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Optimal integrability of the Jacobian of orientation preserving maps

Andrea Cianchi — 1999

Bollettino dell'Unione Matematica Italiana

Dato un qualsiasi spazio invariante per riordinamenti X Ω su un insieme aperto Ω R n , si determina il più piccolo spazio invariante per riordinamenti Y Ω con la proprietà che se u : Ω R n è una applicazione che mantiene l'orientamento e D u n X Ω , allora det D u appartiene localmente a Y Ω .

A sharp iteration principle for higher-order Sobolev embeddings

Andrea CianchiLuboš PickLenka Slavíková — 2014

Banach Center Publications

We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving...

Gradient regularity via rearrangements for p -Laplacian type elliptic boundary value problems

Andrea CianchiVladimir G. Maz'ya — 2014

Journal of the European Mathematical Society

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.

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