@

## QOOSNRQS@13:30--17:00

uҋyё
u
13:30?14:30
uҁFX c (k , M1)
ځFʒȖʏ Fourier ϊ̌]Ɋւ_Љ

u
14:45?15:45
uҁFrc K (k , M1)
ځFlK̍܂񂾊g^ Fisher-Kolmogorov ̒ɂ

Ou
16:00?17:00
uҁF쎛 h (k , M1)
ځF Fourier restriction norm method Ɋւ_Љ

## QOOSNPPU@15:30--17:40

uҋyё
u
15:30?16:30
uҁFK t (k )
ځFwɌ鎞Ԓx̎̕

u
16:40?17:40
uҁF ^ (k )
ځFِ̂锼gU̒Ɖ̏ŌۂƂ̊֌W

uҋyё
u
16:00?17:00
uҁFs mS (k )
ځF3 dߓ_𔺂ϕ݂̉̑ɂ

## QOOSNPW@15:30--16:30

u
FV (k )
Navier-Stokes ̎̐ƈ͋yщQx̊֌Wɂ

## QOORNPQPP@16:00--17:30

u
{ aL (L )
Removable sets for curvature equations of order $k$
v|
In this talk we study the removability of singular sets for the curvature equations of the form $H_k[u]=\psi$, which is determined by the $k$-th elementary symmetric function, in an $n$-dimensional domain. We prove the removability of isolated singularities of viscosity solutions to the curvature equations for $1 \le k \le n-1$. We also consider the class of generalized solutions" and prove the removability of a singular set which is a compact set of vanishing $(n-k)$-dimensional Hausdorff measure.

## QOORNPQS@13:30--14:30

u
Igor Rodnianski (Princeton University)
Stability of N-soliton states of NLS

## QOORNPPQV@16:00--17:30

u
~ (k )
On some generalization of the weighted Strichartz estimates for the wave equation and self-similar solutions to nonlinear wave equations
v|
dݕtStrichartz]ƌĂ΂Ag̉ɑ΂ dݕt̎]ɂčl@.̌^̕]́A g̏l̏ȏlɑ΂鎞ԑ ߂̏dvȓł邱ƂmĂ. {uł͓ɁA[Nbhԏ̋ɍWɊÂāAa ƋʕňقȂwׁ̃[OԂp^̕] člE@.̂悤ȋԂp邱ƂŁA܂ŋΏ̐ ̉̉ŎĂ]Ώ̐̉ȂɎ. ؖɂ͔g̉̋ʒa֐ɂWJp. ܂A̕]̉pƂĔg̎ȑ̑ .

## QOORNPPQO@16:00--17:30

u

Multi-bump standing waves with a critical frequency for nonlinear Schrodinger equations ijoint work with J. Byeon (POSTECH))
v|
VfBK[̔ÓTݔgɑΉ ݂ɂčl@.񕉃|eV֐ $V$ i$\liminf_{|x|\to \infty} V(x) > 0$)@ $\varepsilon^2 \Delta u - V(x)u +u^p=0$, $x\in \bf{R}^N$@ ̌̑ȉ~^ɑ΂āA $V$ _ꍇɁẢŁA ̋ɑ_A̋ɒlA$\varepsilon\to 0$ ̂ƂA قȂXP[悤ȁAV^݂̉̑ؖ.

## QOORNPPPS@13:30--15:00

u
14:30?15:30
uҁF Y (c H)
Incompressible ideal fluid motion with free boundary far from equilibrium
v|
̔ǵAɍLя̈ɂA񈳏kẑ REƂĒ莮.̎̎RÉA ̂QȂ^Ƃ̉ōl@邱Ƃ. ɑ΂A{uł͉QƂl.āAʂƐꂪ Rɋ߂ȂꍇłAԋǏӂɑ݂邱Ƃ.

u
15:30?17:00
uҁFΈ mi (c )
ځFHamilton-Jacobi ɑ΂ Relaxation
v|
AuXgNgFHamiltonian ʊ֐łȂꍇɁCHamilton-Jacobi a.e.̈ӖłLipschitzAȉ̑ɂĊe_ł̏鑀{ƂC ̏ꍇɁC֐͓ʉꂽHamilton-Jacobi̔SɂȂD Hamilton-Jacobiɑ΂RelaxationƌĂсCɂďЉD

u
Tj (k (w))
񐬕nt̑_Ci~NX

## QOORNPORO@16:00--17:30

u
a Iq (w|)
ՊEwM̉̋ɂ
v|
̔M@$u_t = \Delta u + u^p$@̙̔p$p$Sobolev̖ߍ݂̎w傫ꍇ́@ԑ̗LEAL Ŕ̐ڑi@ÓTɂȂ莞ő傷₠ ɎA@ÓTɂȂx̑݁j ēŋ߂̌ʂЉ.

