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