Please use this identifier to cite or link to this item: https://doi.org/10.1016/j.jde.2017.05.001
Title: Boundedness and regularity of solutions of degenerate elliptic partial differential equations
Authors: CHUA SENG KEE @ SAI SENG KEE 
Keywords: degenerate elliptic equations
hormander condition
Poincare inequality
Issue Date: 31-May-2017
Publisher: Elsevier
Citation: CHUA SENG KEE @ SAI SENG KEE (2017-05-31). Boundedness and regularity of solutions of degenerate elliptic partial differential equations. Journal of differential equations 263 (7) : 3714-3736. ScholarBank@NUS Repository. https://doi.org/10.1016/j.jde.2017.05.001
Abstract: We study quasilinear degenerate equations with their associated operators satisfying Sobolev and Poincaré inequalities. Our problems are similar to those in Sawyer and Wheeden (2006), but we consider more general integral equations on abstract Sobolev spaces. Indeed, we provide a unified approach via abstract setting. Our main tool is Moser iteration. We obtain boundedness (assuming a weighted Sobolev embedding) of weak solutions and also Harnack inequalities for nonnegative solutions (assuming further that a weighted Poincaré inequality holds for their logarithm and existence of a sequence of cutoff functions) and hence Hölder's continuity of solutions. Our assumption is weaker than the above mentioned paper and other existing papers even in the case where the setting is on Rn ; for example, we do not assume any doubling condition and we are able to deal with weighted estimates. Our results are new even when the leading term in the equations are vector fields that satisfy Hörmander's conditions.
Source Title: Journal of differential equations
URI: https://scholarbank.nus.edu.sg/handle/10635/171860
ISSN: 0022-0396
DOI: 10.1016/j.jde.2017.05.001
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