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Weak solutions of inverse mean curvature flow for hypersurfaces with boundary

  • Thomas Marquardt EMAIL logo
Published/Copyright: January 20, 2015
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2017 Issue 728

Abstract

We consider the evolution of hypersurfaces with boundary under inverse mean curvature flow. The boundary condition is of Neumann type, i.e. the evolving hypersurface moves along, but stays perpendicular to, a fixed supporting hypersurface. In this setup, we prove existence and uniqueness of weak solutions. Furthermore, we indicate the existence of a monotone quantity which is the analog of the Hawking mass for closed hypersurfaces.

Funding statement: This work was mainly supported by the Max Planck Institute for Gravitational Physics in Potsdam. The manuscript was prepared while the author was at ETH in Zurich where he received support from the Swiss National Science Foundation SNF 200021-140467.

A Linear mixed Dirichlet–Neumann problems

Definition A.1.

Let Ω⊂ℝn be a bounded Lipschitz domain. We denote by Σ a relatively open part of ∂⁡Ω and write σ=∂⁡Ω∖Σ¯. Let μ be the outward pointing unit normal to Ω on Σ. We consider the following mixed Dirichlet–Neumann boundary value problem:

(A.1){L⁢u:=ai⁢j⁢Di⁢j⁢u+bk⁢Dk⁢u=fin ⁢Ω,μk⁢Dk⁢u=0on ⁢Σ,u=von ⁢σ¯,

where L is assumed to be uniformly elliptic and Σ,σ are supposed to be subsets of C2,α-hypersurfaces. Since the domain might have corners, we introduce weighted Hölder spaces to allow for less regular solutions. For δ>0 sufficiently small we define

Ωδ:={x∈Ω∣dist⁡(x,∂⁡Ω∖Σ)>δ}.

Using the classical Hölder norms ∥⋅∥k,α;Ω¯ as they appear in [17] we define

∥u∥k,α;Ω(b):=supδ>0⁡δb+k+α⁢∥u∥k,α;Ωδ¯,Hk,α(b)⁢(Ω):={u∣∥u∥k,α;Ω(b)<∞}

for k∈ℕ,α∈(0,1) and b>-k-α.

These norms have the following useful properties.

Lemma A.2.

Let k1,k2,k,l∈N and α,β∈(0,1). If k+α≥l+β, then

Hk,α(-l-β)⁢(Ω)⊂Cl,β⁢(Ω¯)∩Ck,α⁢(Ω).

Let k1+α≥b>0. If (un)n∈N⊂Hk1,α(-b)⁢(Ω) is bounded, then there is a subsequence (unk)k∈N such that

unk→Hk2,β(-b′)⁢(Ω)u (k→∞)

for 0<b′<b, 0<k2+β<k1+α and k2+β≥b′.

Proof.

See [18, Section 1] and the introduction of [19]. ∎

Now we can state the existence and regularity result for mixed elliptic boundary value problems which is due to Lieberman [20, 21].

Theorem A.3.

Let Σ,σ be subsets of C2,α-hypersurfaces. Let Ω⊂Rn be a bounded Lipschitz domain with boundary ∂⁡Ω=σ¯∪Σ¯ where σ and Σ are relatively open in ∂⁡Ω. Assume that ai⁢j is uniformly continuous in Ω and that L is uniformly elliptic. Furthermore, assume that for all x∈V:=σ¯∩Σ¯ the boundary parts σ and Σ enclose the domain at an angle 0<θ⁢(x)≤θmax<π2. Then there exists some β⁢(θmax)∈(0,1) such that if

ai⁢j∈H0,α(0)⁢(Ω),bi∈H0,α(1-β)⁢(Ω),f∈H0,α(1-β)⁢(Ω),v∈C1,β⁢(Ω¯),

then there exists a unique solution u∈C0⁢(Ω¯)∩C2⁢(Ω∪Σ) of (A.1). Furthermore, each such solution of (A.1) satisfies the estimate

∥u∥2,α;Ω(-1-β)≤C⁢(∥f∥0,α;Ω(1-β)+∥v∥1,β;Ω¯).

Proof.

The existence and uniqueness result can be found in [20, Theorem 2]. The regularity result is a variant of [21, Theorem 4]. It relies on a modification of the height estimate [21, Lemma 3.3]. This modification is necessary in order to match with the definition of the weighted norm which is used in [20]. ∎

Acknowledgements

The author wants to thank Gerhard Huisken for acquainting him with inverse mean curvature flow and for all the support and valuable discussions during the time the author spent at the Max Planck Institute for Gravitational Physics in Potsdam.

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Received: 2014-4-30
Revised: 2014-8-31
Published Online: 2015-1-20
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Formal pseudodifferential operators and Witten’s r-spin numbers
  3. Characteristic classes of symmetric products of complex quasi-projective varieties
  4. The ε - εβ property, the boundedness of isoperimetric sets in ℝN with density, and some applications
  5. Segre numbers, a generalized King formula, and local intersections
  6. The μ-ordinary Hasse invariant of\break unitary Shimura varieties
  7. Equivariant character correspondences and inductive McKay condition for type A
  8. Euler characteristic on noncommutative polyballs
  9. Weak solutions of inverse mean curvature flow for hypersurfaces with boundary
  10. Galois lattices and strongly divisible lattices in the unipotent case
  11. Erratum to Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces (J. reine angew. Math. 697 (2014), 1–14)
https://doi.org/10.1515/crelle-2014-0116

Articles in the same Issue

  1. Frontmatter
  2. Formal pseudodifferential operators and Witten’s r-spin numbers
  3. Characteristic classes of symmetric products of complex quasi-projective varieties
  4. The ε - εβ property, the boundedness of isoperimetric sets in ℝN with density, and some applications
  5. Segre numbers, a generalized King formula, and local intersections
  6. The μ-ordinary Hasse invariant of\break unitary Shimura varieties
  7. Equivariant character correspondences and inductive McKay condition for type A
  8. Euler characteristic on noncommutative polyballs
  9. Weak solutions of inverse mean curvature flow for hypersurfaces with boundary
  10. Galois lattices and strongly divisible lattices in the unipotent case
  11. Erratum to Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces (J. reine angew. Math. 697 (2014), 1–14)
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