Titles and Short Summaries of the Talks

45 Functional Equations
Final: 2013/2/7
36 Norisuke Ioku (Ehime Univ.) ♯ On the best constant for the Hardy inequality in the limiting case with
scale invariance · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 12
Summary: We are concerned with Hardy type inequalities in limiting cases with scale invariance.
Using the oscillation of rearranged function which is defined by Bennett–Devore–Sharpley, we show
Hardy type inequalities in limiting cases with scale invariance. We also show that the Hardy
inequality with logarithmic term can be proved as corollary of the result.
37 Hiroya Ito (Univ. of Electro-Comm.) ♯ A generalization of the Korn inequality · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
14:15–16:15
Summary: We consider a generalization of Korn’s inequality, which is essential in mathematical
elasticity, from the viewpoint of the coercivity for first order elliptic systems with constant coeffi-
cients. Some necessary and sufficient conditions for such coercivity to hold are known from Nečas’s
work. The main purpose of this talk is to present a new necessary and sufficient condition, which
is in general easier to check than the known conditions. Although the establishment of the new
condition is not so easy in the sense that it relies on a certain tool in algebraic geometry, the
condition provides some useful information about what are vague in Nečas’s argument. Moreover,
an inequality stronger than the usual Korn inequality is presented.
38 Naoyuki Ichihara (Hiroshima Univ.) ∗ On the criticality of viscous Hamilton–Jacobi equations · · · · · · · · · · · · · 15
Summary: We deal with a nonlinear additive eigenvalue problem for viscous Hamilton–Jacobi
equations. Qualitative properties of the principal eigenvalue and associated eigenfunction are
studied. Such analysis plays a key role in studying the corresponding stochastic optimal control
problem. Our results can be regarded as a nonlinear extension of the criticality theory for
Schrödinger operators with decaying potentials.
39 Tomoyuki Niizato (Osaka Univ.) ∗ Almost global existence of solutions to the short-pulse equation · · · · · · 10
Summary: We consider the Cauchy problem for the short-pulse equation: ∂tu − ∂−1 (
3
x u = ∂x u ) .
We prove an almost global existence of solutions to the short-pulse equation. More precisely, we
have a lower bound of maximal existence time T as follows: T ≥ exp ( B
ε2 initial data.
) , where ε is the size of
40 Takamori Kato (Kyoto Univ.) ♯ Unconditional well-posedness of the fifth order KdV equation with
Kotaro Tsugawa (Nagoya Univ.) periodic boundary condition · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: We consider the well-posedness of the Cauchy problem for the fifth order KdV equation
in the periodic setting when initial data is given in the Sobolev space H s . This equation is one
of the KdV hierarchies. The direct iteration method does not work for any s ∈ R because three
derivatives are included in the resonant parts. To overcome this difficulty, we use the algebraic
structure of the nonlinear terms. Strong nonlinear interactions are canceled by symmetry of the
nonlinear terms and we obtain the unconditional well-posedness for s ≥ 1. This result is optimal in
some sense.
41 Nakao Hayashi (Osaka Univ.) ∗ Logarithmic time decay and cubic nonlinear Schrödinger equations · · · 10
Summary: We consider the Cauchy problem for cubic nonlinear Schrödinger equations. Our
purpose is to study the influence of the resonance cubic term to another type of cubic nonlinearity.
Under some conditions, we prove time decay of small solutions with a logarithmic correction.