Polar sets for supersolutions of degenerate elliptic equations. (English) Zbl 0706.31015

The paper is devoted to the nonlinear potential theory connected with the equation \(\nabla \cdot A(x,\nabla u)=0\), where A: \({\mathbb{R}}^ n\times {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) is an elliptic operator with \(A(x,h)\cdot h\approx | h|^ p\), \(1<p<\infty\). Weak solutions of the above equation are called A-harmonic; they are always continuous. Analogously to the classical potential theory the authors define A-superharmonic functions, A-polar sets, the balayage of an A-superharmonic function and study these notions.
They prove a fundamental convergence theorem for A-superharmonic functions and the equivalence of the following statements: (i) E is A- polar; (ii) there is an open neighborhood G of E such that if u is a positive A-superharmonic function in G, then the balayage \(\hat R^ u_ E\) vanishes identically in G; (iii) E is of (outer) p-capacity zero; (iv) there is a nonnegative lower semicontinuous function w in \(W^ 1_ p({\mathbb{R}}^ n)\) such that \(E\subset w^{-1}(\infty)\).
Reviewer: R.Semerdjieva


31C45 Other generalizations (nonlinear potential theory, etc.)
35J70 Degenerate elliptic equations
31C15 Potentials and capacities on other spaces
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