| Abstract: |
| In a celebrated paper published in 1979, Gidas, Ni \& Nirenberg proved a symmetry result for a rigidity problem. With minimal hypotheses, the authors showed that positive solutions of semilinear elliptic equations in the unit ball are radial and radially decreasing.\
This result had a big impact on the PDE community and stemmed several generalizations. In a recent work in collaboration with G. Ciraolo, M. Cozzi \& M. Perugini this problem was investigated from a quantitative viewpoint, starting with the following question: given that the rigidity condition implies symmetry, is it possible to prove that if said condition is \emph{almost} satisfied the problem is \emph{almost} symmetrical?\
With the employment of the method of moving planes and quantitative maximum principles we are able to give a positive answer to the question, proving approximate radial symmetry and almost monotonicity for positive solutions of the perturbed problem. |
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