Abstract
Let X={X_{1} ,ldots ,X_{m} } be a system of smooth real vector fields satisfying Hörmander’s rank condition. We consider the interior regularity of weak solutions to an obstacle problem associated with the nonhomogeneous nondiagonal quasilinear degenerate elliptic system \t\t\tXα∗(Aijαβ(x,u)Xβuj)=Bi(x,u,Xu)+Xα∗giα(x,u,Xu).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$X_{\\alpha }^{\\ast } \\bigl( {A_{ij}^{\\alpha \\beta } (x,u)X_{\\beta }u ^{j}} \\bigr)= B_{i}(x,u,Xu)+X_{\\alpha }^{\\ast }g_{i}^{\\alpha }(x,u,Xu). $$\\end{document} After proving the higher integrability and a Campanato type estimate for the weak solutions to the obstacle problem for the homogeneous nondiagonal quasilinear degenerate elliptic system, the interior Morrey regularity and Hölder continuity of weak solutions to the obstacle problem for the nonhomogeneous system are obtained.
Highlights
Using the techniques that appeared in these papers, the local Morrey regularity and Hölder continuity of weak solutions to the obstacle problems associated with elliptic equations with constant coefficients or continuous coefficients have been obtained in [4, 13, 14], and [15]
Let us recall that a function u ∈ SX1,loc(Ω, RN ) is called a weak solution to (3.1) if Aαijβ (x, u)Xβ ujXαφi dx = 0
In order to prove the higher integrability for gradients of weak solutions to the Kθψ obstacle problem for (3.1), we need the Gehring lemma on the metric measure space (Y, d, μ), where d is a metric and μ is a doubling measure
Summary
Using the techniques that appeared in these papers, the local Morrey regularity and Hölder continuity of weak solutions to the obstacle problems associated with elliptic equations with constant coefficients or continuous coefficients have been obtained in [4, 13, 14], and [15]. Let u ∈ SX1 ,loc(Ω, RN ) be a weak solution to the Kθψ -obstacle problem for system (1.1), Xu ∈ L2X,,λloc(Ω, RmN ). 3, we first prove the higher integrability for gradients of weak solutions to the Kθψ -obstacle problem for the homogeneous system (see (3.1) below) by constructing suitable test functions and using the Gehring lemma on the metric measure space.
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