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

By Rolando Magnanini, Shigeru Sakaguchi, Angelo Alvino

The learn of qualitative facets of PDE's has regularly attracted a lot consciousness from the early beginnings. extra lately, as soon as easy concerns approximately PDE's, reminiscent of lifestyles, distinctiveness and balance of suggestions, were understood really good, study on topological and/or geometric houses in their options has turn into extra extreme. The research of those concerns is attracting the curiosity of progressively more researchers and is now a large and well-established learn zone, with contributions that regularly come from specialists from disparate components of arithmetic, similar to differential and convex geometry, sensible research, calculus of adaptations, mathematical physics, to call a couple of.

This quantity collects a variety of unique effects and informative surveys by way of a bunch of foreign experts within the box, analyzes new traits and methods and goals at selling clinical collaboration and stimulating destiny advancements and views during this very lively sector of research.

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The map (−c0 , +∞) c → h(c) is nondecreasing (increasing when achieved), concave, hence, continuous and h(c) = HN for every c ≥ (N − 1)K − d1 (HN ). Moreover, if Ω satisfies (27) and HN − δ < h < HN , then h(c) ≤ h for every −c0 < c ≤ (N − 1)K − d1 (h). Proof The properties of h(c) follow from its definition, we only need to prove the estimates. By (11), the infimum in (21) may be rewritten as h(c) = inf Ω u∈H 2 ∩H01 (Ω)\{0} |Δu|2 dx − 2 ∂Ω αc (x)(uν ) dσ 2 u Ω |x|4 dx , (35) where αc (x) = (N − 1)K(x) − c, as defined in Sect.

R) → R, for a small enough r > 0, Now we introduce the function ϕp,λ : B(x, defined as follows ϕp,λ (x) = λ1 u x1,p + a1,p (x − x) ¯ p + ··· ¯ + λn+1 u xn+1,p + an+1,p (x − x) p 1/p (24) where ai,p = u(xi,p )p , up,λ (x) ¯ p for i = 1, . . , n + 1. (25) The following facts trivially hold: n+1 (A) i=1 λi ai,p = 1 by (15); ¯ for every x ∈ B(x, ¯ r), thanks to (A) and the (B) x = n+1 i=1 λi (xi,p + ai,p (x − x)) first equation in (15); ¯ = up,λ (x); ¯ (C) ϕp,λ (x) ¯ r) (this follows from (B) and from the definition (D) ϕp,λ (x) ≤ up,λ (x) in B(x, of up,λ ).

N + 1 and θ = we notice that the points to the 0-level set of Gθ,0 . Lemma 5 Let Ω ⊂ Rn be an open convex set of class C 1 , λ ∈ Λn+1 , F a degenerate elliptic operator, p ∈ (0, 1), such that (6) holds. Let u ∈ C(Ω) ∩ C 2 (Ω) be a classical solution of (1) such that (7) holds. Then up,λ is a viscosity subsolution of (1). Proof The proof of this lemma is the same as the proof of the previous lemma using Lemma 3 instead of Lemma 2. With these lemmata at hands, the proof of Theorem 1 is now very easy.

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