Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points. (English) Zbl 0851.35047

Malý, Jan

Commentat. Math. Univ. Carol. 37, No. 1, 23-42 (1996).

The author studies quasilinear elliptic equations of type \(- \text{div } A(x, u, \nabla u)+ B(x, u, \nabla u)= \mu\), where \(A\) and \(B\) are Carathéodory functions with growth exponent \(p\in (1, \infty)\) and \(\mu\in (W^{1, p}_0(\Omega))^*\) is a nonnegative Radon measure. Let \(u\) be a weak solution of such equation; the author estimates \(u(x)\) at an interior point \(x\in \Omega\) or an irregular point \(x\in \partial \Omega\), in terms of a norm of \(u\), a nonlinear potential of \(\mu\) and the Wiener integral of \(\mathbb{R}^n\backslash \Omega\). This result quantifies the result on necessity of the Wiener criterion.

Reviewer: Z.G.Sheftel’ (Clausthal-Zellerfeld)

MSC:

35J70	Degenerate elliptic equations
35R05	PDEs with low regular coefficients and/or low regular data
35J67	Boundary values of solutions to elliptic equations and elliptic systems
35J65	Nonlinear boundary value problems for linear elliptic equations
35B45	A priori estimates in context of PDEs

Keywords:

measure data; Wiener integral; Wiener criterion