Maximum Principles and Applications: for a Class of Degenerate Elliptic Linear Operators - Softcover

Monticelli, Dario Daniele

Softcover

ISBN 10: 3838389301 ISBN 13: 9783838389301

Publisher: LAP LAMBERT Academic Publishing, 2010

Synopsis

We study maximum principles for a class of linear, degenerate elliptic differential operators of the second order. The Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypotheses on the principal part and on the degeneracy set of the operator. We prove a Poincaré inequality, which then allows to define the functional setting where to study weak solutions for equations and inequalities involving this class of operators. A good example of such an operator is the Grushin operator, to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space. We use here the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

About this title

About the Author

Dario Monticelli completed his Ph.D. in Mathematics in 2007 at the Università degli Studi in Milano, under the supervision of Prof. Kevin Payne. His research interests involve nonlinear analysis and calculus of variations, partial differential equations and inequalities involving elliptic and degenerate elliptic operators and maximum principles.

Edition details

• Publisher: LAP LAMBERT Academic Publishing
• Publication date: 2010
• ISBN 10: 3838389301
• ISBN 13: 9783838389301
• Binding: Paperback
• Language: English
• Number of pages: 92