By Rolando Magnanini, Shigeru Sakaguchi, Angelo Alvino

ISBN-10: 884702840X

ISBN-13: 9788847028401

The research of qualitative features of PDE's has continuously attracted a lot realization from the early beginnings. extra lately, as soon as simple matters approximately PDE's, corresponding to life, distinctiveness and balance of ideas, were understood particularly good, study on topological and/or geometric houses in their strategies has turn into extra excessive. The research of those concerns is attracting the curiosity of progressively more researchers and is now a wide and well-established learn region, with contributions that frequently come from specialists from disparate parts of arithmetic, equivalent to differential and convex geometry, useful research, calculus of adaptations, mathematical physics, to call a couple of.

This quantity collects a range of unique effects and informative surveys by way of a gaggle of foreign experts within the box, analyzes new traits and strategies and goals at selling clinical collaboration and stimulating destiny advancements and views during this very energetic region of research.

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2. We prove Theorem 1 Let Ω ⊂ RN (N ≥ 5) be a bounded domain such that 0 ∈ Ω and ∂Ω ∈ C 2 . Let c0 be as in (4) and h(c) be as in (21). If c > −c0 , then h(c) > 0 and 2 D 2 u dx + c Ω ∂Ω u2ν dS ≥ h(c) Ω u2 dx |x|4 for all u ∈ H 2 ∩ H01 (Ω). (24) Furthermore, there exists CN = CN (Ω) ∈ (−c0 , (N − 1)K − d1 (HN )], where K is as in (9) and d1 (h) is as in (20), such that (i) h(c) is increasing, concave and continuous with respect to c ∈ (−c0 , CN ]; (ii) h(c) = HN for every c ≥ CN . A Family of Hardy-Rellich Type Inequalities 25 Moreover, the infimum in (21) is not achieved if c > CN , achieved if −c0 < c < CN and the minimizer uc ∈ H 2 ∩ H01 (Ω) solves (22) with h = h(c).

Am } p = −∞ is the (λ-weighted) p-mean of a. For a ≥ 0, we define Mp (a, λ) as above if p ≥ 0 and we set Mp (a, λ) = 0 if p < 0 and ai = 0 for some i = 1, . . , m. A simple consequence of Jensen’s inequality is that, for a fixed 0 ≤ a ∈ Rm and λ ∈ Λm , Mp (a, λ) ≤ Mq (a, λ) if p ≤ q. (10) 40 M. Bianchini and P. Salani Moreover, it is easily seen that lim Mp (a, λ) = max{a1 , . . , am } (11) lim Mp (a, λ) = min{a1 , . . , am }. (12) p→+∞ and p→−∞ Notice that the definition of p-concavity can be now equivalently restated in the following way.

Ii) If ♥(K) has dimension k (1 ≤ k ≤ N − 1), then there exists at least a direction θ ∈ SN −1 such that RK (θ ) = MK , θ , and K is θ -symmetric. Proof (i) The assertion is a straightforward consequence of Proposition 1. (ii) By Lemma 1, the function ω → RK (ω) − MK , ω attains its minimum for some θ ∈ SN −1 . Set r = RK (θ ) − MK , θ and suppose that r > 0. Then, for every x ∈ B(MK , r), we have that x, ω = MK , ω + x − MK , ω < MK , ω + r ≤ RK (ω) − r + r = RK (ω), for every ω ∈ SN −1 , and hence x ∈ ♥(K) by (1).

### Geometric Properties for Parabolic and Elliptic PDE's by Rolando Magnanini, Shigeru Sakaguchi, Angelo Alvino

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