Validated numerical solutions for some semilinear elliptic equations on the disk

TL;DR

This paper develops a validated numerical method to rigorously prove the existence of solutions to a semilinear elliptic equation on a disk, using computer-assisted techniques and Zernike polynomials.

Contribution

It introduces a novel approach combining computer-assisted proofs with Zernike polynomial expansions to validate solutions of semilinear elliptic equations.

Findings

Existence of solutions near approximate ones is rigorously established.

Symmetry properties of solutions are proven.

Morse index of solutions is determined.

Abstract

Starting with approximate solutions of the equation $-\Delta u=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.