Comptes Rendus
no. 1
Partial differential equations
Existence of a renormalized solution to the quasilinear Riccati-type equation in Lorentz spaces
[Existence d'une solution renormalisée des équations quasi linéaires de type Riccati dans les espaces de Lorentz]

Présenté par : the Editorial Board

Minh-Phuong Tran1 ; Thanh-Nhan Nguyen2
1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
2 Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Viet Nam
Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 59-65.

Nous prouvons dans cet article l'existence d'une solution renormalisée des équations quasi linéaires de type Riccati avec des données de mesure d'intégrabilité faibles sur les espaces de Lorentz. Le résultat est établi dans les cas réguliers et singuliers. La preuve est basée sur les estimations du gradient pour une solution d'une classe d'équations quasi linéaires elliptiques.

In this paper, we prove the existence of a renormalized solution for the quasilinear Riccati-type equation with low integrability-measure data in Lorentz spaces. The result is established in both regular and singular cases. Our proof is based on the gradient estimates for a solution to a class of quasilinear elliptic equations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.12.001
Minh-Phuong Tran 1 ; Thanh-Nhan Nguyen 2

1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
2 Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Viet Nam 
@article{CRMATH_2019__357_1_59_0,
     author = {Minh-Phuong Tran and Thanh-Nhan Nguyen},
     title = {Existence of a renormalized solution to the quasilinear {Riccati-type} equation in {Lorentz} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {59--65},
     publisher = {Elsevier},
     volume = {357},
     number = {1},
     year = {2019},
     doi = {10.1016/j.crma.2018.12.001},
     language = {en},
}
TY  - JOUR
AU  - Minh-Phuong Tran
AU  - Thanh-Nhan Nguyen
TI  - Existence of a renormalized solution to the quasilinear Riccati-type equation in Lorentz spaces
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 59
EP  - 65
VL  - 357
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2018.12.001
LA  - en
ID  - CRMATH_2019__357_1_59_0
ER  - 
%0 Journal Article
%A Minh-Phuong Tran
%A Thanh-Nhan Nguyen
%T Existence of a renormalized solution to the quasilinear Riccati-type equation in Lorentz spaces
%J Comptes Rendus. Mathématique
%D 2019
%P 59-65
%V 357
%N 1
%I Elsevier
%R 10.1016/j.crma.2018.12.001
%G en
%F CRMATH_2019__357_1_59_0
Minh-Phuong Tran; Thanh-Nhan Nguyen. Existence of a renormalized solution to the quasilinear Riccati-type equation in Lorentz spaces. Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 59-65. doi : 10.1016/j.crma.2018.12.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.12.001/

[1] K. Adimurthi; N.C. Phuc Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers, Calc. Var. Partial Differ. Equ., Volume 57 (2018), p. 74

[2] G. Dal Maso; F. Murat; L. Orsina; A. Prignet Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa (5) (IV), Volume 28 (1999), pp. 741-808

[3] F. Duzaar; G. Mingione Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., Volume 259 (2010), pp. 2961-2998

[4] F. Duzaar; G. Mingione Gradient estimates via non-linear potentials, Amer. J. Math., Volume 133 (2011), pp. 1093-1149

[5] L. Grafakos Classical and Modern Fourier Analysis, Pearson/Prentice Hall, 2004

[6] M. Kardar; G. Parisi; Y.-C. Zhang Dynamic scaling of growing interfaces, Phys. Rev. Lett., Volume 56 (1986), pp. 889-892

[7] J. Krug; H. Spohn Universality classes for deterministic surface growth, Phys. Rev. A (3), Volume 38 (1988), pp. 4271-4283

[8] T. Kuusi; G. Mingione Guide to nonlinear potential estimates, Bull. Math. Sci., Volume 4 (2014), pp. 1-82

[9] T. Kuusi; G. Mingione Vectorial nonlinear potential theory, J. Eur. Math. Soc., Volume 20 (2018), pp. 929-1004

[10] O. Martio Quasilinear Riccati type equations and quasiminimizers, Adv. Nonlinear Stud., Volume 11 (2011), pp. 473-482

[11] T. Mengesha; N.C. Phuc Quasilinear Riccati-type equations with distributional data in Morrey space framework, J. Differ. Equ., Volume 260 (2016), pp. 5421-5449

[12] G. Mingione The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 6 (2007), pp. 195-261

[13] G. Mingione Gradient estimates below the duality exponent, Math. Ann., Volume 346 (2010), pp. 571-627

[14] Q.-H. Nguyen Potential estimates and quasilinear parabolic equations with measure data | arXiv

[15] Q.-H. Nguyen Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati-type parabolic equations with distributional data, Calc. Var. Partial Differ. Equ., Volume 54 (2015), pp. 3927-3948

[16] Q.-H. Nguyen; N.C. Phuc Good-λ and Muckenhoupt–Wheeden type bounds, with applications to quasilinear elliptic equations with gradient power source terms and measure data, Math. Ann. (2018) (in press) | DOI

[17] Q.-H. Nguyen Gradient estimates for singular quasilinear elliptic equations with measure data | arXiv

[18] N.C. Phuc Quasilinear Riccati-type equations with super-critical exponents, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 1958-1981

[19] N.C. Phuc Global integral gradient bounds for quasilinear equations below or near the natural exponent, Ark. Mat., Volume 52 (2014), pp. 329-354

[20] N.C. Phuc Morrey global bounds and quasilinear Riccati-type equations below the natural exponent, J. Math. Pures Appl., Volume 102 (2014), pp. 99-123

[21] N.C. Phuc Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati-type equations, Adv. Math., Volume 250 (2014), pp. 387-419

[22] M.-P. Tran Good-λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal., Volume 178 (2019), pp. 266-281

Cité par Sources :

Commentaires - Politique