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Zbl 0685.35046
Franchi, B.; Serapioni, R.
Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No.4, 527-568 (1987). ISSN 0391-173X

The authors extend to a class of strongly degenerate, second order, elliptic operators the Harnack inequality and the Hölder continuity of weak solutions. Reference is made to previous work in this general direction and it is pointed out that two of the main, and independent, approaches can be unified in the setting of homogeneous metric spaces. This unified approach enables the proposed extensions to be made. The proofs of the main results are obtained by adjusting the classical Moser technique to the geometry of the homogeneous space. An essential tool is a weighted Sobolev embedding theorem which is obtained by using a representation formula for a function u closely fitting the geometry of the operator and substituting the usual representation of u as a fractional integral of its gradient. The technical difficulties are eased by some well chosen examples.
[G.F.Roach]

MSC 2000:: *35J70 Elliptic equations of degenerate type
35B45 A priori estimates
35D10 Regularity of generalized solutions of PDE
46E35 Sobolev spaces and generalizations

Keywords: Harnack inequality; Hölder continuity; Moser technique; weighted Sobolev embedding

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