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EnglishLet L be a possibly degenerate second order differential operator and let Γη = d^(2−Q) be its fundamental solution at η; here d is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of −Lu ≥ f (ξ, u) ≥ 0 on RN to satisfy the representation formula (R) u(η) ≥ integral RN Γη f (ξ, u) dξ. We prove that (R) holds provided f (ξ, ·) is superlinear, without any as- sumption on the behavior of u at infinity. On the other hand, if u satisfies the condition |u(ξ)|dξ = 0, lim inf − R→∞ R≤d(ξ)≤2R then (R) holds with no growth assumptions on f (ξ, ·).
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| Titolo: | Representation Formulae and Inequalities for Solutions of a Class of Second Order Partial Differential Equations | |
| Autori: | ||
| Data di pubblicazione: | 2006 | |
| Rivista: | ||
| Abstract: | Let L be a possibly degenerate second order differential operator and let Γη = d^(2−Q) be its fundamental solution at η; here d is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of −Lu ≥ f (ξ, u) ≥ 0 on RN to satisfy the representation formula (R) u(η) ≥ integral RN Γη f (ξ, u) dξ. We prove that (R) holds provided f (ξ, ·) is superlinear, without any as- sumption on the behavior of u at infinity. On the other hand, if u satisfies the condition |u(ξ)|dξ = 0, lim inf − R→∞ R≤d(ξ)≤2R then (R) holds with no growth assumptions on f (ξ, ·). | |
| Handle: | http://hdl.handle.net/11586/106035 | |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
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