Abstract
The so-called \(\infty \)-Poisson equation \(\Delta _{\infty }v(x) = F(x)\) has received much attention. It has to be observed that it is not the limit of the corresponding p-Poisson equations \(\Delta _pv = F.\) This can be seen from the construction of Jensen’s auxiliary equations in Chap. 8. Another example is that the limit of the equations \(\Delta _pv_p = -1\) with zero boundary values often yields the distance function as the limit solution. (The limit equation is not \(\Delta _{\infty }v = -1.\)) Thus there is no proper variational solution.
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Notes
- 1.
If the test function \(\Phi =\Phi (x,x_{n+1})\) touches V from below, then \(\Phi (x,x_{n+1}) +\alpha x_{n+1}^{\frac{4}{3}}\) touches v from below. Now \(\frac{\partial v}{\partial x_{n+1}} = 0.\) Thus
$$\begin{aligned} \frac{\partial {xx} }{\partial x_{n\!+\!1}}\left( \Phi (x,x_{n+1}) +\alpha x_{n+1}^{\frac{4}{3}}\right) = 0 \end{aligned}$$at the touching point. Therefore
$$\begin{aligned} \Delta _{\infty }^{(n+1)}(\Phi (x,x_{n+1}) +\alpha x_{n+1}^{\frac{4}{3}}) = \Delta _{\infty }^{(n)}(\Phi (x,x_{n+1}) +\alpha x_{n+1}^{\frac{4}{3}}), \end{aligned}$$i.e. at the touching point the \(\infty \)-Laplacian has the same value in n and in \(n+1\) variables. But in n variables \(\Delta ^{(n)}_{\infty }(...) \le F(x)\) since v was a viscosity supersolution. The desired inequality
$$\begin{aligned} \Delta _{\infty }^{(n+1)}\Phi (x,x_{n+1}) \le F(x) - \frac{64\alpha ^3}{81} \end{aligned}$$follows.
- 2.
There is a pedantic comment. Since \(\Delta _{\infty }\psi < F\) holds pointwise in \(B(x_0,2\delta ),\) it also holds in the viscosity sense, because \(\psi \) has continuous second derivatives. Hence \(\psi \) is a viscosity supersolution.
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Lindqvist, P. (2016). The Equation \(\Delta _{\infty }v = F\) . In: Notes on the Infinity Laplace Equation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31532-4_10
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DOI: https://doi.org/10.1007/978-3-319-31532-4_10
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