Mathematics Colloquium

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Free boundary problems naturally arise in a variety of applications and research in this area has been very fruitful and active for several decades. Using the Bernoulli one-phase problem as a basic elliptic model, we will highlight main contributions and open questions often originating from a striking resemblance with the regularity theory for minimal surfaces. If time permits it, we will further describe certain degenerate free boundary problems which can be interpreted as an interpolation between the Bernoulli problem and a class of energy functionals which generalize the classical perimeter functional.