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Parabolic equations with rough data

Herbert KochTobias Lamm — 2015

Mathematica Bohemica

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local...

Two-dimensional curvature functionals with superquadratic growth

Ernst KuwertTobias LammYuxiang Li — 2015

Journal of the European Mathematical Society

For two-dimensional, immersed closed surfaces f : Σ n , we study the curvature functionals p ( f ) and 𝒲 p ( f ) with integrands ( 1 + | A | 2 ) p / 2 and ( 1 + | H | 2 ) p / 2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2 . Our main result asserts that W 2 , p critical points are smooth in both cases. We also prove a compactness theorem for 𝒲 p -bounded sequences. In the case of p this is just Langer’s theorem [16], while for 𝒲 p we have to impose a bound for the Willmore energy strictly below 8 π as an additional condition....

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