Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach. (English) Zbl 0685.35046

Franchi, B.; Serapioni, R.

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 4, 527-568 (1987).

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.

Reviewer: G.F.Roach

MSC:

35J70	Degenerate elliptic equations
35B45	A priori estimates in context of PDEs
35D10	Regularity of generalized solutions of PDE (MSC2000)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

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

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