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Harmonic maps between two concentric annuli in 𝐑3

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Published/Copyright: July 12, 2019
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 14 Issue 3

Abstract

Given two annuli 𝔸 ⁒ ( r , R ) and 𝔸 ⁒ ( r βˆ— , R βˆ— ) , in 𝐑 3 equipped with the Euclidean metric and the weighted metric | y | - 2 , respectively, we minimize the Dirichlet integral, i.e., the functional

β„± ⁒ [ f ] = ∫ 𝔸 ⁒ ( r , R ) βˆ₯ D ⁒ f βˆ₯ 2 | f | 2 ,

where f is a homeomorphism between 𝔸 ⁒ ( r , R ) and 𝔸 ⁒ ( r βˆ— , R βˆ— ) , which belongs to the Sobolev class 𝒲 1 , 2 . The minimizer is a certain generalized radial mapping, i.e., a mapping of the form f ⁒ ( | x | ⁒ Ξ· ) = ρ ⁒ ( | x | ) ⁒ T ⁒ ( Ξ· ) , where T is a conformal mapping of the unit sphere onto itself and ρ ⁒ ( t ) = R βˆ— ⁒ ( r βˆ— R βˆ— ) R ⁒ ( r - t ) ( R - r ) ⁒ t . It should be noticed that, in this case, no Nitsche phenomenon occurs.

Keywords: Minimizers; Nitsche phenomenon; annuli
MSC 2010: 31A05; 42B30

Communicated by Frank Duzaar

A Appendix

The next corollary follows from Theorem 1.1.

Corollary A.1.

Let f ∈ W 1 , 2 be a homeomorphism between A ⁒ ( r , R ) and A ⁒ ( r βˆ— , R βˆ— ) . Then

(A.1) β„° ⁒ [ f ] = ∫ 𝔸 ⁒ ( r , R ) βˆ₯ D ⁒ f βˆ₯ 2 ⁒ d ⁒ x β©Ύ 4 ⁒ Ο€ R βˆ— 2 ⁒ ( 2 ⁒ ( R - r ) + r R log [ R βˆ— r βˆ— ] 2 R - r ) .

It seems that (A.1) is not sharp, but it shows that the minimizer of Dirichlet energy is not zero for the case of non-degenerated annuli. This is somehow complementary to the result for the case of degenerated annuli, where the infimum of the Dirichlet energy of Sobolev homomorphisms with free boundary condition is zero [14, Theorem 1.6]. It should be noticed that the solution to the equation Ξ” ⁒ h = 0 if h ⁒ ( x ) = H ⁒ ( r ) ⁒ x | x | , according to (2.6), is given by

H ⁒ ( t ) = a ⁒ t + b t 2 .

Now the solution to the boundary value problem

{ Ξ” ⁒ h = 0 if ⁒ h = H ⁒ ( | x | ) ⁒ x | x | , H ⁒ ( r ) = r βˆ— , H ⁒ ( R ) = R βˆ— , where ⁒ β€…0 < r < R ⁒ and ⁒ β€…0 < r βˆ— < R βˆ— ,

is given by

H ⁒ ( t ) = r 2 ⁒ R 2 ⁒ ( - R ⁒ r βˆ— + r ⁒ R βˆ— ) ( r 3 - R 3 ) ⁒ t 2 + ( r 2 ⁒ r βˆ— - R 2 ⁒ R βˆ— ) ⁒ t r 3 - R 3 .

Then

H β€² ⁒ ( t ) = r 2 ⁒ r βˆ— - R 2 ⁒ R βˆ— r 3 - R 3 + 2 ⁒ ( r 2 ⁒ R 3 ⁒ r βˆ— - r 3 ⁒ R 2 ⁒ R βˆ— ) ( r 3 - R 3 ) ⁒ t 3 .

So H β€² ⁒ ( t ) > 0 for t ∈ [ r , R ] if and only if

( - 2 ⁒ r 2 ⁒ R 3 ⁒ r βˆ— + 2 ⁒ r 3 ⁒ R 2 ⁒ R βˆ— ) + ( - r 2 ⁒ r βˆ— + R 2 ⁒ R βˆ— ) ⁒ t 3 β©Ύ 0 , t ∈ [ r , R ] .

