Abstract
The modern theory of viscosity solutions was developed by Crandall, Evans, Jensen, Ishii, Lions, and others. First, it was designed for first order equations. Later, it was extended to second order equations. The solutions must obey a Comparison Principle.
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Notes
- 1.
A few chapters from [K] are enough for our purpose.
- 2.
Here the word “viscosity” is only a label. It comes from the method of vanishing viscosity. In our case one would replace \(\Delta _pu=0\) by \(\Delta _pu_{\varepsilon }+\varepsilon \Delta u_{\varepsilon } = 0\) and send \(\varepsilon \) to zero, so that the artificial viscosity term \(\varepsilon \Delta u_{\varepsilon }\) vanishes. So \(\lim u_{\varepsilon } = u\) is reached. Properly arranged, this is the same concept.
- 3.
It is also sufficient provided that the closures \(\overline{J^{2,-}v}\) of the subjets are evoked. See [K].
- 4.$$\begin{aligned}&\int \!|\nabla v -\nabla v_{\varepsilon }|^p\, dx\\ =&\int \!|\nabla v -\nabla v_{\varepsilon }|^p\bigl (1+|\nabla v|^2+|\nabla v_{\varepsilon }|^2\bigr )^{p(p-2)/4} \bigl (1+|\nabla v|^2+|\nabla v_{\varepsilon }|^2\bigr )^{p(2-p)/4}\, dx\\ \le&\Bigl \{\int \!|\nabla v -\nabla v_{\varepsilon }|^2\bigl (1+|\nabla v|^2+|\nabla v_{\varepsilon }|^2\bigr )^{(p-2)/2}\Bigr \}^{\frac{p}{2}}\Bigl \{\int \!\bigl (1+|\nabla v|^2+|\nabla v_{\varepsilon }|^2\bigr )^{p/2}\, dx\Bigr \}^{1-\frac{p}{2}}\\ \le&\Bigl \{\tfrac{1}{p-1}\int \!\langle |\nabla v|^{p-2}\nabla v - |\nabla v_{\varepsilon }|^{p-2}\nabla v_{\varepsilon },\nabla v -\nabla v_{\varepsilon }\rangle \, dx\Bigr \}^{\frac{p}{2}}\Bigl \{\int \!\bigl (1+|\nabla v|^2+|\nabla v_{\varepsilon }|^2\bigr )^{p/2}\, dx\Bigr \}^{\frac{2-p}{2}} \end{aligned}$$
where inequality (VII) in Sect. 12 was used at the last step.
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Lindqvist, P. (2019). Viscosity Solutions. In: Notes on the Stationary p-Laplace Equation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-14501-9_9
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