On uniqueness of solutions to viscous HJB equations with a subquadratic nonlinearity in the gradient

Abstract

Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form −∆u(x) + 1 γ |Du(x)| γ = f (x) − λ, with f a coercive function and λ a constant, in the subquadratic case, that is, γ ∈ (1, 2), appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that f (x) ≈ |x| β and |Df (x)| |x| (β−1) + for some β > 0, essentially matching earlier results of Ichihara, who considered more general Hamiltonians but with better regularity for f . Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature. In this short article, we show that the equation has a unique positive solution for any locally Lipschitz continuous, coercive f which satisfies Df (x) ≤ κ 1 + |f (x)| 2−1/γ for some positive constant κ. Since 2 − 1 γ > 1, this assumption imposes very mild restrictions on the growth of the potential f . We also show that this solution fully characterizes optimality for the associated ergodic problem. Our method involves the study of an infinite dimensional linear program for elliptic equations for measures, and is very different from earlier approaches. It also applies to the larger class of Hamiltonians studied by Ichihara, and we show that it is well suited to provide verification of optimality results for the associated ergodic control problems, even in a pathwise sense, and without resorting to the parabolic problem.

Large time behavior of unbounded solutions of first-order Hamilton–Jacobi equations in RN

“…Therefore, combining the results of Proposition 3 and Theorem 4, there exists a unique solution of (7)- (8). Since T > 0 is arbitrary (in particular, with no dependence on the data) the solution can be uniquely extended to a solution of (1)- (2), which is also nonnegative. Uniqueness then follows immediately from Theorem 4.…”

“…The work [26] addresses the problem of large-time behavior of unbounded solutions of (1)-(2) mainly with probabilistic techniques with x-dependent Hamiltonians. It is based on explicit representation formulas for the solution of (1)- (2) which comes from the underlying stochastic optimal control problem (see, e.g. [3], [23] for standard references on this topic).…”

Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi equation: quadratic and subquadratic cases

“…In contrast, in a recent work by Arapostathis, Biswas and Caffarelli [1], uniqueness of solutions for (EP ) is obtained assuming only a technical hypothesis which relates f and Df , but still allows f to have arbitrary growth-see (H2) below. This is accomplished by means of the uniqueness of a measure related to the underlying dynamics of the problem.…”

“…There are several works that address the large-time behavior of solutions of (1) set in the whole space in the context of weak solutions, but by necessity these work within the class of bounded solutions; we refer the reader to [9,10,12,14,16,20] and the works cited therein, though this list is by no means exhaustive.…”

“…For unbounded solutions over the whole space, Ichihara [17] obtains results on large-time behavior for a class of equations which includes the model (1), for all m > 1, by combining probabilistic and PDE techniques. Additionally, these results make use of precise growth assumptions for both f and u 0 : roughly, it is assumed in [17] that f (x) ≈ |x| m * , where m * = m m−1 (the conjugate exponent of m) and that u 0 has at most polynomial growth.…”