Multivalued solutions of first-order partial differential equations

Abstract

The concept of multivalued solution of a first-order partial differential equation is introduced. This is a development of the concept of a minimax solution, the definition of which is based on the weak invariance property with respect to the characteristic inclusions. The need to consider such solutions occurs, for example, when first-order partial differential equations do not satisfy the known conditions ensuring the existence of unique continuous viscosity and minimax solutions. As an illustration of the concept of multivalued solutions, the Cauchy problem for the Hamilton-Jacobi equation is discussed. By contrast to results known in the theory of viscosity and minimax solutions, it is not required here that the Hamiltonian satisfy the Lipschitz condition with respect to the phase variable or some modification of this condition. A boundary-value problem of Dirichlet kind for first-order partial differential equations is also considered. As is known, a minimax solution of it exists under certain assumptions and is unique in the class of discontinuous functions. In place of discontinuous solutions one can consider the corresponding multivalued solutions, which makes the study of such problems easier.