Abstract
This chapter discusses the history, examples, motivation and first definitions of viscosity solutions. Viscosity solutions were introduced first by Crandall and Lions in 1983 as a uniqueness criterion for 1st order PDE. The essential idea regarding passage to limits was observed earlier by Evans in 1978 and 1980.
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- 1.
It is customary to call \(F\) the “coefficients” and we will occasionally follow this convention. This terminology is inherited from the case of linear equations, where \(F\) consists of functions multiplying the solution and its derivatives. We will also consistently use the notation “\(\cdot \)” for the argument of \(x\), namely \(F(\cdot , u,Du,D^2u)|_{x}=F\big (x,u(x),Du(x),D^2u(x)\big )\). We will not use the common clumsy notation \(F(x, u,Du,D^2u)\).
- 2.
In other texts, ellipticity is defined with the opposite inequality. This choice of convention that we make is more convenient, since we consider non-divergence operators like \(\varDelta _\infty \). We do not use integration by parts, hence there is no reason to consider “minus the operator”. According to our convention, \(\varDelta \) and \(\varDelta _\infty \) are elliptic, while for text using the opposite convention, \(-\varDelta \) and \(-\varDelta _\infty \) are elliptic.
- 3.
In other texts Viscosity sub/super solutions are defined with the opposite inequalities. The direction of the inequalities corresponds to the choice of convention in the ellipticity notion. However, no confusion should arise for the readers because the definition is essentially the same in both cases: when it comes down to writing the inequalities for, say, \(\varDelta _\infty \), in either choice of conventions we have
$$ D\psi (x)\otimes D\psi (x):D^2\psi (x)\ge 0 $$when \(\psi \) touches \(u\) from above at \(x \in \varOmega \) and
$$ D\phi (y)\otimes D\phi (y):D^2\phi (y)\le 0 $$when \(\phi \) touches \(u\) from below at \(y \in \varOmega \).
References
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Katzourakis, N. (2015). History, Examples, Motivation and First Definitions. In: An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-12829-0_1
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