Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 3 (2014), 627-671.

The 1-harmonic flow with values in a hyperoctant of the $N$-sphere

Lorenzo Giacomelli, Jose Mazón, and Salvador Moll

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We prove the existence of solutions to the 1-harmonic flow — that is, the formal gradient flow of the total variation of a vector field with respect to the L2-distance — from a domain of m into a hyperoctant of the N-dimensional unit sphere, S+N1, under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler–Lagrange formulation in terms of the “geodesic representative” of a BV-director field on its jump set. Such characterization relies on a lower semicontinuity argument which leads to a nontrivial and nonconvex minimization problem: to find a shortest path between two points on S+N1 with respect to a metric which penalizes the closeness to their geodesic midpoint.

Article information

Anal. PDE, Volume 7, Number 3 (2014), 627-671.

Received: 18 April 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 49Q20: Variational problems in a geometric measure-theoretic setting 53C22: Geodesics [See also 58E10] 35K67: Singular parabolic equations 35K92: Quasilinear parabolic equations with p-Laplacian
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 58E20: Harmonic maps [See also 53C43], etc. 68U10: Image processing

harmonic flows total variation flow nonlinear parabolic systems lower semicontinuity and relaxation nonconvex variational problems geodesics Riemannian manifolds with boundary image processing


Giacomelli, Lorenzo; Mazón, Jose; Moll, Salvador. The 1-harmonic flow with values in a hyperoctant of the $N$-sphere. Anal. PDE 7 (2014), no. 3, 627--671. doi:10.2140/apde.2014.7.627.

