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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
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References:

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