3 problems

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Berestycki–Caffarelli–Nirenberg conjecture for overdetermined elliptic problems
Berestycki–Caffarelli–Nirenberg conjecture. If $\mathbb{R}^n\setminus\overline{\Omega}$ is connected, then the existence of a bounded solution implies that $\Omega$ is a ball, a ha…
Conjecturerefutedelliptic-partial-differential-equations overdetermined-problems symmetry-breakingarXiv posted 2 days ago
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de Figueiredo–Lions–Nussbaum conjecture on the necessity of their a priori bound condition
de Figueiredo–Lions–Nussbaum conjecture. This condition is not necessary for the uniform a priori bounds established for positive solutions, although it is essential in their proof…
Conjectureopena-priori-estimates elliptic-partial-differential-equations nonlinear-elliptic-equationsarXiv posted 2 days ago
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Cabré–Terra conjecture on global minimality of saddle-shaped solutions
Cabré–Terra conjecture. The saddle-shaped solutions should be global minimizers of the energy functional.
Conjectureopenelliptic-partial-differential-equations saddle-shaped-solutions variational-methodsarXiv posted 2 days ago

Berestycki–Caffarelli–Nirenberg conjecture for overdetermined elliptic problems