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  • S. B. Alexander, I. D. Berg, and R. L. Bishop, “Cut loci, minimizers, and wavefronts in Riemannian manifolds with boundary”, Michigan Math. J. 40:2 (1993), 229–237.
  • R. Alicandro, A. Corbo Esposito, and C. Leone, “Relaxation in BV of integral functionals defined on Sobolev functions with values in the unit sphere”, J. Convex Anal. 14:1 (2007), 69–98.
  • L. Ambrosio and G. Dal Maso, “A general chain rule for distributional derivatives”, Proc. Amer. Math. Soc. 108:3 (1990), 691–702.
  • L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Clarendon Press, New York, 2000.
  • L. Ambrosio, G. Crippa, and S. Maniglia, “Traces and fine properties of a $BD$ class of vector fields and applications”, Ann. Fac. Sci. Toulouse Math. $(6)$ 14:4 (2005), 527–561.
  • F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, “Minimizing total variation flow”, Differential Integral Equations 14:3 (2001), 321–360.
  • F. Andreu-Vaillo, V. Caselles, and J. M. Mazón, Parabolic quasilinear equations minimizing linear growth functionals, Progress in Mathematics 223, Birkhäuser, Basel, 2004.
  • G. Anzellotti, “Pairings between measures and bounded functions and compensated compactness”, Ann. Mat. Pura Appl. $(4)$ 135 (1983), 293–318.
  • J. W. Barrett, X. Feng, and A. Prohl, “On $p$-harmonic map heat flows for $1\leq p<\infty$ and their finite element approximations”, SIAM J. Math. Anal. 40:4 (2008), 1471–1498.
  • M. Bertsch, R. Dal Passo, and R. van der Hout, “Nonuniqueness for the heat flow of harmonic maps on the disk”, Arch. Ration. Mech. Anal. 161:2 (2002), 93–112.
  • M. Bertsch, R. Dal Passo, and A. Pisante, “Point singularities and nonuniqueness for the heat flow for harmonic maps”, Comm. Partial Differential Equations 28:5-6 (2003), 1135–1160.
  • M. Bonforte and A. Figalli, “Total variation flow and sign fast diffusion in one dimension”, J. Differential Equations 252:8 (2012), 4455–4480.
  • H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011.
  • V. Caselles, “On the entropy conditions for some flux limited diffusion equations”, J. Differential Equations 250:8 (2011), 3311–3348.
  • Y. M. Chen, “The weak solutions to the evolution problems of harmonic maps”, Math. Z. 201:1 (1989), 69–74.
  • G.-Q. Chen and H. Frid, “Divergence-measure fields and hyperbolic conservation laws”, Arch. Ration. Mech. Anal. 147:2 (1999), 89–118.
  • Y. M. Chen, M. C. Hong, and N. Hungerbühler, “Heat flow of $p$-harmonic maps with values into spheres”, Math. Z. 215:1 (1994), 25–35.
  • R. Dal Passo, L. Giacomelli, and S. Moll, “Rotationally symmetric 1-harmonic maps from $D\sp 2$ to $S\sp 2$”, Calc. Var. Partial Differential Equations 32:4 (2008), 533–554.
  • R. W. R. Darling, Differential forms and connections, Cambridge University Press, 1994.
  • A. DeSimone and P. Podio-Guidugli, “On the continuum theory of deformable ferromagnetic solids”, Arch. Rational Mech. Anal. 136:3 (1996), 201–233.
  • J. Diestel and J. J. Uhl, Jr., Vector measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, RI, 1977.
  • J. Eells, Jr. and J. H. Sampson, “Harmonic mappings of Riemannian manifolds”, Amer. J. Math. 86 (1964), 109–160.
  • L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 1992.
  • H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York, 1969.
  • X. Feng, “Divergence-$L\sp q$ and divergence-measure tensor fields and gradient flows for linear growth functionals of maps into the unit sphere”, Calc. Var. Partial Differential Equations 37:1-2 (2010), 111–139.
  • I. Fonseca and S. Müller, “Relaxation of quasiconvex functionals in ${\rm BV}(\Omega,{\mathbb R}\sp p)$ for integrands $f(x,u,\nabla u)$”, Arch. Rational Mech. Anal. 123:1 (1993), 1–49.
  • I. Fonseca and P. Rybka, “Relaxation of multiple integrals in the space ${\rm BV}(\Omega,{\mathbb R}\sp p)$”, Proc. Roy. Soc. Edinburgh Sect. A 121:3-4 (1992), 321–348.
  • L. Giacomelli and S. Moll, “Rotationally symmetric 1-harmonic flows from $D\sp 2$ to $S\sp 2$: local well-posedness and finite time blowup”, SIAM J. Math. Anal. 42:6 (2010), 2791–2817.
  • L. Giacomelli, J. M. Mazón, and S. Moll, “The 1-harmonic flow with values into $\mathbb{S}\sp 1$”, SIAM J. Math. Anal. 45:3 (2013), 1723–1740.
  • L. Giacomelli, J. M. Mazón, and S. Moll, “Solutions to the 1-harmonic flow with values into a hyper-octant of the $N$-sphere”, Appl. Math. Lett. 26:11 (2013), 1061–1064.
  • M. Giaquinta and D. Mucci, “The BV-energy of maps into a manifold: Relaxation and density results”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 5:4 (2006), 483–548.
  • Y. Giga and R. Kobayashi, “On constrained equations with singular diffusivity”, Methods Appl. Anal. 10:2 (2003), 253–277.
  • Y. Giga and H. Kuroda, “On breakdown of solutions of a constrained gradient system of total variation”, Bol. Soc. Parana. Mat. $(3)$ 22:1 (2004), 9–20.
  • Y. Giga, Y. Kashima, and N. Yamazaki, “Local solvability of a constrained gradient system of total variation”, Abstr. Appl. Anal. 8 (2004), 651–682.
  • Y. Giga, H. Kuroda, and N. Yamazaki, “An existence result for a discretized constrained gradient system of total variation flow in color image processing”, Interdiscip. Inform. Sci. 11:2 (2005), 199–204.
  • Y. Giga, H. Kuroda, and N. Yamazaki, “Global solvability of constrained singular diffusion equation associated with essential variation”, pp. 209–218 in Free boundary problems, edited by I. N. Figueiredo et al., Internat. Ser. Numer. Math. 154, Birkhäuser, Basel, 2007.
  • R. van der Hout, “Flow alignment in nematic liquid crystals in flows with cylindrical symmetry”, Differential Integral Equations 14:2 (2001), 189–211.
  • N. Hungerbühler, “Heat flow into spheres for a class of energies”, pp. 45–65 in Variational problems in Riemannian geometry, edited by P. Baird et al., Progr. Nonlinear Differential Equations Appl. 59, Birkhäuser, Basel, 2004.
  • R. Kobayashi, J. A. Warren, and W. C. Carter, “A continuum model of grain boundaries”, Phys. D 140:1-2 (2000), 141–150.
  • M. Misawa, “On the $p$-harmonic flow into spheres in the singular case”, Nonlinear Anal. 50:4, Ser. A: Theory Methods (2002), 485–494.
  • G. Sapiro, Geometric partial differential equations and image analysis, Cambridge University Press, 2001.
  • J. Simon, “Compact sets in the space $L\sp p(0,T;B)$”, Ann. Mat. Pura Appl. $(4)$ 146 (1987), 65–96.
  • M. Struwe, “The evolution of harmonic maps: Existence, partial regularity, and singularities”, pp. 485–491 in Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), edited by N. G. Lloyd et al., Progr. Nonlinear Differential Equations Appl. 7, Birkhäuser, Boston, MA, 1992.
  • B. Tang, G. Sapiro, and V. Caselles, “Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case”, Int. J. Comput. Vis. 36:2 (2000), 149–161.
  • B. Tang, G. Sapiro, and V. Caselles, “Color image enhancement via chromaticity diffusion”, IEEE Trans. Image Process. 10:5 (2001), 701–707.
  • W. P. Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer, New York, 1989.