##
**Potential theory for supersolutions of degenerate elliptic equations.**
*(English)*
Zbl 0688.31005

A nonlinear potential theory for the supersolutions of equation
\[
(1)\quad -div A(x,\nabla u)=0
\]
(or more generally, for A-superharmonic functions) is discussed. Here \(A(x,h)\cdot h\approx | h|^ p\), \(1<p<\infty\); and equation (1) is a perturbation of the p-Laplacian \(- div(| \nabla u|^{p-2}| \nabla u)=0\) in \({\mathbb{R}}^ n\), \(n\geq 2.\)

Basic properties of A-harmonic functions (i.e., continuous weak solutions of equation (1)) and A-superharmonic functions (defined by the aid of comparison with A-harmonic functions) are derived previously by the author and J. Heinonen [Ark. Mat. 26, No.1, 87-105 (1988; Zbl 0652.31006); Math. Scand. 63, No.1, 136-150 (1988)]. In the present paper the Perron method to solve the generalized Dirichlet problem is studied: a characterization of regular boundary points is established by means of barriers, and it is shown that continuous boundary functions are resolutive (i.e. the upper and the lower Perron solutions coincide and are A-harmonic). A special attention is paid for unbounded open sets. Moreover, it is established that the sets of p-capacity zero are isolated in the fine topology of A-superharmonic functions by showing that if \(E\subset {\mathbb{R}}^ n\) is a set of p-capacity zero and \(x_ 0\not\in E\), then there is an A-superharmonic function u in \({\mathbb{R}}^ n\) such that \(u=\infty\) on E such that \(u(x_ 0)<\infty\). The fine topology in this nonlinear potential theory is studied further by J. Heinonen and O. Martio with the present author [Ann. Inst. Fourier 39, No.2, 293-318 (1989; Zbl 0659.35038)].

The principal tool is an obstacle problem and its counterpart in potential theory: the balayage of a function, the least A-superharmonic majorant of a given function.

Basic properties of A-harmonic functions (i.e., continuous weak solutions of equation (1)) and A-superharmonic functions (defined by the aid of comparison with A-harmonic functions) are derived previously by the author and J. Heinonen [Ark. Mat. 26, No.1, 87-105 (1988; Zbl 0652.31006); Math. Scand. 63, No.1, 136-150 (1988)]. In the present paper the Perron method to solve the generalized Dirichlet problem is studied: a characterization of regular boundary points is established by means of barriers, and it is shown that continuous boundary functions are resolutive (i.e. the upper and the lower Perron solutions coincide and are A-harmonic). A special attention is paid for unbounded open sets. Moreover, it is established that the sets of p-capacity zero are isolated in the fine topology of A-superharmonic functions by showing that if \(E\subset {\mathbb{R}}^ n\) is a set of p-capacity zero and \(x_ 0\not\in E\), then there is an A-superharmonic function u in \({\mathbb{R}}^ n\) such that \(u=\infty\) on E such that \(u(x_ 0)<\infty\). The fine topology in this nonlinear potential theory is studied further by J. Heinonen and O. Martio with the present author [Ann. Inst. Fourier 39, No.2, 293-318 (1989; Zbl 0659.35038)].

The principal tool is an obstacle problem and its counterpart in potential theory: the balayage of a function, the least A-superharmonic majorant of a given function.

Reviewer: T.Kilpeläinen

### MSC:

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

35J70 | Degenerate elliptic equations |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

31C15 | Potentials and capacities on other spaces |

31C99 | Generalizations of potential theory |