Mean values of subsolutions of elliptic and parabolic equations

Ziemer, William P.

doi:10.1090/S0002-9947-1983-0709568-3

Mean values of subsolutions of elliptic and parabolic equations
HTML articles powered by AMS MathViewer

by William P. Ziemer

Trans. Amer. Math. Soc. 279 (1983), 555-568

DOI: https://doi.org/10.1090/S0002-9947-1983-0709568-3

PDF | Request permission

Abstract:

Integral averages of weak subsolutions (and supersolutions) in ${R^n}$ of quasilinear elliptic and parabolic equations are investigated. The important feature is that these integral averages are defined in terms of measures that reflect interesting geometric phenomena. Harnack type inequalities are established in terms of these integral averages.

References

David R. Adams and Norman G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972/73), 169–197. MR 316724, DOI 10.1512/iumj.1972.22.22015
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Wendell H. Fleming, Functions whose partial derivatives are measures, Illinois J. Math. 4 (1960), 452–478. MR 130338
Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. MR 435361, DOI 10.1512/iumj.1972.22.22013
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
Ronald Gariepy and William P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 67 (1977), no. 1, 25–39. MR 492836, DOI 10.1007/BF00280825
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
Norman G. Meyers, Integral inequalities of Poincaré and Wirtinger type, Arch. Rational Mech. Anal. 68 (1978), no. 2, 113–120. MR 493308, DOI 10.1007/BF00281405
Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
Norman G. Meyers and William P. Ziemer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math. 99 (1977), no. 6, 1345–1360. MR 507433, DOI 10.2307/2374028

Local behavior of solutions of quasilinear elliptic equations

111

Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406
Neil S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. MR 226168, DOI 10.1002/cpa.3160210302
William P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 271 (1982), no. 2, 733–748. MR 654859, DOI 10.1090/S0002-9947-1982-0654859-7
William P. Ziemer, Behavior at the boundary of solutions of quasilinear parabolic equations, J. Differential Equations 35 (1980), no. 3, 291–305. MR 563383, DOI 10.1016/0022-0396(80)90030-3