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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

This contribution is part of the Special Issue: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni
Link: www.aimspress.com/mine/article/5754/special-articles

Special Issues: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta

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.
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Keywords degenerate elliptic operators; singular equations; Dirichlet problem; viscosity solutions

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

References

  • 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.    

 

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