Program
- 組織委員：
- 新井仁之（東大），石井仁司（早大），小池茂昭（埼玉大），儀我美一（東大／北大）
- 幹事：
- 河添 健（慶大），剣持信幸（千葉大），酒井 良（都立大），柴田良弘（早大）， 望月 清（中央大），宮地晶彦（東女大），山崎昌男（早大）
- 日 時：
- 2005年3月2日(水) 10:30-11:30 11:45-12:45
- 場 所：
- 東京大学大学院 数理科学研究科270号室
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- 講 演 者：
- I : Italo Capuzzo-Dolcetta (Universita di Roma)
II: Antonio Siconolfi (Universita di Roma) - 演 題：
- I : "The maximum principle in unbounded domains"
II: "Aubry set and applications" - ABSTRACT:
-
I : "The maximum principle in unbounded domains" : Italo Capuzzo-Dolcetta
The issue of the talk is the validity of the Weak Maximum Principle for functions u satisfying a second-order partial differential inequality of the form
(*) F(x,u,Du,D^2u) ≧ 0
in a domain A of the n-dimensional euclidean space.
The main result presented in the lecture is that for bounded above upper semicontinuous functions verifying
(*) in the viscosity sense, the inequality u≦ 0 on the boundary ∂A is propagated in the interior of the domain itself, under suitable conditions on F and A.
These conditions include ellipticity of F, a general geometric condition on the (possibly) unbounded domain A and a joint requirement involving the spread of A and the decay of first order terms at infinity.
This result, contained in I.C.D, A.Leoni, A.Vitolo "The Alexandrov-Bakelman-Pucci weak Maximum Principle for fully nonlinear equations in unbounded domains", to appear in Comm.in PDE's, extends previous results due to X.Cabré and L.Caffarelli-X.Cabré.
In the second part of the talk we present different versions of Weak Maximum Principle, namely for solutions growing exponentially fast of (*) in narrow domains and for solutions of
(**) F(x,u,Du,D^2u) + c(x)u ≧ 0
(c changing sign) in domains of small measure. - II : "Aubry set and applications" : Antonio Siconolfi
For given Hamiltonian H(x, p) continuous and quasiconvex in the second argument, defined in Rn × Rn or on the cotangent bundle of a compact boundaryless manifold, we consider the equation
H= c
with c critical value, i.e. for which the equation admits locally Lipschitzcontinuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by A, where the obstruction to the existence of such subsolutions is concentrated. We give a metric characterization of A, and we discuss its main properties.
They are applied to a projection problem in a Banach space, to the study of the largetime behaviour of subsolutions to a timedependent HamiltonJacobi equation, and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.
