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Results tagged with elliptic-pde
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user 101776
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
2
votes
1
answer
759
views
Optimal $L_p$-estimate for elliptic operator
Let us start with the Poisson equation
\begin{equation}
-\Delta u=f
\end{equation}
on a domain $\Omega$. The classical regularity says if the the boundary of the domain is sufficiently smooth, then th …
- 333
2
votes
$C^2$-solution of Lane-Emden equation with positive frequency
The answer to my questions is indeed negative. In this paper it is shown that solutions to the second equation do exist, which are periodic in $x_1$ (but not constant) and decay to zero when $|x'|\to\ …
- 333
4
votes
1
answer
100
views
$C^2$-solution of Lane-Emden equation with positive frequency
Consider the Lane-Emden equation
$$-\Delta u=u^{\frac{d+2}{d-2}} $$
in $\mathbb{R}^d$ with $d\geq 3$ and $u>0$ a positive $C^2$-solution. It is well-known, due to [Caffarelli et al., CPAM '89] that $u …
- 333