Median values, 1-harmonic functions, and functions of least gradient

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March 2013, 12(2): 711-719. doi: 10.3934/cpaa.2013.12.711

Median values, 1-harmonic functions, and functions of least gradient

Matthew B. Rudd ^1, and Heather A. Van Dyke ^2,

1.	Department of Mathematics, Sewanee: The University of the South, Sewanee, TN 37383, United States
2.	Department of Mathematics, Washington State University, Pullman, WA 99164, United States

Received October 2011 Revised June 2012 Published September 2012

Abstract
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Motivated by the mean value property of harmonic functions, we introduce the local and global median value properties for continuous functions of two variables. We show that the Dirichlet problem associated with the local median value property is either easy or impossible to solve, and we prove that continuous functions with this property are $1$-harmonic in the viscosity sense. We then close with the following conjecture: a continuous function having the global median value property and prescribed boundary values coincides with the function of least gradient having those same boundary values.

Keywords: functions of least gradient, 1-Laplacian, statistical functional equation, strict convexity., Mean value property, median value property, 1-harmonic.

Mathematics Subject Classification: Primary: 35J92, 39B22, 35A35; Secondary: 35B05, 35D40, 35A1.

Citation: Matthew B. Rudd, Heather A. Van Dyke. Median values, 1-harmonic functions, and functions of least gradient. Communications on Pure and Applied Analysis, 2013, 12 (2) : 711-719. doi: 10.3934/cpaa.2013.12.711

References:

[1]	F. Cao, "Geometric Curve Evolution and Image Processing," Lecture Notes in Mathematics 1805, Springer-Verlag, Berlin, 2003. doi: 10.1007/b10404.
[2]	F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), 1895-1909. doi: 10.1137/0732085.
[3]	L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[4]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.
[5]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics 80, Birkhäuser Verlag, Basel, 1984. doi: 775682 (87a:58041).
[6]	D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables, Rocky Mountain J. Math., 41 (2011), 493-504. doi: 10.1216/RMJ-2011-41-2-493.
[7]	D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions,, preprint., ().
[8]	P. Juutinen, $p$-Harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1029. doi: 10.1512/iumj.2005.54.2658.
[9]	P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.
[10]	R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101.
[11]	S. G. Noah, The median of a continuous function, Real Analysis Exchange, 33 (2008), 269-74.
[12]	A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), 365-379. doi: 10.1007/s00211-004-0566-1.
[13]	S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion, SIAM J. Appl. Math., 60 (2000), 868-890. doi: 10.1137/S003613999833397X.
[14]	D. Stroock, "Probability Theory, An Analytic View," Cambridge UP, Cambridge, 1993.
[15]	Z. Waksman and J. Wasilewsky, A theorem on level lines of continuous functions, Israel J. Math., 27 (1977), 247-251.
[16]	W. P. Ziemer, "Weakly Differentiable Functions," Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
[17]	W. P. Ziemer, Functions of least gradient and BV functions, in "Nonlinear Analysis, Function Spaces and Applications," Vol. 6, Acad. Sci. Czech Repub., Prague, 1999, 270-312.

show all references

References:

[1]	F. Cao, "Geometric Curve Evolution and Image Processing," Lecture Notes in Mathematics 1805, Springer-Verlag, Berlin, 2003. doi: 10.1007/b10404.
[2]	F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), 1895-1909. doi: 10.1137/0732085.
[3]	L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[4]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.
[5]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics 80, Birkhäuser Verlag, Basel, 1984. doi: 775682 (87a:58041).
[6]	D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables, Rocky Mountain J. Math., 41 (2011), 493-504. doi: 10.1216/RMJ-2011-41-2-493.
[7]	D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions,, preprint., ().
[8]	P. Juutinen, $p$-Harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1029. doi: 10.1512/iumj.2005.54.2658.
[9]	P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.
[10]	R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101.
[11]	S. G. Noah, The median of a continuous function, Real Analysis Exchange, 33 (2008), 269-74.
[12]	A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), 365-379. doi: 10.1007/s00211-004-0566-1.
[13]	S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion, SIAM J. Appl. Math., 60 (2000), 868-890. doi: 10.1137/S003613999833397X.
[14]	D. Stroock, "Probability Theory, An Analytic View," Cambridge UP, Cambridge, 1993.
[15]	Z. Waksman and J. Wasilewsky, A theorem on level lines of continuous functions, Israel J. Math., 27 (1977), 247-251.
[16]	W. P. Ziemer, "Weakly Differentiable Functions," Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
[17]	W. P. Ziemer, Functions of least gradient and BV functions, in "Nonlinear Analysis, Function Spaces and Applications," Vol. 6, Acad. Sci. Czech Repub., Prague, 1999, 270-312.

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