On the boundary behavior of positive solutions of elliptic differential equations

A. A. Logunov

Algebra i analiz, Tome 27 (2015) no. 1, pp. 125-148.

Bibliographie

[1] Allen A. C., Kerr E., “The converse of Fatou's theorem”, J. London Math. Soc., 28 (1953), 80–89 | DOI | MR | Zbl

[2] Ancona A., “Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien”, Ann. Inst. Fourier (Grenoble), 28:4 (1978), 169–213 | DOI | MR | Zbl

[3] Beurling A., “A minimum principle for positive harmonic functions”, Ann. Acad. Sci. Fenn. Ser. AI, 372 (1965), 1–7 | MR

[4] Brossard J., Chevalier L., “Problème de Fatou ponctuel et dérivabilitè des mesures”, Acta Math., 164:1 (1990), 237–263 | DOI | MR | Zbl

[5] Carmona J. J., Donaire J. J., “The converse of Fatou's theorem for Zygmund measures”, Pacific J. Math., 191:2 (1999), 207–222 | DOI | MR | Zbl

[6] Dahlberg B., “A minimum principle for positive harmonic functions”, Proc. London Math. Soc. (3), 33:2 (1976), 238–250 | DOI | MR | Zbl

[7] Dahlberg B., “Estimates of harmonic measure”, Arch. Rational Mech. Anal., 65:3 (1977), 275–288 | DOI | MR | Zbl

[8] Dubtsov E. S., “Obratnaya teorema Fatu dlya gladkikh mer”, Zap. nauch. semin. POMI, 315, 2004, 90–95 | MR | Zbl

[9] Dubtsov E. S., “Proizvodnye regulyarnykh mer”, Algebra i analiz, 19:2 (2007), 86–104 | MR | Zbl

[10] Fatou P., “Séries trigonometriques et séries de Taylor”, Acta Math., 30:1 (1906), 335–400 | DOI | MR | Zbl

[11] Gehring F. W., “The Fatou theorem for functions harmonic in a half-space”, Proc. London Math. Soc. (3), 8 (1958), 149–160 | DOI | MR | Zbl

[12] Gehring F. W., “The Fatou theorem and its converse”, Trans. Amer. Math. Soc., 85 (1957), 106–121 | DOI | MR | Zbl

[13] Gruter M., Widman K.-O., “The Green function for uniformly elliptic equations”, Manuscripta Math., 37:3 (1982), 303–342 | DOI | MR

[14] Hardy G. H., Divergent series, Clarendon Press, Oxford, 1949 | MR | Zbl

[15] Hueber H., Sieveking M., “Uniform bounds for quotients of Green functions on $C^{1,1}$-domains”, Ann. Inst. Fourier (Grenoble), 32:1 (1982), 105–117 | DOI | MR | Zbl

[16] Hueber H., Sieveking M., “Continuous bounds for quotients of Green functions”, Arch. Rational Mech. Anal., 89:1 (1985), 57–82 | DOI | MR | Zbl

[17] Ifra A., Riahi L., “Estimates of Green functions and harmonic measures for elliptic operators with singular drift terms”, Publ. Mat., 49:1 (2005), 159–177 | DOI | MR | Zbl

[18] Kheifits A. I., “Pointwise Fatou theorem for generalized harmonic functions – normal boundary values”, Potential Anal., 3:4 (1994), 379–389 | DOI | MR | Zbl

[19] Loomis L. H., “The converse of the Fatou theorem for positive harmonic functions”, Trans. Amer. Math. Soc., 53 (1943), 239–250 | DOI | MR | Zbl

[20] Mazya V. G., “K teoreme Berlinga o printsipe minimuma dlya polozhitelnykh garmonicheskikh funktsii”, Zap. nauch. semin. LOMI, 30, 1972, 76–90 | MR | Zbl

[21] Pan Y., Wang M., “An application of the Hardy–Littlewood Tauberian theorem to harmonic expansion of a complex measure on the sphere”, Real Analysis Exchange, 35:2 (2009), 517–524 | MR

[22] Ramey W., Ullrich D., “On the behavior of harmonic functions near a boundary point”, Trans. Amer. Math. Soc., 305:1 (1988), 207–220 | DOI | MR | Zbl

[23] Rudin W., “Tauberian theorems for positive harmonic functions”, Nederl. Akad. Wetensch. Indag. Math., 40:3 (1978), 376–384 | DOI | MR

[24] Serrin J., “On the Harnack inequality for linear elliptic equations”, J. Analyse Math., 4:1 (1955/56), 292–308 | DOI | MR

[25] Widman K.-O., “Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations”, Math. Scand., 21 (1967), 17–37 | MR | Zbl

[26] Zhao Z. X., “Green function for Schrödinger operator and conditioned Feynman–Kac gauge”, J. Math. Anal. Appl., 116:2 (1986), 309–334 | DOI | MR | Zbl