It follows that r 3 ⁒ r βˆ— + 2 ⁒ R 3 ⁒ r βˆ— - 3 ⁒ r ⁒ R 2 ⁒ R βˆ— β©½ 0 , i.e., the condition

r βˆ— R βˆ— β©½ 3 ⁒ r ⁒ R 2 r 3 + 2 ⁒ R 3

is sufficient and necessary for existence of radial Euclidean harmonic mappings between given annuli (the generalized Nitsche condition). In this case, the harmonic mapping h ⁒ ( x ) = H ⁒ ( r ) ⁒ x | x | satisfies the equation

(A.2) β„° ⁒ [ h ] = 4 ⁒ Ο€ ⁒ ( r ⁒ ( r 3 + 2 ⁒ R 3 ) ⁒ r βˆ— 2 - 6 ⁒ r 2 ⁒ R 2 ⁒ r βˆ— ⁒ R βˆ— + R ⁒ ( 2 ⁒ r 3 + R 3 ) ⁒ R βˆ— 2 ) R 3 - r 3 .

It is clear that the quantity X on the right-hand side of (A.2) is bigger than the quantity Y on the right-hand side of (A.1). It is also clear that

Z = inf ⁑ { β„° ⁒ [ h ] : h ∈ 𝒲 1 , 2 ⁒ ( 𝔸 ⁒ ( r , R ) , 𝔸 ⁒ ( r βˆ— , R βˆ— ) ) } ∈ ( Y , X ] ,

and probably Z < X , in view of [14, Theorem 1.6], but the right value of Z remains so far unknown.

Motivated by the case n = 3 , we state the following conjecture.

Conjecture A.2.

Let n β‰  3 . Assume that β„± is the family of homeomorphisms between spherical rings 𝔸 ⁒ ( r , R ) and 𝔸 ⁒ ( r βˆ— , R βˆ— ) in 𝐑 n that belongs to 𝒲 1 , n - 1 . Then the Dirichlet integral of f ∈ β„± with respect to the weight β„˜ ⁒ ( y ) = | y | 1 - n , i.e., the integral

β„± ⁒ [ f ] = ∫ 𝔸 ⁒ ( r , R ) βˆ₯ D ⁒ f βˆ₯ n - 1 | f | n - 1 ⁒ d ⁒ x ,

achieves its minimum for generalized-radial diffeomorphisms between annuli.

Acknowledgements

I am thankful to the referee for providing constructive comments that helped improving the paper.

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Received: 2018-11-08
Revised: 2019-05-05
Accepted: 2019-06-06
Published Online: 2019-07-12
Published in Print: 2021-07-01

Β© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Harmonic maps between two concentric annuli in 𝐑3
  3. An existence theorem for non-homogeneous differential inclusions in Sobolev spaces
  4. On the supremal version of the Alt–Caffarelli minimization problem
  5. Strong approximation in h-mass of rectifiable currents under homological constraint
  6. Harnack inequality and an asymptotic mean-value property for the Finsler infinity-Laplacian
  7. Sticky-disk limit of planar N-bubbles
  8. On different notions of calibrations for minimal partitions and minimal networks in ℝ2
  9. Non-axially symmetric solutions of a mean field equation on π•Š2
  10. Homogenization in BV of a model for layered composites in finite crystal plasticity
https://doi.org/10.1515/acv-2018-0074
Keywords for this article
Minimizers; Nitsche phenomenon; annuli

Articles in the same Issue

  1. Frontmatter
  2. Harmonic maps between two concentric annuli in 𝐑3
  3. An existence theorem for non-homogeneous differential inclusions in Sobolev spaces
  4. On the supremal version of the Alt–Caffarelli minimization problem
  5. Strong approximation in h-mass of rectifiable currents under homological constraint
  6. Harnack inequality and an asymptotic mean-value property for the Finsler infinity-Laplacian
  7. Sticky-disk limit of planar N-bubbles
  8. On different notions of calibrations for minimal partitions and minimal networks in ℝ2
  9. Non-axially symmetric solutions of a mean field equation on π•Š2
  10. Homogenization in BV of a model for layered composites in finite crystal plasticity
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