# zbMATH — the first resource for mathematics

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:
##### References:
 [1] Di Benedetto, E. andTrudinger, N. S., Harnack inequalities for quasi-minima of variational integrals,Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 295–308. [2] Gilbarg, D. andTrudinger, N. S.,Elliptic partial differential equations of second order, Second Edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. · Zbl 0562.35001 [3] Granlund, S., Lindqvist, P. andMartio, O., Conformally invariant variational integrals,Trans. Amer. Math. Soc. 277 (1983), 43–73. · Zbl 0518.30024 [4] Granlund, S., Lindqvist, P. andMartio, O., Note on thePWB-method in the non-linear case.Pacific J. Math. 125 (1986), 381–395. · Zbl 0633.31004 [5] Hedberg, L. I., Removable singularities and condenser capacities,Ark. Mat. 12 (1974), 181–201. · Zbl 0297.30017 [6] Hedberg, L. I. andWolff, Th. H., Thin sets in nonlinear potential theory,Ann. Inst. Fourier (Grenoble) 33:4 (1983), 161–187. · Zbl 0508.31008 [7] Iwaniec, T., Some aspects of partial differential equations and quasiregular mappingsProceedings of the International Congress of Mathematicians, Warszawa 1983, 1193–1208. [8] Kilpeläinen, T., On the uniqueness of the solutions of degenerate elliptic equations (to appear). · Zbl 1008.31005 [9] Kinderlehrer, D. andStampacchia, G.,An introduction to variational inequalities and their applications, Academic Press, New York-London-Toronto-Sydney-San Francisco, 1980. · Zbl 0457.35001 [10] Lehtola, P., An axiomatic approach to non-linear potential theory,Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 62 (1986), 1–40. · Zbl 0695.31014 [11] Leray, J. andLions, J.-L., Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder,Bull. Soc. Math. France 93 (1965), 97–107. [12] Lindqvist, P., On the definition and properties ofp-superharmonic functions,J. Reine Angew. Math. 365 (1986), 67–79. · Zbl 0572.31004 [13] Lindqvist, P. andMartio, O., Two theorems of N. Wiener for solutions of quasilinear elliptic equations,Acta Math. 155 (1985), 153–171. · Zbl 0607.35042 [14] Martio, O., Counterexamples for unique continuation (to appear). · Zbl 0653.30013 [15] Maz’ya, V. G., On the continuity at a boundary point of solutions of quasi-linear elliptic equations,Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13 (1970), 42–55, (Russian).Vestnik Leningrad Univ. Math. 3 (1976), 225–242, (English translation). [16] Maz’ya, V. G.,Sobolev spaces, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. [17] Meyers, N. G. andElcrat, A., Some results on regularity for solutions of non-linear elliptic systems and quasiregular functions,Duke Math. J. 42 (1975), 121–136. · Zbl 0347.35039 [18] Michael, J. H. andZiemer, W. P., Interior regularity for solutions to obstacle problems,Nonlinear Anal. 10 (1986), 1427–1448. · Zbl 0603.49006 [19] Reshetnyak, Yu. G., The concept of capacity in the theory of functions with generalized derivatives,Sibirsk. Mat. Zh. 10 (1969), 1109–1138, (Russian). [20] Serrin, J., Local behavior of solutions of quasi-linear equations,Acta Math. 111 (1964), 247–302. · Zbl 0128.09101 [21] Trudinger, N. S., On Harnack type inequalities and their application to quasilinear elliptic equations,Comm. Pure Appl. Math. 20 (1967), 721–747. · Zbl 0153.42703 [22] Ziemer, W. P., Mean values of solutions of elliptic and parabolic equations,Trans. Amer. Math. Soc. 279 (1983), 555–568. · Zbl 0552.35008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.