Seminario / Cozzi
Lunedì 18 dicembre 2017, alle ore 16.00, in Sala di Rappresentanza del Dipartimento di Matematica (Via Saldini, 50),
Matteo Cozzi (BGSMath e UPC, Barcelona) terra' un seminario dal titolo:
"A gradient estimate for nonlocal minimal graphs"
"We consider the class of nonlocal minimal surfaces that can be written as the global graph of a measurable function defined in \R^n. We establish that any such function is smooth wherever it is bounded, and that its gradient can be locally controlled in terms of its oscillation. As an application, we deduce a flatness result for nonlocal minimal graphs that grow at most linearly at infinity.
Our result extends to the fractional setting the celebrated gradient bound of Finn and Bombieri, De Giorgi, and Miranda for solutions of the classical mean curvature equation. We remark that our gradient estimate depends on the oscillation in a power-like fashion, and not exponentially as the classical one.
In order to establish the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of an equation driven by a truncated fractional Jacobi operator. Furthermore, we establish new fractional Sobolev inequalities and Harnack-type estimates for solutions of singular integral equations posed on nonlocal minimal surfaces and on more general subsets of the Euclidean space satisfying suitable density hypotheses.
This is a joint work with X. Cabre' (ICREA and UPC, Barcelona)."
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13 dicembre 2017