Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions

  • Tuoc Phan


This paper studies regularity estimates in Lebesgue spaces for gradients of weak solutions of a class of general quasilinear equations of p-Laplacian type in bounded domains with inhomogeneous conormal boundary conditions. In the considered class of equations the leading terms are vector-valued functions that are measurable in the x-variable and that depend in a nonlinear way on the solution and on its gradient. This class of equations consists of the well-known class of degenerate p-Laplace equations for \(p >1\). Under some sufficient conditions, we establish local interior, local boundary, and global \(W^{1,q}\)-regularity estimates for weak solutions with \(q>p\), assuming that the weak solutions are in the John–Nirenberg BMO space. The paper therefore improves available results because it removes the boundedness or continuity assumptions on solutions. Our results also unify and cover known results for equations in which the principals are only allowed to depend on x-variable and gradient of solution variable. More than that, this paper gives a method to treat non-homogeneous boundary value problems directly without using any form of translations that is sometimes complicated due to the nonlinearities.


Quasilinear elliptic equations Quasilinear p-Laplacian type equations Calderón–Zygmund regularity estimates Conormal boundary value problems 

Mathematics Subject Classification

35J92 35J62 35J66 35J60 35B45 



This research is supported in part by the Simons Foundation, Grant # 354889. The author also would like to thank anonymous referees for their valuable comments.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Tennessee, KnoxvilleKnoxvilleUSA

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