### Mathematics in Engineering

2021, Issue 3: 1-9. doi: 10.3934/mine.2021026
Research article Special Issues

# Weak maximum principle for Dirichlet problems with convection or drift terms

• Received: 08 January 2020 Accepted: 21 February 2020 Published: 24 July 2020
• In this paper, dedicated to Italo Capuzzo Dolcetta, a maximum principle for some linear boundary value problems with lower order terms of order one is proved: the aim of this paper is the proof that the solutions can be zero at most in a zero measure set, if we assume that the data are greater or equal than zero (but not identically zero).

Citation: Lucio Boccardo. Weak maximum principle for Dirichlet problems with convection or drift terms[J]. Mathematics in Engineering, 2021, 3(3): 1-9. doi: 10.3934/mine.2021026

### Related Papers:

• In this paper, dedicated to Italo Capuzzo Dolcetta, a maximum principle for some linear boundary value problems with lower order terms of order one is proved: the aim of this paper is the proof that the solutions can be zero at most in a zero measure set, if we assume that the data are greater or equal than zero (but not identically zero).

 [1] Berestycki H, Capuzzo Dolcetta I, Porretta A, et al. (2015) Maximum principle and generalized principal eigenvalue for degenerate elliptic operators. J Math Pure Appl 103: 1276-1293. [2] Capuzzo Dolcetta I, Vitolo A (2019) Directional ellipticity on special domains: Weak maximum and Phragmen-Lindelöf principles. Nonlinear Anal 184: 69-82. [3] Capuzzo Dolcetta I, Vitolo A (2018) The weak maximum principle for degenerate elliptic operators in unbounded domains. Int Math Res Not 2018: 412-431. [4] Capuzzo Dolcetta I (2008) On the weak maximum principle for fully nonlinear elliptic pde's in general unbounded domains. Lect Notes Semin Interdiscip Mat 7: 81-92. [5] Capuzzo Dolcetta I, Vitolo A (2007) On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains. Matematiche 62: 69-91. [6] 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. [7] Boccardo L (2009) Some developments on Dirichlet problems with discontinuous coefficients. Boll Unione Matematica Italiana 2: 285-297. [8] Boccardo L (2015) Dirichlet problems with singular convection terms and applications. J Differ Equations 258: 2290-2314. [9] Boccardo L (2019) Stampacchia-Calderon-Zygmund theory for linear elliptic equations with discontinuous coefficients and singular drift. ESAIM: COCV 25:1-13. [10] Boccardo L (2019) Finite energy weak solutions to some Dirichlet problems with very singular drift. Differ Integral Equ 32: 409-422. [11] Boccardo L, Buccheri S, Cirmi GR (2018) Two linear noncoercive Dirichlet problems in duality. Milan J Math 86: 97-104. [12] Brezis H, Ponce AC (2003) Remarks on the strong maximum principle. Differ Integral Equ 16: 1-12. [13] Orsina L, Ponce AC (2016) Strong maximum principle for Schrödinger operators with singular potential. Ann I H Poincare Anal Non Lineaire 33: 477-493. [14] Porretta A (2019) Elliptic equations with first order terms - Rough notes of the course at Alexandria, Ecole Cimpa 1/2009. Available from: http://archive.schools.cimpa.info/anciensite/NotesCours/PDF/2009/Alexandrie_Porretta.pdf. [15] Stampacchia G (1965) Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann I Fourier 15: 189-258. [16] Vazquez JL (1984) A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim 12: 191-202.
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