By Goro Akagi (auth.), Rolando Magnanini, Shigeru Sakaguchi, Angelo Alvino (eds.)

The examine of qualitative features of PDE's has consistently attracted a lot cognizance from the early beginnings. extra lately, as soon as simple concerns approximately PDE's, reminiscent of life, strong point and balance of options, were understood really good, study on topological and/or geometric houses in their suggestions has turn into extra excessive. The learn of those concerns is attracting the curiosity of increasingly more researchers and is now a wide and well-established study quarter, with contributions that regularly come from specialists from disparate components of arithmetic, akin to differential and convex geometry, practical research, calculus of diversifications, mathematical physics, to call a number of.

This quantity collects a range of unique effects and informative surveys through a bunch of overseas experts within the box, analyzes new tendencies and methods and goals at selling clinical collaboration and stimulating destiny advancements and views during this very lively quarter of research.

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Notice that assumption (7) is needed only for p > 0 and it is in general provided by a suitable version of the Hopf’s Lemma. The idea of the proof is to show that the function v = up (or v = log u is p = 0) in fact coincides with its concave envelope v ∗ and the role of (7) is to guarantee that the the contact set of v (that is the set of points where v and v ∗ coincide) does not touch the boundary of the domain. In the case p ≤ 0 this requirement is automatically satisfied since |v| → +∞ as x → ∂Ω; when p ∈ (0, 1), assumption (7) implies ∂v(x) = +∞ (8) ∂ν for every x0 ∈ ∂Ω and every inward direction ν of Ω at x0 , and the latter forces the contact set of v to stay away from ∂Ω.

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