×

Pucci’s conjecture and the Alexandrov inequality for elliptic PDEs in the plane. (English) Zbl 1147.35021

Summary: The inequality of Alexandrov, Bakel’man and Pucci is a basic tool in the theory of linear elliptic partial differential equations (PDEs) which are not in divergence form as well as in the more general theory of nonlinear elliptic PDEs. Here, in two dimensions, we prove the sharp form of the maximum principle as conjectured by C. Pucci [Ann. Mat. Pura Appl., IV. Ser. 72, 141–170 (1966; Zbl 0154.12402)], give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting. These results make use of the substantial recent advances in the planar theory of quasiconformal mappings.

MSC:

35J15 Second-order elliptic equations
35B50 Maximum principles in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0154.12402
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1007/BF02392634 · Zbl 0041.20301
[2] Alexandrov A. D., Vestnik Leningrad University 21 pp 5– (1966)
[3] DOI: 10.1007/BF02392568 · Zbl 0815.30015
[4] DOI: 10.1215/S0012-7094-01-10713-8 · Zbl 1009.30015
[5] DOI: 10.1002/cpa.3160480504 · Zbl 0828.35017
[6] Ca{\currency}arelli L. A., Amer. Math. Soc. Colloqu. Publ. pp 43– (1995)
[7] David G., Ann. Acad. Sci. Fenn. Ser. A I Math. 13 pp 25– (1988) · Zbl 0619.30024
[8] DOI: 10.1016/S0294-1449(03)00014-3 · Zbl 1029.30012
[9] Goldstein V. M., Sb. Mat. Z. 17 pp 515– (1976)
[10] Iwaniec T., J. London Math. Soc. 67 pp 123– (2) · Zbl 1047.30010
[11] DOI: 10.1007/s002220100130 · Zbl 1006.30016
[12] Iwaniec T., Ann. Acad. Sci. Fenn. Ser. A I Math. 10 pp 267– (1985) · Zbl 0588.30023
[13] DOI: 10.2307/2160025 · Zbl 0784.30015
[14] Moscariello G., Math. Japonica 40 pp 323– (1992)
[15] DOI: 10.1090/S0273-0979-1989-15818-7 · Zbl 0689.49006
[16] DOI: 10.1215/S0012-9074-02-11223-X · Zbl 1025.30018
[17] Pucci C., Ann. Mat. Pura. Appl. 74 pp 15– (4) · Zbl 0144.35801
[18] Pucci C., Ann. Mat. Pura. Appl. 72 pp 141– (4)
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.