we can find a Calderón projector C for ∆g on Ω, and if we have

N = C−101 (id− C00) = (id− C11)−1C10 ,

we could study related problems (plasmonic, Robin, Steklov etc.).

Alas: Due to the singularity at p0, Γ is not a manifold!




Quasi-homogeneous blow-up of p0 yields a φ-manifold with boundary X:

Here, ∂sX is the front face of the blow-up and ∂BVX is the lift of the“old boundary”.

Moreover, if g lifts to a φ-metric on X, ∆g lifts to be a φ-elliptic φ-DO!(E.g. true in the Euclidean setting on Rn, away from infinity.)



Classical Construction Revisited

Now: X a compact φ-manifold with boundary, P ∈ Diffmφ (X) φ-elliptic.Essentially as before:

i. Take the double of X across ∂BVXii. Construct φ-elliptic extension P1 of Piii. From a fully elliptic extension P2: construct a Calderón projector C

Why does this work as before? Uses standard properties of ΨDO-calculi Boundary faces ∂BVX and ∂sX are transversal!

We can, e.g., formulate the transmission condition in terms of conormaldistributions and apply it just the same.

Alas: The φ-elliptic extension P1 will in general not be fully elliptic!



Fully Elliptic Extension

To obtain a fully elliptic extension, we need the following assumption:

∀ (τ, y, η) : kerN(P )(τ, y, η) ∩ C∞(Fy) = 0 , (ucnf)

where C∞(Fy) =u ∈ C∞(Fy)

∣∣u|∂Fy ≡ 0.

Proposition (F.-Grieser-Schrohe):Let X be a φ-manifold with boundary and suppose P ∈ Diffmφ (X) satisfies(ucnf). Then, there is a fully elliptic operator P2 ∈ Ψm

φ (X), defined onthe BV-double X of X, so that P2u = Pu for all u ∈ C∞(X).



Fully Elliptic Extension

Outline of Proof: Assuming (ucnf), locally choose dual bases ωµj for kerN(P1)(µ)with suppωµj ⊂ F−y , where µ = (τ, y, η),

Glue these together to obtain a family of projections

Πµ : L2(Fy) −→ C∞(F−y )

with rg Πµ + rgN(P1)(µ) = L2(Fy). Then, N(P1)(µ) + Πµ isinvertible for all µ.

Extend the Schwartz-kernel of this family to the φ-double space ofX to obtain a φ-elliptic perturbation ΠN ∈ Ψ−∞φ (X).

P2 = P1 + ΠN is then a fully elliptic extension of P .



φ-Calderón Projectors

Specify growth conditions at the singular boundary ∂sX by introducingthe sum of weighted φ-Sobolev spaces

Hα,s =m−1⊕j=0

xαHs−j− 1

2φBV

.

Theorem (F.-Grieser-Schrohe):With X and P as before, there is an operator C ∈ Ψ∗φ(∂BVX) which is aHα,s-Calderón projector for any choice of α, s ∈ R, i.e., a projection

C : Hα,s −→ γ kerP ,

from Hα,s onto the Cauchy data space γ kerP of solutions to Pu = 0.C = (Ckl) is a matrix of φ-ΨDOs of order k − l.



Closing Remarks

We’re still working on getting rid of Assumption (ucnf).

Having chosen a background φ-metric, we can also construct theorthogonal Hα,m− 1

2 –Calderón projector.

Earlier and explicit calculations in [Fri19] suggested that C is not anelement of any small calculus xαΨ∗φ(∂BVX). But there, no exactinverse was being used.

To construct the Dirichlet-to-Neumann map N for P = ∆g, weneed to invert either C01 or id− C00 and show that the result isagain a φ-ΨDO. This is work in progress.

This can be considered a limiting problem in the situation when thedistance of two domains tends to 0. Plasmonic Eigenvalues for twospheres in R3: Schnitzer [Sch19].



ReferencesI Albin, P., Leichtnam, E., Mazzeo, R., and Piazza, P. The Signature Package on Witt

Spaces. Ann. Sci. Éc. Norm. Supér. (4) 45, 2 (2012), 241–310.

I Fritzsch, K. Full Asymptotics and Laurent Series of Layer Potentials for Laplace’sEquation on the Half-Space. Math. Nachr. (2019), 1–33.

I Grieser, D. The Plasmonic Eigenvalue Problem. Reviews in Mathematical Physics 26, 3(2014), 1450005.

I Grieser, D., Hunsicker, E. A Parametrix Construction for the Laplacian on Q-Rank 1Locally Symmetric Spaces. In: Fourier Analysis (Cham) (Ruzhansky and Turunen, eds.),Birkhäuser, Basel, 2014, 149–186.

I Grubb, G. Functional Calculus of Pseudodifferential Boundary Problems. Birkhäuser, Basel,1996.

I Hörmander, L. The Analysis of Linear Partial Differential Operators III. Springer, Berlin,1985.

I Mazzeo, R., Melrose, R. B. Pseudodifferential Operators on Manifolds with FibredBoundaries. Asian J. Math. 2, 4 (1998), 833–866.

I Schnitzer, O. Asymptotic Approximations for the Plasmon Resonances of Nearly TouchingSpheres. Euro. Jnl of Applied Mathematics. DOI: 10.1017/S0956792518000712

I Seeley, R. Singular Integrals and Boundary Value Problems. Am. J. Math. 88, 4 (1966),781–809.

I Seeley, R. Topics in Pseudo-Differential Operators. In Pseudodifferential Operators(Stresa) (Nirenberg ed.), CIME Summer Schools, Springer, Berlin, 1985, 169–305.

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