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Additional info for Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective
Sample text
Notions of the theory of viscosity solutions were reconsidered and simplified in [54). The theory of viscosity solutions has advanced investigations of firstorder partial differential and elliptic equations. Theorems of uniqueness and existence for various types of equations and boundary-value problems have been proven in the framework of this theory and also some applications to problems of control and differential games have been studied. A review of the results of the theory of viscosity solutions can be found in [56).
It is easy to establish that (jv+) (respectively, (jv-)) implies (iv+) (respectively, (iv-)). § = ((, s) E Rm x Rn. ;, x, z, 1/J)} = F(x, z, s). ;, x, z, s)} satisfies all the conditions (j), (jj), (jjj+), (jjj-), (jv+), (jv-). In this case 1/J = s E Rn = I}!. 4 Criteria of Weak Invariance for Minimax Solutions Investigations of minimax solutions are essentially based on methods of nonsmooth analysis and simultaneously enrich it with new results. At the infinitesimal level one can take notice of duality of various forms of approaches to defining generalized solutions, including the duality of definitions of minimax and viscosity solutions.
Obviously, the last condition is equivalent to the equality F(x, n(x), Dn(x)) = 0. Thus we have that minimax and classical solutions are compatible. 7:, u(x), Du(x)) = 0. As is shown above, at the points x E G at which the function u is differentiable, the considered equation and condition (M6) are equivalent. 1). G. -L. Lions (cf. [57, 58]). 12) F(xo,u(xo),Drp(x 0 ));::: 0. 13) A function is called a viscosity solution if it is a supersolution and a subsolution simultaneously. In the theory of viscosity solutions for the purpose of defining [:mpersolutions and subsolutions, the relations of the form (U5) and (L5) were also introduced (cf.