Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups
Rendiconti del Seminario Matematico della Università di Padova, Tome 141 (2019), pp. 65-105.
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Let {X 1 ,...,X m } be a basis of the space of horizontal vector fields on the Carnot group 𝔾=( N ,)(m<N). We establish regularity results for solutions to the following quasilinear degenerate elliptic obstacle problem

Ω A X u , X u p - 2 2 A X u , X ( v - u ) d x Ω B ( x , u , X u ) ( v - u ) d x + Ω f ( x ) , X ( v - u ) d x , for all v 𝒦 ψ θ ( Ω ) ,
where A=(a ij (x)) m×m is a symmetric positive-definite matrix with measurable coefficients, p is close to 2, 𝒦 ψ θ (Ω)={vHW 1,p (Ω):vψa.e. in Ω,v-θHW 0 1,p (Ω)}, ψ is a given obstacle function, θ is a boundary value function with θψ. We first prove the C X 0,α regularity of solutions provided that the coefficients of A are of vanishing mean oscillation (VMO). Then the C X 1,α regularity of solutions is obtained if the coefficients belong to the class BMO ω which is a proper subset of VMO.

DOI : 10.4171/RSMUP/15
Classification : 35, 00
Mots clés : Carnot group, quasilinear degenerate elliptic obstacle problem, $C_X^{0,\alpha}$ regularity, $C_X^{1,\alpha}$ regularity
Du, Guangwei 1 ; Niu, Pengcheng 1 ; Han, Junqiang 1

1 Northwestern Polytechnical University, Xi'an, Shaanxi, China
@article{RSMUP_2019__141__65_0,
     author = {Du, Guangwei and Niu, Pengcheng and Han, Junqiang},
     title = {Regularity results for quasilinear degenerate elliptic obstacle problems in {Carnot} groups},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {65--105},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {141},
     year = {2019},
     doi = {10.4171/RSMUP/15},
     mrnumber = {3962821},
     zbl = {1422.35064},
     url = {http://www.numdam.org/articles/10.4171/RSMUP/15/}
}
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Du, Guangwei; Niu, Pengcheng; Han, Junqiang. Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups. Rendiconti del Seminario Matematico della Università di Padova, Tome 141 (2019), pp. 65-105. doi : 10.4171/RSMUP/15. http://www.numdam.org/articles/10.4171/RSMUP/15/

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