Remarks on viscosity solutions for evolution equations

Abstract

By Yun-Gang CHEN,*) Yoshikazu GIGA,**) and Shun'ichi GOTO***) (Communicated by Kunihiko KODAIRA, M. J. A., Dec. 12, 1991) 1o Introduction. We consider a degenerate parabolic equation

Viscosity solutions with shocks

“…We first recall properties that follow from accessibility in [8]. The accessibility property is valid for every subsolution that may not be proper.…”

Surface Evolution Equations

“…We have to restrict F or modify the notion of solutions as in §2.1.3 to conclude the left accessibility of solutions. The next lemma is essentially found in a paper by Y.-G. Chen, Y. Giga and S. Goto (1991b). We do not give the proof here since we won't use this result.…”

“…The conterexample for the comparison principle for a very singular equation is due to Y.-G. Chen, Y. Giga and S. Goto (1991b).…”

“…Extension of a solution defined in Ω×(0, T ) to Ω×(0, T ] is often important as discussed in §3.2.2. It is stated in several papers including Y.-G. Chen, Y. Giga and S. Goto (1991b) and H. Ishii and P. E. Souganidis (1995) (for F -solutions). Most results in §3.2.2 are taken from the work of Y.-G. Chen, Y.…”

Central limit theorem under variance uncertainty

“…Consider a viscosity subsolution u and a viscosity supersolution w of (1.7), (1.8). Since we require (1.7) to be satisfied in the viscosity sense at the lower boundary of Q, by the accessibility theorem of [6], we have and by the comparison result of [7] (Theorem 1) it follows that u ≤ w on Q.…”