Nonlinear Analysis 51 (2002) 239–248 www.elsevier.com/locate/na Regularity of perturbed Hamilton–Jacobi equations Jerome A. Goldstein a , Yudi Soeharyadi b; ∗ a Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA b Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901, USA Received 8 June 2001; accepted 22 June 2001 Keywords: Hamilton–Jacobi; Perturbed Hamilton–Jacobi; m-dissipative operator; Invariant set; Second centered dierence; Elliptic regularization 1. Introduction Nonlinear hyperbolic problems are notorious for destroying or at least not preserving regularity. Burch [3] proved an interesting regularity result in this context. Consider the Hamilton–Jacobi equation u t + F (∇u)=0; x ∈ R N ;t ¿ 0; u(x; 0)= u 0 (x); x ∈ R N ; (1) where u t = @u=@t; ∇ is the spatial derivative operator, F ∈ C 2 (R N ) is weakly convex, i.e. ∑ N i;j F xi xj (x) i  j ¿ 0, for all x; ∈ R N , and all functions are real valued. Further- more, F is normalized to satisfy F (0) = 0. By working in the Banach space B ∪ C (R N ) of all bounded uniformly continuous functions from R N to R (with the supremum norm), one can show that the operator A 0 = - F ◦∇ dened by A 0 u = - F (∇u) (on a suitable domain) is densely dened and m-dissipative on X . Hence problem (1) is governed (according to the Crandall–Liggett theorem) by a strongly continuous non- expansive (or contractive) nonlinear semigroup T 0 = {T 0 (t ): t ¿ 0} on X . In particular u(t )= T 0 (t )u 0 is the unique mild solution of (1), for any initial data u 0 ∈ X . See Burch [3] and Aizawa [1,2]. For background in semigroup theory, see for example [5]. * Corresponding author. Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia. E-mail addresses: jgoldste@memphis.edu (J.A. Goldstein), yudish@dns.math.itb.ac.id (Y. Soeharyadi). 0362-546X/02/$-see front matter c  2002 Elsevier Science Ltd. All rights reserved. PII:S0362-546X(01)00828-8