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Computer-assisted proofs for radially symmetric solutions of PDEs

  • * Corresponding author: Jean-Philippe Lessard

    * Corresponding author: Jean-Philippe Lessard 
The first, fourth and sixth authors were supported by the Hungarian Scientific Research Fund (NKFIH-OTKA), Grant No. K109782. The second author was supported in part by NWO-Vici grant 639.033.109. The fifth and the seventh authors were supported by NSERC.
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  • We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

    Mathematics Subject Classification: 58J05, 58C40, 65G40, 41A58, 58J20, 65M99, 65G40, 35K57.

    Citation:

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  • Figure 1.  (Left) Ten relative equilibria of (CR4BP) with equal masses. (Right) Eight relative equilibria of (CR4BP) with masses $m_1 = 0.9987451087$, $m_2 = 0.0010170039$ and $m_3 = 0.0002378873$. In both plots, some level sets of the effective potential $\Omega$ are depicted.

    Figure 2.  (Left) The first solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11). Since $r_{\min}<10^{-8}$, the true solution lies with the line-width by Theorem 2.1.

    Figure 3.  The second solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).

    Figure 4.  (Left) The third solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).

    Figure 5.  Six solutions of (22) for $\lambda \in \{118.2,120,250,350,450,500\}$.

    Figure 6.  (Left) A stationary solution of the Swift-Hohenberg equation (20) on the unit ball in $\mathbb{R}^3$ at $\lambda = 500$. (Right) The corresponding graph of $u(s) = u(\sqrt{x^2+y^2+z^2})$.

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