American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval
  • CPAA Home
  • This Issue
  • Next Article
    Note on evolutionary free piston problem for Stokes equations with slip boundary conditions
July  2014, 13(4): 1641-1652. doi: 10.3934/cpaa.2014.13.1641

The classification of constant weighted curvature curves in the plane with a log-linear density

Doan The Hieu 1, and  Tran Le Nam 2,

1. 

Departement of Mathematics, College of Education, Hue University, Hue, Vietnam
2. 

Departement of Mathematics,Dong Thap University, Dong Thap, Vietnam

Received  May 2013 Revised  January 2014 Published  February 2014

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^y.$
Keywords: planes with a log-linear density, self similar translator, traveling curved front, weighted curvature., Curve flow with a forcing term.
Mathematics Subject Classification: Primary 53C44, 53A04; Secondary 53C21, 35J6.
Citation: Doan The Hieu, Tran Le Nam. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1641-1652. doi: 10.3934/cpaa.2014.13.1641
References:
[1]

M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations, 25 (2006), 205-246. doi: 10.1007/s00526-005-0351-8.

[2]

C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc., 78 (2008), 177-197. doi: 10.1017/S000497270800052X.

[3]

A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geo. Anal., 20 (2010), 243-290. doi: 10.1007/s12220-009-9109-4.

[4]

I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7 (2006).

[5]

I. Corwin and F. Morgan, The Gauss-Bonnet formula on surfaces with densities, Involve, 4 (2011), 199-202. doi: 10.2140/involve.2011.4.199.

[6]

J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density $r^p$, New York J. Math., 16 (2010) 31-51.

[7]

K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys., 135 (1991), 595-613.

[8]

D. T. Hieu and N. M. Hoang, Ruled minimal surfaces in $R^3$ with density $e^z$, Pacific J. Math., 243 (2009), 277-285. doi: 10.2140/pjm.2009.243.277.

[9]

D. T. Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys., 61 (2011), 1625-1629. doi: 10.1016/j.geomphys.2011.04.005.

[10]

G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE, 8 (1999), 1-14. doi: 10.1007/s005260050113.

[11]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70. doi: 10.1007/BF02392946.

[12]

H. Jian, H. Ju, Y. Liu and W. Sun, Symmetry of translating solutions to mean curvature flows, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 2006-2016. doi: 10.1016/S0252-9602(10)60191-9.

[13]

H. Jian, H. Ju, Y. Liu and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975.

[14]

H. Ju, J. Lu and H. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963.

[15]

Q. Maurmann and F. Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE, 36 (2009), 1-5. doi: 10.1007/s00526-008-0219-9.

[16]

F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858.

[17]

F. Morgan, Myers' Theorem with density, Kodai Math. J., 29 (2006), 454-460. doi: 10.2996/kmj/1162478772.

[18]

F. Morgan, Geometric Measure Theory: a Beginner's Guide, $4^{th}$ edition, Academic Press, 2009.

[19]

F. Morgan, Manifolds with density and Perelman's proof of the Poincaré Conjecture, Amer. Math. Monthly, 116 (2009), 134-142. doi: 10.4169/193009709X469896.

[20]

H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: theory and applications, I (Chiba, 1999). 206-221, GAKUTO Internat. Ser. Math. Sci. Appl., 13, Gakkotosho, Tokyo, 2000.

[21]

C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE, 31 (2008), 27-46. doi: 10.1007/s00526-007-0104-y.

show all references

References:
[1]

M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations, 25 (2006), 205-246. doi: 10.1007/s00526-005-0351-8.

[2]

C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc., 78 (2008), 177-197. doi: 10.1017/S000497270800052X.

[3]

A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geo. Anal., 20 (2010), 243-290. doi: 10.1007/s12220-009-9109-4.

[4]

I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7 (2006).

[5]

I. Corwin and F. Morgan, The Gauss-Bonnet formula on surfaces with densities, Involve, 4 (2011), 199-202. doi: 10.2140/involve.2011.4.199.

[6]

J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density $r^p$, New York J. Math., 16 (2010) 31-51.

[7]

K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys., 135 (1991), 595-613.

[8]

D. T. Hieu and N. M. Hoang, Ruled minimal surfaces in $R^3$ with density $e^z$, Pacific J. Math., 243 (2009), 277-285. doi: 10.2140/pjm.2009.243.277.

[9]

D. T. Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys., 61 (2011), 1625-1629. doi: 10.1016/j.geomphys.2011.04.005.

[10]

G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE, 8 (1999), 1-14. doi: 10.1007/s005260050113.

[11]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70. doi: 10.1007/BF02392946.

[12]

H. Jian, H. Ju, Y. Liu and W. Sun, Symmetry of translating solutions to mean curvature flows, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 2006-2016. doi: 10.1016/S0252-9602(10)60191-9.

[13]

H. Jian, H. Ju, Y. Liu and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975.

[14]

H. Ju, J. Lu and H. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963.

[15]

Q. Maurmann and F. Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE, 36 (2009), 1-5. doi: 10.1007/s00526-008-0219-9.

[16]

F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858.

[17]

F. Morgan, Myers' Theorem with density, Kodai Math. J., 29 (2006), 454-460. doi: 10.2996/kmj/1162478772.

[18]

F. Morgan, Geometric Measure Theory: a Beginner's Guide, $4^{th}$ edition, Academic Press, 2009.

[19]

F. Morgan, Manifolds with density and Perelman's proof of the Poincaré Conjecture, Amer. Math. Monthly, 116 (2009), 134-142. doi: 10.4169/193009709X469896.

[20]

H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: theory and applications, I (Chiba, 1999). 206-221, GAKUTO Internat. Ser. Math. Sci. Appl., 13, Gakkotosho, Tokyo, 2000.

[21]

C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE, 31 (2008), 27-46. doi: 10.1007/s00526-007-0104-y.

[1]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
[2]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[3]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[4]

Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure and Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975
[5]

Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785
[6]

Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
[7]

Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003
[8]

Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058
[9]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[10]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231
[11]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[12]

Zhi-Cheng Wang. Traveling curved fronts in monotone bistable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2339-2374. doi: 10.3934/dcds.2012.32.2339
[13]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[14]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[15]

T. Candan, R.S. Dahiya. Oscillation of mixed neutral differential equations with forcing term. Conference Publications, 2003, 2003 (Special) : 167-172. doi: 10.3934/proc.2003.2003.167
[16]

Mia Jukić, Hermen Jan Hupkes. Curvature-driven front propagation through planar lattices in oblique directions. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2189-2251. doi: 10.3934/cpaa.2022055
[17]

Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson, Roger Temam. Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 509-546. doi: 10.3934/dcds.2018023
[18]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819
[19]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
[20]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy