Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 3: 1-16. doi: 10.3934/mine.2021027
Previous Article Next Article
Research article Special Issues

Singular elliptic equations with directional diffusion

  • 1.
    Department of Civil Engineering, University of Salerno, via Giovanni Paolo Ⅱ 132, 84084 Fisciano (SA), Italy
  • 2.
    Istituto Nazionale di Alta Matematica, INdAM - GNAMPA c/o University of Salerno, Italy
  • Received: 14 January 2020 Accepted: 28 April 2020 Published: 24 July 2020

  • We investigate conditions for the existence and uniqueness of viscosity solutions of the Dirichlet problem for a degenerate elliptic equation describing a stationary diffusion, which may take place in a partial number of spatial directions, with a possibly singular reaction term.

    Citation: Antonio Vitolo. Singular elliptic equations with directional diffusion[J]. Mathematics in Engineering, 2021, 3(3): 1-16. doi: 10.3934/mine.2021027

    Related Papers:

  • We investigate conditions for the existence and uniqueness of viscosity solutions of the Dirichlet problem for a degenerate elliptic equation describing a stationary diffusion, which may take place in a partial number of spatial directions, with a possibly singular reaction term.


    加载中


    [1] Amendola ME, Galise G, Vitolo A (2013) Riesz capacity, maximum principle and removable sets of fully nonlinear second order operators. Differ Integral Equ 27: 845-866.
    [2] Amendola ME, Rossi L, Vitolo A (2008) Harnack inequalities and ABP estimates for nonlinear second-order Elliptic equations in unbounded domains. Abstr Appl Anal 2008: 1-19.
    [3] Bardi M, Mannucci P (2006) On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun Pure Appl Anal 5: 709-731.
    [4] Birindelli I, Capuzzo Dolcetta I, Vitolo A (2016) ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains. Commun Contemp Math 18: 1-16.
    [5] Birindelli I, Galise G (2019) The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity. Calc Var 58: 180.
    [6] Birindelli I, Galise G, Ishii H (2018) A family of degenerate elliptic operators: Maximum principle and its consequences. Ann I H Poincaré Anal Non Linéaire 35: 417-441.
    [7] Blanc P, Esteve C, Rossi JD (2019) The evolution problem associated with eigenvalues of the Hessian. Commun Contemp Math, arXiv: 1901.01052.
    [8] Blanc P, Rossi JD (2019) Games for eigenvalues of the Hessian and concave/convex envelopes. J Math Pure Appl 127: 192-215.
    [9] Cabré X (1995) On the Alexandro ff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Commun Pure Appl Math 48: 539-570.
    [10] Cafagna V, Vitolo A (2002) On the maximum principle for second-order elliptic operators in unbounded domains. C R Math Acad Sci Paris 334: 359-363.
    [11] Caffarelli LA (1989) Interior a priori estimates for solutions of fully nonlinear equations. Ann Math 130: 189-213.
    [12] Caffarelli LA, Cabré X (1995) Fully Nonlinear Elliptic Equations, Providence RI: American Mathematical Society.
    [13] Caffarelli LA, Li Y, Nirenberg L (2009) Some remarks on singular solutions of nonlinear elliptic equations. J Fixed Point Theory Appl 5: 353-395.
    [14] Caffarelli LA, Li Y, Nirenberg L (2012) Some remarks on singular solutions of nonlinear elliptic equations. Ⅱ: symmetry and monotonicity via moving planes, In: Advances in Geometric Analysis, Somerville: International Press, 97-105.
    [15] Caffarelli LA, Li Y, Nirenberg L (2013) Some remarks on singular solutions of nonlinear elliptic equations. Ⅲ: viscosity solutions, including parabolic operators. Commun Pure Appl Math 66: 109-143.
