Show Summary Details
More options …

# Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2018: 2.316

CiteScore 2018: 1.77

SCImago Journal Rank (SJR) 2018: 2.350
Source Normalized Impact per Paper (SNIP) 2018: 1.465

Mathematical Citation Quotient (MCQ) 2018: 1.44

Online
ISSN
1864-8266
See all formats and pricing
More options …

# Harmonic maps between two concentric annuli in 𝐑3

David Kalaj
• Corresponding author
• Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro
• Email
• Other articles by this author:
Published Online: 2019-07-12 | DOI: https://doi.org/10.1515/acv-2018-0074

## Abstract

Given two annuli $𝔸\left(r,R\right)$ and $𝔸\left({r}_{\ast },{R}_{\ast }\right)$, in ${𝐑}^{3}$ equipped with the Euclidean metric and the weighted metric ${|y|}^{-2}$, respectively, we minimize the Dirichlet integral, i.e., the functional

$\mathcal{ℱ}\left[f\right]={\int }_{𝔸\left(r,R\right)}\frac{{\parallel Df\parallel }^{2}}{{|f|}^{2}},$

where f is a homeomorphism between $𝔸\left(r,R\right)$ and $𝔸\left({r}_{\ast },{R}_{\ast }\right)$, which belongs to the Sobolev class ${\mathcal{𝒲}}^{1,2}$. The minimizer is a certain generalized radial mapping, i.e., a mapping of the form $f\left(|x|\eta \right)=\rho \left(|x|\right)T\left(\eta \right)$, where T is a conformal mapping of the unit sphere onto itself and $\rho \left(t\right)={R}_{\ast }{\left(\frac{{r}_{\ast }}{{R}_{\ast }}\right)}^{\frac{R\left(r-t\right)}{\left(R-r\right)t}}$. It should be noticed that, in this case, no Nitsche phenomenon occurs.

Keywords: Minimizers; Nitsche phenomenon; annuli

MSC 2010: 31A05; 42B30

## References

• [1]

L. V. Ahlfors, Moebius Transformations in Several Dimensions (in Russian), Mir, Moscow, 1986. Google Scholar

• [2]

S. S. Antman, Nonlinear Problems of Elasticity, Appl. Math. Sci. 107, Springer, New York, 1995. Google Scholar

• [3]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser. 48, Princeton University, Princeton, 2009. Google Scholar

• [4]

K. Astala, T. Iwaniec and G. Martin, Deformations of annuli with smallest mean distortion, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 899–921.

• [5]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1976/77), no. 4, 337–403.

• [6]

F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), no. 3–4, 153–206.

• [7]

J.-C. Bourgoin, The minimality of the map $\frac{x}{\parallel x\parallel }$ for weighted energy, Calc. Var. Partial Differential Equations 25 (2006), no. 4, 469–489. Google Scholar

• [8]

H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), no. 4, 649–705.

• [9]

P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-dimensional Elasticity, Stud. Math. Appl. 20, North-Holland, Amsterdam, 1988. Google Scholar

• [10]

M. Csörnyei, S. Hencl and J. Malý, Homeomorphisms in the Sobolev space ${W}^{1,n-1}$, J. Reine Angew. Math. 644 (2010), 221–235.

• [11]

B. Dacorogna, Introduction to the Calculus of Variations, Imperial College, London, 2004. Google Scholar

• [12]

M.-C. Hong, On the minimality of the p-harmonic map $\frac{x}{|x|}:{B}^{n}\to {S}^{n-1}$, Calc. Var. Partial Differential Equations 13 (2001), no. 4, 459–468. Google Scholar

• [13]

T. Iwaniec, L. V. Kovalev and J. Onninen, The Nitsche conjecture, J. Amer. Math. Soc. 24 (2011), no. 2, 345–373.

• [14]

T. Iwaniec and J. Onninen, p-harmonic energy of deformations between punctured balls, Adv. Calc. Var. 2 (2009), no. 1, 93–107.

• [15]

T. Iwaniec and J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians, Mem. Amer. Math. Soc. 218 (2012), no. 1023, 1–105. Google Scholar

• [16]

J. Jost and X. Li-Jost, Calculus of Variations, Cambridge Stud. Adv. Math. 64, Cambridge University, Cambridge, 1998. Google Scholar

• [17]

D. Kalaj, On the Nitsche conjecture for harmonic mappings in ${ℝ}^{2}$ and ${ℝ}^{3}$, Israel J. Math. 150 (2005), 241–251. Google Scholar

• [18]

D. Kalaj, Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture, J. Lond. Math. Soc. (2) 93 (2016), no. 3, 683–702.

• [19]

D. Kalaj, $\left(n,\rho \right)$-harmonic mappings and energy minimal deformations between annuli, Calc. Var. Partial Differential Equations 58 (2019), no. 2, Article ID 51.

• [20]

A. Koski and J. Onninen, Radial symmetry of p-harmonic minimizers, Arch. Ration. Mech. Anal. 230 (2018), no. 1, 321–342.

• [21]

A. Lyzzaik, The modulus of the image annuli under univalent harmonic mappings and a conjecture of Nitsche, J. London Math. Soc. (2) 64 (2001), no. 2, 369–384.

• [22]

J. C. C. Nitsche, Mathematical notes: On the module of doubly-connected regions under harmonic mappings, Amer. Math. Monthly 69 (1962), no. 8, 781–782.

• [23]

T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis. With an Introduction to Algebraic Topology, Grundlehren Math. Wiss. 75, Springer, Berlin, 1955. Google Scholar

• [24]

R. Schoen and S. T. Yau, Lectures on Harmonic Maps, International Press, Cambridge, 1997. Google Scholar

• [25]

M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 1319, Springer, Berlin, 1988. Google Scholar

• [26]

A. Weitsman, Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche, Israel J. Math. 124 (2001), 327–331.

Revised: 2019-05-05

Accepted: 2019-06-06

Published Online: 2019-07-12

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.