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Variational principles for second-order quasi-linear scalar equations. (English) Zbl 0533.49010
The problem of constructing variational principles for a second-order quasi-linear partial differential equation is considered. In particular, the problem of finding a first-order function f whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian is in the centre of the examination. For such a function f the authors derive two sets of equations. Necessary and sufficient conditions for the integration of the first set are established in the general case, and these lead to a considerable simplification of the second set. In certain special cases, such as the case when the operator is elliptic, the problem is solved completely. The utility of the obtained results is illustrated by a variety of examples.
Reviewer: J.Vaníček

MSC:
49K20 Optimality conditions for problems involving partial differential equations
35A15 Variational methods applied to PDEs
35G20 Nonlinear higher-order PDEs
49L99 Hamilton-Jacobi theories
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