Perron’s method for Hamilton-Jacobi equations

$W^{2,\infty }$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

“…We just remark that the first thing to do is to prove a so-called strong comparison principle for (1.1) [5,3]. Existence of a bounded solution then follows by Perron's method [10,5]. Uniqueness and regularity are then (for this problem) rather simple consequences of the comparison principle.…”

“…Then by Perron's method [10,5], the following function is a discontinuous viscosity solution of this problem:…”

Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations

“…We see that u is locally Lipschitz continuous on Ω in view of (A2) and u is a subsolution (5). We argue as in the proof of Perron's method for viscosity solutions (see [25,1,18]), to find that u is a solution of (5) …”

“…Therefore, we have z ∈ Ω. Moreover, we may assume that dist (z m , ∂Ω) ≥ α(n) > 0 for all suitable large m ∈ N and some constant α(n) > 0 which is independent of m. We also have z m ̸ ∈ ∂A α due to (25). Thus, we can choose r(= r(n, λ)) > 0 which is independent of m such that B(z m , r) ⊂ Ω n \ A α for all m ∈ N, where…”

Existence of the Solution of an Implicit Integro-Differential Equation of Order One

“…A good amount of nice works can be found in the literature concerning the existence and uniqueness results for implicit di¤erential equations, see [2,5,6,7,10] and the references cited in the papers. H. Ishii [8] concerned with the existence of solutions of nonlinear first order scalar partial di¤erential equations (Hamilton-Jacobi equations) F ðx; u; DuÞ ¼ 0; in W; ð1:3Þ where W is an open subset of R N , F A CðW Â R Â R N ; RÞ, u : W ! R is the unknown and Du denotes the gradient of u.…”