A boundary estimate for nonlinear equations with discontinuous coefficients. (English) Zbl 1161.35394

Kinnunen, Juha; Zhou, Shulin

Differ. Integral Equ. 14, No. 4, 475-492 (2001).

This paper considers the following \(p\)-Laplacian type equation: \(\text{div} \bigl((ADu \cdot Du) ^{(p-2)/2}ADu\bigr) = \text{div} \bigl(| F| ^{p-2} F\bigr)\), where \(A\) is a symmetric \(n\times n\) matrix with measurable coefficients satisfying a uniform ellipticity condition and with “vanishing mean oscillation” (VMO), and \(F\) is \(q\)-integrable for some \(q>p\). The authors establish boundary and global \(W^{1,q}\)-estimates for weak solutions. This is a continuation of an earlier work by the same authors [Commun. Partial Differ. Equations 24, 2043–2068 (1999; Zbl 0941.35026)], where interior \(q\)-integrability for the gradient of weak solutions is derived.

Reviewer: Ning Su (Beijing)

MSC:

35J60	Nonlinear elliptic equations
35B45	A priori estimates in context of PDEs

Keywords:

\(p\)-Laplacian type equation; integral estimate; maximal function

Citations:

Zbl 0941.35026