## PDE Real Analysis Seminar

Seminar information archive ～11/01｜Next seminar｜Future seminars 11/02～

Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|

### 2005/03/02

10:30-11:30 Room #270 (Graduate School of Math. Sci. Bldg.)

The maximum principle in unbounded domains

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

Aubry set and applications

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Italo Capuzzo-Dolcetta**(Universita di Roma) 10:30-11:30The maximum principle in unbounded domains

[ Abstract ]

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.

[ Reference URL ]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.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Antonio Siconolfi**(Universita di Roma) 11:45-12:45Aubry set and applications

[ Abstract ]

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.

[ Reference URL ]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.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html