Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

  • Guangwei Du

    Northwestern Polytechnical University, Xi'an, Shaanxi, China
  • Pengcheng Niu

    Northwestern Polytechnical University, Xi'an, Shaanxi, China
  • Junqiang Han

    Northwestern Polytechnical University, Xi'an, Shaanxi, China
Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups cover
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Abstract

Let {X1,,Xm}\{X_1 ,\ldots ,X_m \} be a basis of the space of horizontal vector fields on the Carnot group G=(RN,)(m<N)\mathbb{G}=(\mathbb{R}^N, \circ)(m < N). We establish regularity results for solutions to the following quasilinear degenerate elliptic obstacle problem \begin{align*} &\int\limits_\Omega\! {\langle \langle AXu,Xu\rangle ^{\frac{p-2}{2}}AXu,X(v-u)\rangle } dx\\ &\quad\geq\int\limits_\Omega\! {B(x,u,Xu)(v-u)}dx +\int\limits_\Omega \!{\langle f(x),X(v-u)\rangle } dx,\quad \text{for all } v\in \mathcal{K}_\psi^\theta (\Omega), \end{align*} where A=(aij(x))m×mA=(a_{ij}(x))_{m\times m} is a symmetric positive-definite matrix with measurable coefficients, pp is close to 2, Kψθ(Ω)={vHW1,p(Ω) ⁣:vψ  \mboxa.e.  in  Ω,vθHW01,p(Ω)}\mathcal{K}_\psi ^\theta(\Omega )=\{v\in HW^{1,p}(\Omega)\colon v\geq \psi \;\mbox{a.e.}\;{\rm in}\;\Omega ,v-\theta\in HW_0^{1,p} (\Omega )\}, ψ\psi is a given obstacle function, θ\theta is a boundary value function with θψ\theta \geq \psi . We first prove the CX0,αC_X^{0,\alpha} regularity of solutions provided that the coefficients of AA are of vanishing mean oscillation (VMO). Then the CX1,αC_X^{1,\alpha } regularity of solutions is obtained if the coefficients belong to the class \mboxBMOω\mbox{BMO}_\omega which is a proper subset of VMO.

Cite this article

Guangwei Du, Pengcheng Niu, Junqiang Han, Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups. Rend. Sem. Mat. Univ. Padova 141 (2019), pp. 65–105

DOI 10.4171/RSMUP/15