## QOORNPOQR@16:00--17:30

u
@u (k@)
Structure of the solution set of a vector valued elliptic boundary value problem with a Lipschitz nonlinear term
v|
We consider the boundary value problem with the Dirichlet condition in a Banach space for a semilinear elliptic equation on a bounded domain in ${\bf R}^n$ whose nonlinear term satisfies the Lipschitz condition. If the Lipschitz constant $L$ is less than $\lambda_1$, then this problem has a unique solution, where $\lambda_1$ is the least eigenvalue of the corresponding (real valued) eigenvalue problem. On the other hand, for any $L>\lambda_1^{}$ we can consrtuct a nonlinear term with the Lipcshitz constant $L$ such that the solution set is homeomorphic to any prescribed closed subset of the Banach space.

## QOORNPOX@16:00--17:30

u
OY pV (k )
Remark on uniqueness of mild solutions to the Navier-Stokes equations
v|
nNavier-StokesɂẮAKatoAGiga- Miyakawaɂ邎ϕ ֐lƂ̋Ǐ݂悭mĂ܂A̋Ǐ̈Ӑ 藧NX̌ɂĂ[j܂.́A]̈Ӑ ̃NX̊gƂɊ֘AbbƎv܂.

## QOORNVR@16:00-17:30

u
x Y ( )
On the minimal solution for quasilinear degenerate elliptic equation and its blow-up
v|
\subsection{Introduction} Let $\Omega$ be a bounded open set of $\Bbb R^N \,(N\ge2)$ whose boundary $\partial\Omega$ is of class $C^{2}$. In connection with combustion theory and other applications, we are interested in the study of positive solutions of the quasi-linear elliptic boundary value problem $$\begin{cases}&L_p(u)= \lambda f(u) \text{ in } \Omega ,\\ & u=0\qquad\qquad \text{ on } \partial \Omega,\end{cases}\label{1.1}$$ where %$L_p(\cdot)$ is the $p$-Laplace operator defined by $L_p(\cdot)=$ $-\div \big(|\nabla \cdot|^{p-2}\nabla \cdot)\big)$. Here $p>1$, $\lambda$ is a nonnegative parameter and the nonlinearity $f$ is, roughly speaking, continuously differentiable, positive, increasing and strictly convex on $[0,+\infty)$. Typical examples are $f(t)= e^t$ and $(1+t)^q$ for $q>p-1$. When $p=2$, it is known that there is a finite number $\lambda^*$ such that (\ref{1.1}) has a classical positive solution $u\in C^2(\overline\Omega)$ if $0\lambda\lambda^*$. On the other hand no solution exists, even in the weak sense, for $\lambda>\lambda^*$. This value $\lambda^*$ is often called the extremal value and solutions for this extremal value are called extremal solutions. %It has been a very interesting problem to find and study the properties of these extremal solutions. In this talk we treat similar problems for the quasilinear operator $L_p(u)$. The minimal solution $u_\lambda\in C^{1}(\overline \Omega)$ is defined by as the smallest solution among all possible bounded solutions The linearized operator is given by $$L_p'(u)(\cdot) =-\div\Big(|\nabla u|^{p-2}\big(\nabla \cdot+(p-2)\frac{(\nabla u,\nabla \cdot)} {|\nabla u|^2}\nabla u\big)\Big).\label{1.2}$$ Note that $L_p(u)$ is not always differentiable at any point $u\in W^{1,p}_0(\Omega)$ in the sense of Frechet. We introduce a Hilbert space $V_{\lambda,p}(\Omega)$ and an admissible class of directions $\tilde V_{\lambda,p}(\Omega)\subset V_{\lambda,p}(\Omega)$ which depend essentially upon $u_\lambda$. Then the operator $L_p(\cdot)$ becomes differentiable at $u_\lambda$ in the direction to $\tilde V_{\lambda,p}(\Omega).$ Although $L_p'(u_\lambda)(\cdot)$ is a degenerate elliptic operator, it is shown that $L_p'(u_\lambda)(\cdot)$ has a compact inverse from $L^2(\Omega)$ to itself. This crucial property is based on the compactness of the imbedding; $V_{\lambda,p}(\Omega)\longrightarrow L^2(\Omega)$ for $\lambda\in (0,\lambda^*)$. \begin{df}\label{extremal}{\bf( Extremal value $\lambda^*$)} The extremal value $\lambda^*$ is defined as the supremum of $\mu$ such that: \par\noindent $(a)$ For any $\lambda \in (0,\mu]$ there exists the minimal solution $u_\lambda$ of (\ref{1.1}).