Viscosity solutions

A notion of solutions of fully non-linear second-order partial differential equations of the form $ F( x, u( x), Du( x), D ^ {2} u( x)) = 0 $, where $ u $ is a real-valued function defined on a set $ \Omega \subset \mathbf R ^ {n} $ and $ F: \Omega \times \mathbf R ^ {n} \times \mathbf R ^ {n} \times {\mathcal S} ^ {n} \rightarrow \mathbf R $ is continuous ( $ {\mathcal S} ^ {n} $ is the space of real symmetric $ ( n \times n ) $- matrices). This notion is relevant when $ F $ satisfies

$$ \tag{a1 } F( x, r, p, X) \geq F( x, s, p, Y) $$

$$ \textrm{ whenever } r \geq s \textrm{ and } X \leq Y , $$

with the usual ordering on symmetric matrices. The anti-monotonicity in $ X $ is a very weak ellipticity condition, which is satisfied, in particular, by equations of first order. Examples include classical Hamilton–Jacobi equations, Hamilton–Jacobi–Bellman equations from optimal control, Isaacs equations from differential games, possibly degenerate linear elliptic and parabolic equations, various equations of differential geometry (Monge–Ampère, minimal surfaces), etc.

An upper (respectively, lower) semi-continuous function $ u: \Omega \rightarrow \mathbf R $ is a viscosity subsolution (respectively, viscosity supersolution of $ F = 0 $ in $ \Omega $ if for every $ \phi \in C ^ {2} ( \mathbf R ^ {n} ) $ and local maximum (respectively, minimum) point $ z $ of $ u - \phi $ in $ \Omega $ one has $ F( z, u( z), D \phi ( z), D ^ {2} \phi ( z)) \leq 0 $( respectively, $ F( z, u( z), D \phi ( z), D ^ {2} \phi ( z)) \geq 0 $). A continuous function $ u: \Omega \rightarrow \mathbf R $ is a viscosity solution of $ F = 0 $ in $ \Omega $ if it is both a viscosity subsolution and a viscosity supersolution of $ F = 0 $ in $ \Omega $. The inequalities defining viscosity sub- and supersolutions are a consequence of the structure condition (a1) and the necessary conditions for extremals if $ u $ is a classical solution of $ F \leq 0 $ or $ F \geq 0 $ in an open set, a fact which shows a connection between the notion of viscosity solutions and the classical maximum principle for second-order elliptic equations.

The importance of this notion lies in the fact that very general uniqueness and existence theorems are valid for viscosity solutions. A typical example is the existence and uniqueness of a unique bounded and uniformly continuous function $ u( x) $, $ x = ( t, y) \in [ 0, T] \times \mathbf R ^ {m} $, which is a viscosity solution of $ u _ {t} + G( D _ {y} u , D _ {y} ^ {2} u ) = 0 $ on $ ( 0, T ] \times \mathbf R ^ {m} $ and satisfies $ u ( 0, y) = \psi ( y) $ for $ y \in \mathbf R ^ {m} $ whenever $ T > 0 $, $ G( q, Z) $ is continuous in $ ( q, Z) \in \mathbf R ^ {m} \times {\mathcal S} ^ {m} $ and anti-monotone in $ Z $, and $ \psi $ is bounded and uniformly continuous on $ \mathbf R ^ {m} $. In fact, existence is essentially a consequence of the proof of uniqueness, which also establishes monotone and continuous dependence of the solution with respect to $ \psi $ and may be proved by an adaptation of the Perron method.

In addition to many existence, uniqueness and comparison results, the theory of viscosity solutions now includes a treatment of other basic problems, such as the correct formulation of various boundary conditions, including the classical Dirichlet, Neumann and oblique derivative conditions; the convergence of numerical approximations; the study of regularity and other qualitative properties of solutions; the analysis of many asymptotic problems, including large deviations and homogenization problems; extensions to discontinuous data; weak passages to the limit; and extensions to certain integro-differential operators.

Primary application of viscosity solutions is in the theory of optimal control and differential games for deterministic and stochastic evolutions. In particular, the uniquely defined viscosity solutions of the associated Hamilton–Jacobi–Bellman and Isaacs equations are the corresponding value functions, and this fact provides a complete mathematical justification of dynamic programming arguments.

Extensions of the theory include the study of problems in infinite-dimensional spaces for both first- and second-order equations, one of the goals being to provide a theoretical foundation for dynamic programming approaches to optimal control by partial differential equations.

The references provide some basic information about the theory and contain many references to the various topics described above.

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