Seminars on Applied Analysis
Date: March 2 (Thu.) – March 3 (Fri.), 2017
Venue: Room 002, Graduate School of Mathematical Sciences, The Univ. of Tokyo

ProgramMarch 2 (Thu.)
15:30-16:15

Prof. Vladimir Romanov (Sobolev Institute of Mathematic)
Phaseless inverse problems for Schrödinger and Helmholtz equations

16:15-17:00

Prof. Mourad Bellassoued (Université de Tunis El Manar)
Carleman estimates and applications

17:10-17:55

Prof. Yves. Dermenjian (Aix-Marseille Université)Some applications of Carleman estimates for parabolic operators in an isotropic
discontinuous media

March 3 (Fri.)
10:00-10:45

Prof. Eric Soccorsi (Aix-Marseille Université)
On Carleman estimates in unbounded cylindrical quantum domains andapplication to inverse problems

10:45-11:30

Prof. Adam Kubica (Warsaw University of Technology)
Fractional diffusion equation

11:40-12:25

Prof. Piermarco Cannarsa (University of Rome Tor Vergata)
Invariance for quasi-dissipative systems in Banach spaces

Supported partly by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for thePromotion of Science-1-Phaseless inverse problems for Schrodinger and Helmgoltz equations
Vladimir Romanov
Sobolev Institute of Mathematics
Novosibirsk, Russia
For the Schrödinger equation−∆x u − k 2 u + q(x)u = δ(x − y),x ∈ R3 ,(1)with the scattering conditions( ) ∂uu (x, y, k) = O r−1 ,+ iku = o(r−1 ), r = |x − y| → ∞.
∂r(2)

an inverse problem of recovering the potential q(x) is considered. Let u(x, y, k) be the solutionof problem (1)-(2) and u0 (x, y, k) be the solution of the same problem with q(x) ≡ 0. Denotethe scattered field by usc (x, y, k) = u(x, y, k) − u0 (x, y, k).We assume that supp q(x) ⊂ B ⊂ R3 ,where B = {x| |x| < R}, R > 0, with the boundary S = {x| |x| = R}. The following inverse
problem is considered: given f (x, y, k) = |usc (x, y, k)| for (x, y) ∈ S × S and k ≥ k0 , k0 > 0,
find q(x) in B. The main result consists in reduction of this problem to the usual tomography
problem for all straight lines crossing B.
The second inverse problem is related to the Helmgoltz equation∆u + k 2 n2 (x)u = −δ(x − y),x ∈ R3 ,(3)where n(x) = 1 outside of B. We denote solution of (3) satisfying (2) as u(x, y, k) and by
u0 (x, y, k) = eik|x−y| /(4π|x − y|) the incident wave in the homogeneous media with n ≡ 1. Let
usc (x, y, k) = u(x, y, k)−u0 (x, y, k) be the scattering wave. Consider the inverse problem similar
to the previous one: given f (x, y, k) = |usc (x, y, k)| for (x, y) ∈ S × S and k ≥ k0 , k0 > 0, findn(x) in B. The main result in this case consists in reduction of this problem to the inverse
kinematic problem.

-2-Some applications of Carleman estimates for parabolic operators in
an isotropic discontinuous media
Assia Benabdallah, Yves Dermenjian & Masahiro Yamamoto
Abstract
We consider• the cylinder Ω := (0, 1) × (−H, H), Ω+ := (0, 1) × (0, H), Ω− := (0, 1) × (−H, 0)• a real diffusion coefficient a such that the restrictions a± to Ω± belong to C 2 (Ω̄± ) and0 < cmin ≤ a ≤ cmax < +∞.
• the positive selfadjoint operator A = −∇ · (a∇) with Dirichlet boundary condition.
So, the discontinuities of the coefficient a are included in the interface S = (0, 1) × {0}. We
prove a Carleman estimate for the heat operator ∂t + A on Ω × (0, T ) when the localization of theobservation is on the boundary of ∂Ω. Precisely, we prove
2Theorem 1. There exist a weight function ϕ(x, t) := ed(x)−β(t−t0 ) and positive constants C, s0 , λ0such that, for s > s0 and λ > λ0 , we haveZCZ+1|∂t u|2 dxdtsϕΩ×(0,T )Z2sϕ22 22sλϕe|(∇u)|Si ×(0,T ) | +(sλ) ϕ |u| dγS dt−
e2sϕ (s3 λ4 ϕ3 |u|2 + sλ2 ϕ|∇u|2 +e2sϕ sλϕa2 (∂ν d)(∂ν u)2 dγdt
Z≤
f 2 e2sϕ dxdt,∂Ω×(0,T )S×(0,T )for u(·, t) ∈ D(A) and u(·, 0) = u(·, T ) = 0.

Ω×(0,T )This kind of inequality for a parabolic operator is well known when the diffusion coefficient isregular or when the interface S is far from the boundary1. Here, the interface is tranverse tothe boundary and the weight function ϕ is not usual for a parabolic operator. Starting from thisestimate, we shall develop applications that estimate the restriction of u on some subsets of Ω ×(0, T ) from the knowledge of the normal derivative of u on a part of ∂Ω.

1

Le Rousseau J. & Robbiano L., Local and global Carleman estimates for parabolic operators with coefficients withjumps at interfaces. Inv. math (2011) 183:245-336.

-3-

Fractional diffusion equation
Adam Kubica
Warsaw University of Technology
A problem with the time fractional Caputo derivative and general elliptic operator is studied. We
prove the existence of weak and regular solutions and investigate the issue of continuity of solution
at initial time.

-4-Invariance for quasi-dissipative systems in Banach spaces
Piermarco Cannarsa
University of Rome Tor Vergata
In a separable Banach space, we study the invariance of a closed set K under the action of theevolution equation associated with a maximal dissipative linear operator perturbed by aquasi-dissipative term. Using the distance to the closed set, we give a general necessary andsufficient condition for the invariance of K. Then, we apply our result to several examples of partial
differential equations.

-5-