\par\noindent $(b)$ The following Hardy type inequality is valid : $$\int_\Omega |\nabla u_\lambda|^{p-2} \Big(|\nabla \varphi|^2+(p-2)\frac{(\nabla u_\lambda,\nabla \varphi)^2} {|\nabla u_\lambda|^2}\Big)\,dx\ge \lambda \int_\Omega f'(u_\lambda)\varphi^2\,dx\label{1.3}$$ for any $\varphi\in V_{\lambda,p}(\Omega)$. Here $V_{\lambda,p}(\Omega)$ is defined by \begin{align} &V_{\lambda,p}(\Omega)= \{\varphi : || \varphi||_{V_{\lambda,p}} +\infty , \varphi = 0 \text{ on }\partial \Omega\},\\ & || \varphi||_{V_{\lambda,p}}= \bigg(\int_\Omega |\nabla u_\lambda(x)| ^{p-2}|\nabla \varphi|^2 \,dx\bigg)^{\frac12}.\end{align} \end{df} \begin{df}\label{accessibility}{\bf( Accessibility Condition )} The first eigenfunction $\hat \varphi^\lambda \ge 0$ is said to satisfy (AC) if for any $\e>0$ there exists a nonnegative $\varphi\in \tilde V_{\lambda,p}(\Omega)$ such that $$L'_p(u_\lambda)( \varphi-\hat \varphi^\lambda)+ | \varphi-\hat \varphi^\lambda|\le \e \max(\hat \varphi^\lambda,dist(x,\partial\Omega)) \quad\text{in } \Omega.\label{definition 1.3}$$ Here $\tilde V_{\lambda,p}(\Omega)= \{\varphi\in V_{\lambda,p}(\Omega), \nabla \varphi =0 \text{on some neighborhood of } F\}$, $F= \{ x\in \Omega : \nabla u_\lambda =0\}$. \end{df} \begin{df}\label{definition 10.1}{\bf( Growth Condition )} For $p>1$, a function $f(t)\in C^1([0,\infty))$ is said to satisfy (GC) if $f$ is increasing, strictly convex with $f(0)>0$ and $\frac{f'(t)} {f(t)^\frac{p-2}{p-1}}\, \text{is nondecreasing on } [0,\infty).\label{10.2}$ \end{df} \subsection{Results} \begin{thm}\label{theorem 1.1} (1) The extremal value $\lambda^*$ is positive. Moreover the first eigenvalue of $L'_p(u_\lambda)-\lambda f'(u_\lambda)$ is positive provided that $\lambda$ is suficiently small. \par\noindent (2) Let $u_\lambda \in C^1(\overline \Omega)$ be the minimal solution of (\ref{1.1}) for $\lambda\in (0,\lambda^*)$. We have as $\lambda\to \lambda^*$ a finite limit a.e. $u_{\lambda^*}(x) = \lim_{\lambda\to \lambda^*}u_{\lambda} (x).$ Moreover $u_{\lambda^*}\in W^{1,p}_0(\Omega)$ and $u_{\lambda^*}$ is a weak solution of (\ref{1.1}) with $\lambda=\lambda^*$. \end{thm} \begin{thm}\label{theorem 1.2} Assume that $p\in [2,\infty)$ Then for a sufficiently small $\lambda>0$ $u_\lambda$ is left differentiable at $\lambda$ in $V_{\lambda,p}(\Omega)$. Moreover the left derivative $v_\lambda \in V_{\lambda,p}(\Omega)$ satisfies $$\begin{cases}&L'_p(u_\lambda)v_\lambda-\lambda f'(u_\lambda)v_\lambda =f(u_\lambda), \quad\text{ in } \Omega\\ & v_\lambda= 0,\qquad\qquad\qquad\quad\qquad\qquad\text{ on } \partial\Omega.\end{cases} \label{1.5}$$\end{thm} \begin{thm}\label{theorem main } Assume that $1p+\infty$. Let $u_\lambda \in C^1(\overline \Omega)$ be the minimal solution of (\ref{1.1}) for some $\lambda>0$. Assume that $\hat\varphi^\lambda$ satisfies (AC). Then the first eigenvalue of $L'_p(u_\lambda)-\lambda f'(u_\lambda)$ is nonnegative. \end{thm} \begin{thm}\label{theorem 1.4} Let $u_{\lambda^*}$ be a unbounded extremal solution. Assume that (GC). Then there is no solution to (\ref{1.1}) provided that $\lambda> \lambda^*$. \end{thm} \begin{prop}\label{prop 1.1} Assume that $p\ge 2$. Let $u_{\lambda^*}$ be the extremal solution. Then the Hardy type inequality (\ref{1.3}) holds for $u_{\lambda^*}$. (When $1p2$, we also have a somewhat weak result.) \end{prop} \begin{prop}\label{prop 1.2} Assume that $1p\le 2$. For $\lambda>0$, let $u_\lambda$ be the minimal solution or possibly the extremal solution. Let $u\in W_0^{1,p}(\Omega)$ be a weak energy solution of (\ref{1.1}) satisfying the Hardy type inequality for any $\varphi\in V_{\lambda,p}(\Omega)$. Moreover, if $1p2$, then we assume that $|\nabla u|\ge|\nabla u_\lambda|\,\text{ a.e. in } \Omega.$ Then we have $\lambda=\lambda^*$ and $u =u_{\lambda^*}$(When $p>2$, we also have a somewhat weak result.) \end{prop} When $\Omega$ is a ball, we investigate these problems rather precisely by using {\bf the weighted Hardy type inequality with a sharp missing term}. % The extremal solutions are determined in most cases.