    [16] Capuzzo Dolcetta I, Leoni F, Vitolo A (2005) The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains. Commun Part Diff Eq 30: 1863-1881.
    [17] Capuzzo Dolcetta I, Leoni F, Vitolo A (2014) Entire subsolutions of fully nonlinear degenerate elliptic equations. Bull Inst Math Acad Sin 9: 147-161.
    [18] Capuzzo Dolcetta I, Leoni F, Vitolo A (2016) On the inequality F(x, D2u) ≥ f (u)+g(u)|Du|q. Math Ann 365: 423-448.
    [19] Capuzzo Dolcetta I, Vitolo A (2007) A qualitative Phragmèn-Lindelöf theorem for fully nonlinear elliptic equations. J Differ Equations 243: 578-592.
    [20] Capuzzo Dolcetta I, Vitolo A (2018) The weak maximum principle for degenerate elliptic operators in unbounded domains. Int Math Res Notices 2018: 412-431.
    [21] Capuzzo Dolcetta I, Vitolo A (2019) Directional ellipticity on special domains: weak maximum and Phragmén-Lindelöf principles. Nonlinear Anal 184: 69-82.
    [22] Crandall MG (1997) Viscosity solutions: A primer, In: Viscosity solutions and applications, Berlin: Springer.
    [23] Crandall MG, Ishii H, Lions PL (1992) User's guide to viscosity solutions of second order partial differential equations. B Am Math Soc 27: 1-67.
    [24] Crandall MG, Rabinowitz PH, Tartar L (1977) On a Dirichlet problem with a singular nonlinearity. Commun Part Diff Eq 2: 193-222.
    [25] Ferrari F, Vitolo A (2020) Regularity properties for a class of non-uniformly elliptic Isaacs operators. Adv Nonlinear Stud 20: 213-241.
    [26] Galise G, Vitolo A (2017) Removable singularities for degenerate elliptic Pucci operators. Adv Differential Equ 22: 77-100.
    [27] Giarrusso E, Porru G (2006) Problems for elliptic singular equations with a gradient term. Nonlinear Anal 65: 107-128.
    [28] Harvey FR, Lawson HB Jr (2009) Dirichlet duality and the Nonlinear Dirichlet problem. Commun Pure Appl Math 62: 396-443.
    [29] Ishii H, Lions PL (1990) Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J Differ Equations 83: 26-78.
    [30] Koike S (2004) A Beginners Guide to the Theory of Viscosity Solutions. Tokyo: Math Soc Japan.
    [31] Lazer AC, McKenna PJ (1991) On a singular nonlinear elliptic boundary value problem. P Am Math Soc 111: 721-730.
    [32] Nachman A, Callegari A (1986) A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J Appl Math 28: 271-281.
    [33] Porru G, Vitolo A (2007) Problems for elliptic singular equations with a quadratic gradient term. J Math Anal Appl 334: 467-486.
    [34] Oberman AM, Silvestre L (2011) The Dirichlet problem for the convex envelope. T Am Math Soc 11: 5871-5886.
    [35] Sha JP (1986) p-convex Riemannian manifolds. Invent Math 83: 437-447.
    [36] Sha JP (1987) Handlebodies and p-convexity. J Diff Geom 25: 353-361.
    [37] Vitolo A (2003) On the maximum principle for complete second-order elliptic operators in general domains. J Differ Equations 194: 166-184.
    [38] Vitolo A (2004) On the Phragmén-Lindelöf principle for second-order elliptic equations. J Math Anal Appl 300: 244-259.
    [39] Vitolo A (2007) A note on the maximum principle for second-order elliptic equations in general domains. Acta Math Sin 23: 1955-1966.
    [40] Vitolo A (2018) Removable singularities for degenerate elliptic equations without conditions on the growth of the solution. T Am Math Soc 370: 2679-2705.
    [41] Vitolo A (2019) Maximum principles for viscosity solutions of weakly elliptic equations. Bruno Pini Mathematical Analysis Seminar 10: 110-136.
    [42] Wu H (1987) Manifolds of partially positive curvature. Indiana U Math J 36: 525-548.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(3475) PDF downloads(611) Cited by(3)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog