Positive solutions of elliptic equations in nondivergence form and their adjoints. (English) Zbl 0557.35033

Consider a uniformly elliptic second order operator L with only continuous coefficients in the principal part which is not in divergence form. The main result of the paper under review is a ”comparison theorem” for positive L-harmonic functions u in a Lipschitz domain \(\Omega\) stating (roughly) that these may approach zero on a portion of \(\partial \Omega\) only at a ”fixed rate” independent of u. (Similar results - which have applications in potential theory - for more regular operators have been obtained by Hunt, Wheeden and Ancona.) Using standard tools of the theory of L-harmonic functions (maximum principle, Harnack inequality etc.) the comparison theorem is shown to follow from an estimate for Green’s function G. Thus the second part of the paper is devoted to the study of G and of positive solutions for \(L^*v=O\). Due to possibly pathological behavior of these much care has to be exercised. One of the ideas is to work with ”normalized” functions \(\tilde v(\)y):\(= v(y)/G(x,y)\) instead of v.
Reviewer: N.Weck


35J15 Second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B50 Maximum principles in context of PDEs
31B25 Boundary behavior of harmonic functions in higher dimensions
Full Text: DOI


[1] Agmon, S., Douglis, A. andNirenberg, L., Estimates near the boundary for solutions of elliptic P.D.E. satisfying general boundary conditions,Comm. Pure Appl. Math. 12 (1959), 623–727. · Zbl 0093.10401
[2] Ancona, A., Principe de Harnack à la frontier et théorème de Fatou pour un opérateur elliptique dans un domaine Lipschitzien,Ann. Inst. Fourier (Grenoble) 28 (1978), 169–213. · Zbl 0377.31001
[3] Bauman, P., Properties of nonnegative solutions of second-order elliptic equations and their adjoints,Ph. D. thesis, University of Minnesota, Minneapolis, Minnesota (1982).
[4] Bauman, P., Equivalence of the Green’s functions for diffusion operators inR n : A counterexample, to appear inProc. Amer. Math. Soc.
[5] Bony, J. M., Principe du maximum dans les espaces de Sobolev,C. R. Acad. Sci. Paris Sér. A. 265 (1967), 333–336. · Zbl 0164.16803
[6] Coifman, R. andFefferman, C., Weighted norm inequalities for maximal functions and singular integrals,Studia Math. 51 (1974), 241–250. · Zbl 0291.44007
[7] Gilbarg, D. andSerrin, J., On isolated singularities of solutions of second order elliptic differential equations,J. Analyse Math. 4 (1955/56), 309–340. · Zbl 0071.09701
[8] Gilbarg, D. andTrudinger, N. S.,Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, New York (1977). · Zbl 0361.35003
[9] Hunt, R. A. andWheeden, R. L., On the boundary values of harmonic functions,Trans. Amer. Math. Soc.,132 (1968), 307–322. · Zbl 0159.40501
[10] Hunt, R. A. andWheeden, R. L., Positive harmonic functions on Lipschitz domains,Trans. Amer. Math. Soc. 147 (1970), 507–527. · Zbl 0193.39601
[11] Krylov, N. V. andSafanov, M. V., An estimate of the probability that a diffusion process hits a set of positive measure,Dokl. Akad. Nauk SSSR 245 (1979), 253–255. English translation,Soviet Math., Dokl. 20 (1979), 253–255. · Zbl 0459.60067
[12] Miller, K., Barriers on cones for uniformly elliptic operators,Ann. Mat. Pura Appl. 76 (1967), 93–105. · Zbl 0149.32101
[13] Miranda, C.,Partial Differential Equations of Elliptic Type, Springer-Verlag, New York (1970). · Zbl 0198.14101
[14] Pucci, C., Limitazioni per soluzioni di equazioni ellittiche,Ann. Mat. Pura Appl. 74 (1966), 15–30. · Zbl 0144.35801
[15] Sjögren, P., On the adjoint of an elliptic linear differential operator and its potential theory,Ark. Mat. 11 (1973), 153–165. · Zbl 0267.31011
[16] Stroock, D. W. andVaradhan, S. R. S., Diffusion processes with continuous coefficients, I and II,Comm. Pure Appl. Math. 22 (1969), 345–400 and 475–530. · Zbl 0167.43903
[17] Trudinger, N., Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations,Invent. Math. 61 (1980), 67–79. · Zbl 0453.35028
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.