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A-superharmonic functions and supersolutions of degenerate elliptic equations. (English) Zbl 0652.31006
A function $$u\in loc W$$ $$1_ p(G)$$, where G is an open set in R n, $$n\geq 2$$, is a supersolution of the equation (*) $$\nabla \cdot A(x,\nabla u)=0$$ if for all nonnegative $$\phi \in C_ 0^{\infty}(G)$$ one has $$\int_{G}A(x,\nabla u)\cdot \nabla \phi dx\geq 0$$. A prototype for A here is the p-harmonic operator $$A(x,h)=| h|^{p-2}h$$ [see P. Lindqvist, J. Reine Angew. Math. 365, 67-79 (1986; Zbl 0572.31004)]. A lower semicontinuous function u is A-superharmonic if it satisfies the comparison principle: for each domain $$D\subset \subset G$$ and each function $$h\in C(\bar D)$$ which is a solution of (*) in D, the condition $$h\leq u$$ on $$\partial D$$ implies $$h\leq u$$ in D. The authors show that supersolutions of (*) can be redefined in a set of measure zero so that they are A-superharmonic and, conversely, if u is a locally bounded A- superharmonic function, then u belongs locally to $$W$$ $$1_ p$$ and is a supersolution of (*). The analysis depends on solving the obstacle problem with a continuous obstacle. A discussion of removable sets for A- superharmonic functions is also presented.
Reviewer: P.W.Schaefer

MSC:
 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 35J70 Degenerate elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Zbl 0572.31004
Full Text:
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