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Invariant measures for a stochastic Fokker-Planck equation
Regularity theorems for a biological network formulation model in two space dimensions
Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, USA |
We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu [
References:
[1] |
G. Albi, M. Artina, M. Fornasier and P. Markowich,
Biological transportation networks: Modeling and simulation, Anal. Appl. (Singap.), 14 (2016), 185-206.
doi: 10.1142/S0219530515400059. |
[2] |
S. Chanillo and R. L. Wheeden,
Existence and estimates of Green's function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 309-340.
|
[3] |
R. Coifman, P. L. Lions, Y. Meyer and S. Semmes,
Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.
|
[4] |
G. Di Fazio,
$ L^p$ Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996), 409-420.
|
[5] |
L. C. Evans,
Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991), 101-113.
doi: 10.1007/BF00375587. |
[6] |
L. C. Evans and R. F. Gariepy,
Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992. |
[7] |
F. Gehring,
Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353-393.
doi: 10.1090/S0002-9947-1962-0139735-8. |
[8] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. |
[9] |
J. Haskovec, P. Markowich and B. Perthame,
Mathematical analysis of a PDE system for biological network formulation, Comm. Partial Differential Equations, 40 (2015), 918-956.
doi: 10.1080/03605302.2014.968792. |
[10] |
J. Haskovec, P. Markowich, B. Perthame and M. Schlottbom,
Notes on a PDE system for biological network formulation, Nonlinear Anal, 138 (2016), 127-155.
doi: 10.1016/j.na.2015.12.018. |
[11] |
J. Heinonen, T. Kilpeläinen and O. Martio,
Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993. |
[12] |
D. Hu,
Optimization, Adaptation, and Initialization of Biological Transport Networks, Workshop on multi scale problems from physics, biology, and material sciences, May 28-31,2014, Shanghai. |
[13] |
D. Hu and D. Cai,
Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701.
doi: 10.1103/PhysRevLett.111.138701. |
[14] |
R. L. Johnson and J. C. Neugebauer,
Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.
doi: 10.4171/ZAA/583. |
[15] |
J. Kinnunen,
Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994), 355-366.
|
[16] |
J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity,
J. Differential Equations, to appear. |
[17] |
J. J. Manfredi,
Weakly monotone functions, J. Geometric Analysis, 4 (1994), 393-402.
doi: 10.1007/BF02921588. |
[18] |
S. Müller,
A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc., 21 (1989), 245-248.
doi: 10.1090/S0273-0979-1989-15818-7. |
[19] |
J. R. Rodrigues,
Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987. |
[20] |
S. Semmes,
A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations, 19 (1994), 277-319.
doi: 10.1080/03605309408821017. |
[21] |
X. Xu, Existence theorems for the quantum drift-diffusion system with mixed boundary conditions,
Commun. Contemp. Math. , 18 (2016), 1550048, 21 pp. |
show all references
References:
[1] |
G. Albi, M. Artina, M. Fornasier and P. Markowich,
Biological transportation networks: Modeling and simulation, Anal. Appl. (Singap.), 14 (2016), 185-206.
doi: 10.1142/S0219530515400059. |
[2] |
S. Chanillo and R. L. Wheeden,
Existence and estimates of Green's function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 309-340.
|
[3] |
R. Coifman, P. L. Lions, Y. Meyer and S. Semmes,
Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.
|
[4] |
G. Di Fazio,
$ L^p$ Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996), 409-420.
|
[5] |
L. C. Evans,
Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991), 101-113.
doi: 10.1007/BF00375587. |
[6] |
L. C. Evans and R. F. Gariepy,
Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992. |
[7] |
F. Gehring,
Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353-393.
doi: 10.1090/S0002-9947-1962-0139735-8. |
[8] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. |
[9] |
J. Haskovec, P. Markowich and B. Perthame,
Mathematical analysis of a PDE system for biological network formulation, Comm. Partial Differential Equations, 40 (2015), 918-956.
doi: 10.1080/03605302.2014.968792. |
[10] |
J. Haskovec, P. Markowich, B. Perthame and M. Schlottbom,
Notes on a PDE system for biological network formulation, Nonlinear Anal, 138 (2016), 127-155.
doi: 10.1016/j.na.2015.12.018. |
[11] |
J. Heinonen, T. Kilpeläinen and O. Martio,
Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993. |
[12] |
D. Hu,
Optimization, Adaptation, and Initialization of Biological Transport Networks, Workshop on multi scale problems from physics, biology, and material sciences, May 28-31,2014, Shanghai. |
[13] |
D. Hu and D. Cai,
Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701.
doi: 10.1103/PhysRevLett.111.138701. |
[14] |
R. L. Johnson and J. C. Neugebauer,
Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.
doi: 10.4171/ZAA/583. |
[15] |
J. Kinnunen,
Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994), 355-366.
|
[16] |
J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity,
J. Differential Equations, to appear. |
[17] |
J. J. Manfredi,
Weakly monotone functions, J. Geometric Analysis, 4 (1994), 393-402.
doi: 10.1007/BF02921588. |
[18] |
S. Müller,
A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc., 21 (1989), 245-248.
doi: 10.1090/S0273-0979-1989-15818-7. |
[19] |
J. R. Rodrigues,
Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987. |
[20] |
S. Semmes,
A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations, 19 (1994), 277-319.
doi: 10.1080/03605309408821017. |
[21] |
X. Xu, Existence theorems for the quantum drift-diffusion system with mixed boundary conditions,
Commun. Contemp. Math. , 18 (2016), 1550048, 21 pp. |
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