A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

Abstract

Abstract. We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Hölder inequality.Mathematics Subject Classification. 35F20, 49L25.

De Giorgi Techniques Applied to the Hölder Regularity of Solutions to Hamilton–Jacobi Equations

“…The non-convex case is technically more challenging. The first proofs, relying on stochastic methods, were obtained by Cardaliaguet [5], and Cannarsa Cardaliaguet [4]. Cardaliaguet and Silvestre provided a simpler proof in [6].…”

Sobolev regularity for the first order Hamilton–Jacobi equation

“…Next we explain that inequality (5) can also be understood in the distributional sense: Lemma 2.3. If u is continuous on [0, 1] × Q 1 and satisfies (5) in the viscosity sense, then u is of bounded variation (BV) in (0, 1) × Q 1 , Du ∈ L p ((0, 1) × Q 1 ) and (5) holds in the sense of distributions.…”

“…Throughout this part u : [−1/2, 1/2] × Q 1 → R is a continuous map which satisfies (5) and (6) in [−1/2, 1/2] × Q 1 . Our aim is to show that, if Du and f are well estimated in some cube, then Du satisfies a reverse Hölder inequality.…”

“…The result was later extended to equations with unbounded right-hand side by Dall'Aglio and the second author [12]. For HJ equations of evolution type, Hölder bounds were progressively obtained by the first author ( [5], first order case, convex Hamiltonians), then in collaboration with Cannarsa ([3], quasilinear case), with Rainer ( [6], fully nonlinear, nonlocal equations), and with Silvestre ( [9], unbounded right-hand side). The proof in this setting is completely different, and more involved, than for the stationary case: it relies either on suitable one-dimensional reverse Hölder inequality (as in [3,5,6]) or on the method of improvement of oscillations (as in [9]).…”

Homogenization of pathwise Hamilton–Jacobi equations

“…. The results of [16,17,19,50] imply that the Hölder semi-norm of u can be locally controlled in terms of the growth of f ∞ , Ḃ ∞ , u ∞ , and the growth of H in Du. However, none of these works apply to (1.8), where the right-hand side is not only unbounded, but nowhere point-wise defined.…”