On certain degenerate and singular elliptic PDEs I: Nondivergence form operators with unbounded drifts and applications to subelliptic equations |
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Authors: | Diego Maldonado |
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Affiliation: | Kansas State University, Department of Mathematics, 138 Cardwell Hall, Manhattan, KS 66506, USA |
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Abstract: | We prove a Harnack inequality for nonnegative strong solutions to degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts. Our main theorem extends the Harnack inequality for the linearized Monge–Ampère equation due to Caffarelli and Gutiérrez and it is related, although under different hypotheses, to a recent work by N.Q. Le.Since our results are shown to apply to the convex functions with and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge–Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot–Carathéodory metric in the study of certain subelliptic PDEs. |
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Keywords: | primary 35J70 35J96 secondary 35J75 31E05 Degenerate and singular elliptic PDEs Linearized Monge–Ampère operator Grushin and subelliptic operators |
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