## QOORNUQU@16:00--17:30

u
| T (Hw@)
fMcuOE_E^̑AgN^[
v|
^vVAvƂfMcuOE_E^ i镡fMcuOE_E܂ށj̑ AgN^[݂̑ɂďqׂD̈Ӑ҂ ݂̑̂ݕۏ؂悤ȕfp[^[ɑ΂āi ́jʑ\CꂪlƓ֐ԂŃRpNg ȑAgN^[ƂDCӂ̋Ԏɑ΂đ AgN^[݂̑邱ƂɒӂĂDiǐG i򕌑jCcqijƂ̋j

## QOORNUPX@16:00--17:30

u
R\'emi Weidenfeld @(Universities of Paris-Sud and Aix-Marseille III)
Allen-Cahn equation and anisotropic mean curvature flow
v|
In material growth appear models where the state function satisfies an Allen-Cahn equation which contains Finsler metrics to treat the anisotropy of the material. In this work in common with D. Hilhorst (University of Paris-Sud) and M. bene\v{s} (Faculty of Nuclear Science and Physical Engineering, Praha), we consider the sequence $(u^\epsilon)$ solution of $(P^\epsilon)\quad \begin{cases} u_t = \nabla.\left( \Phi^0(\nabla u)\Phi^0_\xi(\nabla u)\right) +\fra{2}{\epsilon^2}u(1-u^2)+F\tilde{\Phi}^0(\nabla u) &\text{in \Omega\times (0,\infty),}\\ \Phi^0(\nabla u)\Phi^0_\xi(\nabla u).\nu &\text{on \partial\Omega\times (0,\infty),}\\ u(x,0)=u_0(x) &\text{for all x\in\Omega,} \end{cases}$ where $\Phi^0$ and $\tilde{\Phi}^0$ are two Finsler metrics, $\epsilon>0$ is a small parameter, $F$ is a function on $\Omega\times (0,\infty)$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$ and $\nu$ its normal unit vector. After giving the notions of normal vector and mean curvature in the context of Finsler metrics, we prove the convergence of $u^\epsilon$ to $\pm 1$ on both sides of an interface moving by mean curvature together with a forcing term. The proofs involve a well-posedness result for the limiting problem as well as generation and propagation results for Problem $(P^\epsilon)$.

## QOORNUPQ@16:00--17:30

u
b (Y)
gJ-{e^gUnɂJڑw̌Ƃ̉^ɂ
v|
{uł͋ԔlȃgJ-{e^nlC ̉Jڑw̐ƋɂĘ_D ɂJڑw͎̋̂Q̉ߒȂ邱Ƃ𐔊wI Ɏ. (i) ŒZԂ̂ɑJڑwiEʁjߒ (ii) ꂽJڑwEʕɏ]ĉ^ߒ PƕAllen-Cahn ɂẮCގ̌܂ł . ԈlȌW̏ꍇCEʕ͕ϋȗ ȂD܂ԔlȌW̏ꍇ́CEʕ͕ϋȗ ڗ̂ɂȂ(Ei-Iida-Yanagida, Nakamura-Matano- Hilhorst-Sch\"atzlet). CnɂẮCԈl̏ꍇ Ei-Yanagida (ii) ̊EʕϋȗɂȂ邱ƂĂ. {ł͂ł͍܂łȂĂȂ (i) ؖ邱Ƃł. ԔlȏꍇlĂ, Ԃߒ(i), (i) (ii) ֐ڂߒ, ߒ(ii) Ƃĉ̋Lq.

