Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation

摘要

For a bounded domain Omega subset of R-n and g is an element of C (partial derivative Omega), assume that H is an element of C-2 (R-n) is convex and coercive, and that {p is an element of R-n parallel to H (p) = min(R)(n) H} has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson's equation: A[u]:= H-p (del u) circle times H-p (del u) : del(2)u = 0, in Omega u = g, on partial derivative Omega. For H = H (p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation and that of the Hamilton-Jacobi equation H (del u, x) = c.