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Regularity theorems for a biological network formulation model in two space dimensions

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  • We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu [13]. A consequence of these results is that a stationary weak solution must be a classical one in two space dimensions. Our mathematical analysis is based upon the weakly monotone function theory and Hardy space methods.

    Mathematics Subject Classification: Primary: 35D30, 35Q99; Secondary: 35A01.


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  •   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.
      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. 
      R. Coifman , P. L. Lions , Y. Meyer  and  S. Semmes , Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993) , 247-286. 
      G. Di Fazio , $ L^p$ Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996) , 409-420. 
      L. C. Evans , Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991) , 101-113.  doi: 10.1007/BF00375587.
      L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992.
      F. Gehring , Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962) , 353-393.  doi: 10.1090/S0002-9947-1962-0139735-8.
      D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
      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.
      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.
      J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993.
      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.
      D. Hu  and  D. Cai , Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013) , 138701.  doi: 10.1103/PhysRevLett.111.138701.
      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.
      J. Kinnunen , Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994) , 355-366. 
      J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, to appear.
      J. J. Manfredi , Weakly monotone functions, J. Geometric Analysis, 4 (1994) , 393-402.  doi: 10.1007/BF02921588.
      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.
      J. R. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987.
      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.
      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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