## QOORNTQX@16:00--17:30

u
Ԗx j (k )
VfBK[g̘An̑ɂ
v|
4ɂ邱̑̕, o],GlM[NX LԂł͒mĂȂ. ,g̏f[^ ${\dot H}^1$ ̂Ƃ, VfBK[̉GlM[NX $H^1$ LƂ, I-methodg 邱Ƃł邱Ƃ.

## QOORNTQQ@16:00--17:30

u
ω KY (Q )
Lotka-Volterra n̒yѐisg̑Iȕ\ɂ
v|
Lotka-Volterra n $$u_t = \varepsilon \, u_{xx} + u \, (1 - u - c \, v), \quad v_t = \varepsilon \, d \, v_{xx} + v \, (a - b \, u - v) \eqno{(1)}$$ Cp[^Ɋւyѐisg̑Iȕ\ɂčl@ Dr藝pāCpf̐𒲂ׂ邱ƂɂC(1) ̒y isg̕\ƁCgU $$u_t = \varepsilon \, u_{xx} + u \, (1 - u) \, (u - a), \quad 0 a 1\eqno{(2)}$$ ̂Ƃ͗ގĂ邱ƂĂĂD{uł́C(2) ̕\ TςɁC̍\ƑΉtȂC(1) ̕\Ɋւē Ă錋ʂЉ\ɂĂD

## QOORNTPET@16:00--17:30

u
} ( )
Ginzburg-Landau-Maxwell̑Qߋɂ
v|
ۂ̃fłGinzburg-Landau Maxwell̘An̑Qߋl. ̕n Q[Wsϐ邽߂ɁAt_ԂɈˑȂ 肷邱ƂłȂ. ̍uł t_̒ GlM[Ċ֐𗘗p邱ƂɂA蕽t_ߖTł Qߋc_A܂ 蕽t_ւ decay rate ɂĂ l@.

## QOORNTW@16:00--17:30

u
ML (k (V))
vY}^ɊÂFvY}ɉMIۂ̌
v|
Fɑ݂V̂́Aiʖ10^33gjA1牭̐̏Wcł́A ̋͂̏Wcł͒cA̋͒cAȂďo͒cA ƊKw\ȂĂ.͒ćA͊wIɕtɒBĂV̂̒ őK͂̕ł. ̋͒cɂ́Ax1xAx0.001 /cc 󔖓dvY}Ă. ̃vY}́AˉߒɂX ˂Ă. č1999NɑłグXϑqChandra ́A܂ł ϑuɔׂĈ|Iɍpx\ĂÅϑ肱܂ \Ȃ͒cvY}̑lȍ\݂̑炩ɂȂ. ̗lȍ\̌Aiߒ𖾂炩ɂŏdvȃL[h́AuMI vY}ہvłƉX͍lĂ. uł́AvY}^_ɊÂ MIvY}ۂ̌̌ЉA͒cvY}̔MIۂ̉𖾂 ɗL]ł邱Ƃ[鎖݂. KRIɔ񕽍tAߒ 舵ȂĂ͂Ȃ炸Ał̂̎̕舵͑Ss\ł. ǂ̗lȐiW]܂邩o邾̓IɏЉ.

## QOORNTP@16:00--17:30

u
g (k )
Weak solution for the Falk model system of shape memory alloys in energy class
v|
L Falk model system ll. ̃GlM[NX 鎞ԑ݂̑ƈӐɂĂ̌ʂ,܂ł̌ʂƔrȂqׂ.

## QOORNSPV@16:00--16:30

u
Hyunseok KIM ()
Existence results for viscous polytropic fluids with vacuum
v|
We study the full Navier-Stokes system for compressible fluids and are interested , in particular , to obtain existence results with nonnegative density. The energy and momentum equations lose the parabolicity in a region (so-called vacuum) where the density vanishes. We show how to overcome this difficulity by using a natural compatibility condition on the initial data.