Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types

• Published in: Duke mathematical journal Vol. 127; no. 2; pp. 251 - 290
• Main Authors: Lupo, Daniela, Payne, Kevin R.
• Format: Journal Article
• Language: English
• Published: DUKE University Press 01-04-2005; Duke University Press
• Subjects: 35A05; 35B33; 35L65; 35M10; 58J70; Conservation laws; Critical exponents; Equations of mixed type; Invariance and symmetry properties [See also 35A30]
• ISSN: 0012-7094; 1547-7398
• DOI: 10.1215/S0012-7094-04-12722-8

For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an associated Lagrangian, invariance with respect to changes in independent and dependent variables is investigated, as are results in the classification of continuous one-parameter symmetry groups. For the variational and divergence symmetries, conservation laws are derived via the method of multipliers. The conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems, and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems.