3rd italian-japanese workshop on geometric properties for...
TRANSCRIPT
3rd Italian-Japanese Workshop
on Geometric Properties
for Parabolic and Elliptic PDE’s
Abstract
September 2–6, 2013
Tokyo Institute of Technology, Tokyo, Japan
3rd Italian-Japanese workshopon geometric properties
for parabolic and elliptic PDE’s
September 2–6, 2013
Tokyo Institute of Technology, Tokyo, Japan
Organizing Committee:
Filippo Gazzola (Politecnico di Milano)Kazuhiro Ishige (Tohoku University)Rolando Magnanini (Univesita di Firenze)Shigeru Sakaguchi (Tohoku University)Paolo Salani (Univesita di Firenze)
Eiji Yanagida (Tokyo Institute of Technology, Japan)
Local organizing Committee:
Junichi Harada, Toru Kan, Yohei Sato,Masataka Shibata, Masahiro Suzuki
This workshop is supported by
• Grant-in-Aid for Scientific Research from the Japan Society for the Promotion ofScience:
– Basic Research (S) No. 25220702 (Takayoshi Ogawa);
– Basic Research (A) No. 24244012 (Eiji Yanagida);
– Basic Research (B) No. 23340035 (Kazuhiro Ishige).
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ABSTRACTS
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Extremal solutions to Liouville-Gelfand typeelliptic problems with nonlinear Neumann
boundary conditions
Futoshi TakahashiDepartment of Mathematics, Osaka City University
& Osaka City University Advanced Mathematical InstituteSumiyoshi-ku, Osaka, 558-8585, Japan
Tel: (+81)(0)6-6605-2508E-mail: [email protected]
Abstract. We consider the Liouville-Gelfand type problems with nonlinearNeumann boundary conditions
−∆u + u = 0 in Ω,∂u∂ν
= λf(u) on ∂Ω,
where Ω ⊂ RN , N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞)is a smooth, strictly positive, convex, increasing function with superlinear at+∞, and λ > 0 is a parameter. It is known that there exists an extremalparameter λ∗ > 0 such that a classical minimal solution exists for λ < λ∗, andthere is no solution for λ > λ∗. In this talk, after introducing an appropriatenotion of weak solutions to our problem, we discuss several properties ofextremal solutions u∗ corresponding to λ = λ∗, such as regularity, uniqueness,and the existence of weak eigenfunctions associated to the linearized extremalproblem.
References
[1] F. Takahashi: Extremal solutions to Liouville-Gelfand type elliptic prob-lems with nonlinear Neumann boundary conditions, preprint availableat
http://www.sci.osaka-cu.ac.jp/math/OCAMI/preprint/index.html
Characterization of ellipsoids via an affine
overdetermined boundary value problem
Carlo NitschUniversita di Napoli “Federico II”
Abstract. The study of the optimal constant in an Hessian-type Sobolev inequalityleads to a fully nonlinear boundary value problem, overdetermined with non standardboundary conditions. We show that all the solutions have ellipsoidal symmetry. Inthe proof we use the maximum principle applied to a suitable auxiliary function inconjunction with an entropy estimate from affine curvature flow.
On the uniqueness of quadrature surfaces
Michiaki Onodera (Kyushu University)
One of the classical problems in potential theory is to specify a surface Γ for a prescribed electriccharge density µ in such a way that the uniform distribution of electric charges on Γ produces thesame potential (at least in a neighborhood of the infinity) as µ. Mathematically, this problem can beformulated as follows: Given a measure µ, find a surface Γ such that
(1)
∫h dµ =
∫Γh dHN−1
holds for all harmonic functions h defined in a neighborhood of Ω, where HN−1 denotes the (N − 1)-dimensional Hausdorff measure. We call such a surface Γ a quadrature surface of µ. The mean valueproperty of harmonic functions implies that (1) is valid when µ = ωNδ0 and Γ = ∂B(0, 1), where ωN isthe area of the unit sphere ∂B(0, 1) in RN and δ0 is the Dirac measure supported at the origin. Thus,the identity (1) can be seen as a generalization of the mean value formula for harmonic functions.
Another characterization of the problem is the following overdetermined problem:
(2)
−∆u = µ in Ω,
u = 0 on ∂Ω,∂u
∂n= −1 on ∂Ω,
where n is the outward unit normal vector to ∂Ω. Namely, the boundary value problem (2) possessessa solution u in Ω if and only if the boundary Γ := ∂Ω satisfies the identity (1). From this point ofview, the uniqueness of a quadrature surface Γ holds in the case where µ = ωNδ0 by the method ofmoving planes.
The existence of a quadrature surface Γ of a prescribed µ has been investigated by several authorswith different approaches. In particular, a variational method was successfully applied to obtain ageneral existence result. However, as a counterexample shows, the uniqueness of a quadrature surfacecannot hold in general. The collapse of uniqueness shown by the example indicates a bifurcationphenomenon of surfaces Γ = Γ(t) to (1) with a parametrized measure µ = µ(t).
Our main purpose is to establish a qualitative result on the uniqueness of a family of quadraturesurfaces Γ(t) of the parametrized measure µ(t) := HN−1⌊Γ(0)+tµ. We prove that, under appropriateregularity assumptions, a continuous family of quadrature surfaces Γ(t) is unique if the mean curva-ture of each Γ(t) is positive everywhere. In other words, the bifurcation of quadrature surfaces occursonly when the surface Γ(t) ceases to have the positivity of the mean curvature.
For the proof, we first establish an equivalent characterization of a continuous family of quadraturesurfaces by a geometric flow describing an evolution of moving surfaces. Indeed, it will be shown thata continuous family of surfaces becomes a family of quadrature surfaces if and only if it is a solution tothe geometric flow. This characterization enables us to reduce the uniqueness of a family of quadraturesurfaces to the unique solvability of the geometric flow. We then prove the unique solvability by adetailed spectral analysis, with the aid of a general existence theory for abstract evolution equations.
References
[1] Onodera, M., Geometric flows for quadrature identities. preprint.
Stability results for solutions of elliptic equations
with a level surface parallel to the boundary
Giulio CiraoloUniversita di Palermo
Abstract. Positive solutions of homogeneous Dirichlet boundary value problems forcertain elliptic equations must be radially symmetric if just one of their level surfacesis parallel to the boundary of the domain. In this talk, we shall review this result andshow its stability counterpart. In fact, we show that if the solution is almost constanton a surface at a fixed distance from the boundary, then the domain is almost radiallysymmetric, in the sense that is contained in and contains two concentric balls with thedifference of the radii (linearly) controlled by a suitable norm of the deviation of thesolution from a constant.
The location and uniqueness of a center of a body
Shigehiro Sakata
Tokyo Metropolitan UniversityE-mail: [email protected]
In this talk, we consider the location and uniqueness of a point which gives the maximum value of apotential with strictly decreasing distance kernel. A maximizer of a potential defines a center of a body(the closure of a bounded open set) in Rm. Our investigation is motivated by [O] and [BM].
Let Ω be a body in Rm (m ≥ 2) with a piecewise C1 boundary. We consider a potential of the form
KΩ(x) =
∫Ω
k(r)dy, x ∈ Rm, r = |x− y| , (1)
where k : (0,+∞) → R is a strictly decreasing C1 function such that KΩ becomes of class C1 on entireRm. Then KΩ has a maximum point only in the convex hull of Ω, and we call it a k-center of Ω ([S]).
In particular, when k(r) is given by
vα(r) :=
rα−m (0 < α = m),
− log r (α = m),(2)
KΩ is the rα−m-potential (a generalization of Riesz potential) V(α)Ω , and its extremum (maximum or
minimum according to the value of α) point is said an rα−m-center of Ω, after J. O’Hara in [O]. Heshowed the following statements in [O]:
(1) There is a body Ω having (at least) two rα−m-centers for some α.
(2) If α ≥ m+ 1, then Ω has a unique rα−m-center.
(3) If α < 1, and if Ω is convex, then Ω has a unique k-center.
The common idea of (2) and (3) is to show the concavity of KΩ on entire the convex hull of Ω.But we can restrict a region containing all centers smaller than the convex hull of Ω by using the
radial symmetry of the kernel. We call such a small region the minimal unfolded region ([O]) or the heart([BMS]) of Ω, denoted by Uf(Ω) or (Ω), respectively. Hence, in order to show the uniqueness of ak-center, it suffices to show the concavity of KΩ on the minimal unfolded region.
Our main results in this talk are to give sufficient conditions for the uniqueness of a k-center in two-dimensional case where KΩ is not always concave on entire the convex hull of Ω. To be precise, if k′(r)/ris increasing, and if any of the following conditions is satisfied, then Ω has a unique k-center:
• If Ω is an axially symmetric convex body.
• If Ω is a non-obtuse triangle.
References
[BM] L. Brasco and R. Magnanini, The heart of a convex body, Geometric properties for parabolic andelliptic PDE’s (R. Magnanini, S. Sakaguchi and A. Alvino eds), Springer INdAM Series 2, 2 (2013),49–66.
[BMS] L. Brasco, R. Magnanini and P. Salani, The location of the hot spot in a grounded convex conductor,Indiana Univ. Math. J. 60 (2011), 633–660.
[O] J. O’Hara, Renormalization of potentials and generalized centers, Adv. in Appl. Math. 48 (2012),365–392.
[S] S. Sakata, Movement of centers with respcet to various potentials, to appear in Trans. Amer. Math.Soc.
Asymptotics of solutions to classes of
nonlinear diffusions
Gabriele GrilloPolitecnico di Milano
Abstract. We shall discuss some variants of the porous media equation and of the fastdiffusion equation, including weighted analogues of such equations and problems posedon Riemannian manifolds. On the one hand, we shall show how smoothing and decayproperties of the evolutions are linked to certain functional inequalities, and sometimesequivalent to them. On the other hand, we shall discuss precise asymptotic results forsome of such equations both in the Euclidean context and on some negative curvaturemanifolds whose model is the hyperbolic space.
Diffusion equation with strong irreversibility
Goro AkagiKobe University
Abstract. In this talk, we discuss the well-posedness and the long-time behavior ofsolutions to the Cauchy-Dirichlet problem for the following strongly irreversible systemof diffusion type:
∂tu =(∆u + f(x, t)
)+,
where ∂tu = ∂u/∂t and (·)+ = max·, 0, with a given function f ∈ L2(0, T ; L2(Ω)) sat-isfying f(x, t) ≤ f∗(x) for a.e. (x, t) ∈ Ω × (0, T ) for some f ∗ ∈ L2(Ω). This problem isoriginally motivated by a study on a phase field model of crack propagation. We developan elliptic estimate for variational inequities of obstacle type and apply it along withtime-discretization technique to prove the existence of solutions and comparison princi-ple for the strongly irreversible diffusion equation. Moreover, the long-time behavior ofsolutions will be also analyzed by employing the devices developed so far.
This talk is based on a joint work with Masato Kimura (Kanazawa University,Japan).
Sharp decay estimates of Lq-norms
for nonnegative Schrodinger heat semigroups
Norisuke IokuEhime University
Abstract. Let H := −∆ + V be a Schrodinger operator on L2(RN), where N ≥ 2 andV ∈ Lr
loc(RN) with r > N/2. Assume that the operator H is nonnegative, that is,∫
RN
|∇φ|2 + V φ2
dx ≥ 0 for all φ ∈ C∞
0 (RN).
In this talk we focus on a nonnegative Schrodinger operator H := −∆+V with a radiallysymmetric potential V = V (|x|) behaving like
V (r) = ωr−2(1 + o(1)) as r → ∞,
whereω > −ω∗ and ω∗ := (N − 2)2/4.
We show the exact and optimal decay rates of the operator norm of the Schrodingerheat semigroup e−tH from Lp(RN) to Lq(RN),
‖e−tH‖(Lp→Lq) := sup
‖e−tHφ‖Lq(RN )
‖φ‖Lp(RN )
: φ ∈ Lp(RN) \ 0
,
as t → ∞, where 1 ≤ p ≤ q ≤ ∞. The decay rates of ‖e−tH‖(Lp→Lq) depend on whetherthe operator H is subcritical or critical and on the behavior of the positive harmonicfunction for the operator H.
This is a joint walk with Kazuhiro Ishige (Tohoku University) and Eiji Yanagida(Tokyo Institute of Technology).
References
[1] N. Ioku, K. Ishige, and E. Yanagida, Sharp decay estimates of Lq-norms for non-negative Schrodinger heat semigroups, J. Funct. Anal. 264 (2013), 2764–2783.
[2] B. Simon, Large time behavior of the Lp norm of Schrodinger semigroups, J. Funct.Anal. 40 (1981), 66–83.
On the structure of solutions to the Liouville equationin non-simply connected domains
Toru KanTokyo Institute of Technology, Japan
We discuss the structure of solutions of the Liouville equation in a planar domain.−∆u = λ eu in Ω,
u = 0 on ∂Ω.
It is known that the topology of domains strongly influences the structure of solutions. Inthis talk we particularly consider a non-simply connected domain defined byΩ = Ω0 \Dε,whereΩ0 is a planar domain andDε is a “hole” with a diameterε > 0. Then, for smallε,we can construct a solution whose profile is near the hole provided that the gradient of theRobin function onΩ0 does not vanish at the center of the hole. As a case where the gradientof the Robin function vanishes, we particularly treat an annular domain, that is,Ω0 andDε
are concentric circles. We also discuss how a mass of the solution behaves as it blows up.
Characterization of ellipsoids as K-dense sets
Michele MariniScuola Normale Superiore Pisa
Abstract. Let K ⊂ RN be any convex body containing the origin. A measurable setG ⊂ RN with finite and positive Lebesgue measure is said to be K-dense if, for anyfixed r > 0, the measure of G ∩ (x + rK) is constant when x varies on the boundaryof G (here, x + rK denotes a translation of a dilation of K). We will show that theabove property characterizes ellipsoids, more precisely we prove that if G is K-dense,then both G and K must be homotetic to the same ellipsoid.
Blow-up solutions for the heat equation with a
supercritical nonlinearity on the boundary
Junichi HaradaTokyo Institute of Technology
Abstract. We consider the following heat equation equation with a nonlinear boundarycondition:
ut = ∆u in Rn+ × (0, T ),
∂νu = uq on ∂Rn+ × (0, T ),
u(x, 0) = u0(x) ≥ 0 in Rn+,
where Rn+ = x = (x′, xn) ∈ Rn; xn > 0, ∂ν = −∂/∂xn and q > n/(n − 2). It is
known that a solution blows up on the boundary in a finite time if the initial data islarge enough. A goal of this talk is to show the existence of blow-up solutions whoseblow-up rate is different from the self-similar rate. We explain how to construct suchnon self-similar blow-up solutions and give their asymptotic behavior.
Fine asymptotics for radial solutions of the fast
diffusion equation on the hyperbolic space
Matteo MuratoriPolitecnico di Milano
Abstract. We consider positive radial solutions of the Fast Diffusion Equation on thehyperbolic space when the spatial dimension is greater than 2 and the exponent appear-ing in such equation is supercritical. By radial solutions we mean solutions dependingonly on the geodesic distance from a given point. We investigate their fine asymptoticproperties near the extinction time in terms of the behaviour of a separable solution,which is known to exist and to be positive and radial. In particular, we show that allpositive radial solutions converge to the separable solution, together with their spatialderivatives, in relative error.
Removable singularities of solutionsof semilinear heat equations
Kentaro Hirata
(Hiroshima University)
In 2010, Hsu [1] investigated removable singularities of bounded solutions of semilin-ear heat equations and of solutions of the heat equation under a suitable growth condition.His proof relied on the integral representation and several estimates of the Green func-tions for a circular cylinder and for the exterior by concrete functions. After that, Hui [2]gave another proof based on the parabolic Schauder estimate and the maximum principle.However his proof is not applicable to semilinear heat equations.
In this talk, I will discuss removable singularities of solutions of semilinear heat equa-tions under a suitable growth condition, i.e., the extension of their results, and will presenta proof based on the parabolic potential theory, particularly Watson’s result [3], and aniteration argument.
References
[1] S. Y. Hsu,Removable singularities of semilinear parabolic equations, Adv. Differen-tial Equations15 (2010), no. 1-2, 137–158.
[2] K. M. Hui, Another proof for the removable singularities of the heat equation, Proc.Amer. Math. Soc.138(2010), no. 7, 2397–2402.
[3] N. A. Watson,Green functions, potentials, and the Dirichlet problem for the heatequation, Proc. London Math. Soc. (3)33 (1976), no. 2, 251–298.
[4] K. Hirata, Removable singularities of semilinear parabolic equations, Proc. Amer.Math. Soc. (to appear)
Address: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
E-mail: [email protected]
Long time behaviour of solution to
degenerate/singular parabolic equations
Vincenzo VespriUnivesita di Firenze
Abstract. We consider a Cuachy- Dirichlet problem in bounded domains and a aCauchy problem in RN . We study the long time behaviour of nonnegative solutionsto degenerate/singular parabolic equations of p-Lapalacean/Porous medium type. Byapplying recent techniques introduced by E. DiBenedetto, U. Gianazza and myself weare able to extend results proved by J. L. Vazquez and his school for the prototypeoperators. All these resulat are obtained in collaboration with S. Piro-Vernier andF. Ragnedda.
Large time behavior of solutions of
a semilinear elliptic equation
with a dynamical boundary condition
Tatsuki KawakamiOsaka Prefecture University
Abstract. We consider the following initial value problem for a semilinear ellipticequation with a dynamical boundary condition:
(P )
−∆u = up, x ∈ RN
+ , t > 0,
∂tu + ∂νu = 0, x ∈ ∂RN+ , t > 0,
u(x, 0) = ϕ(x′) ≥ 0, x = (x′, 0) ∈ ∂RN+ ,
where N ≥ 2, RN+ := x = (x′, xN) : x′ ∈ RN−1, xN > 0, u = u(x, t), ∆ is the N -
dimensional Laplacian (in x), ∂t := ∂/∂t, ∂ν := −∂/∂xN , and p > 1. In this talkwe prove that there is a critical exponent for the existence of positive solutions ofproblem (P ). Furthermore, we show that small solutions behave asymptotically likesuitable multiples of the Poisson kernel.
This is a joint work with Marek Fila (Comenius University) and Kazuhiro Ishige (TohokuUniversity).
References
[1] M. Fila, K. Ishige, T. Kawakami, Large-time behavior of solutions of a semilinearelliptic equation with a dynamical boundary condition, Adv. Differential Equations18 (2013), 69–100.
[2] M. Fila, K. Ishige, T. Kawakami, Large-time behavior of small solutions of atwo-dimensional semilinear elliptic equation with a dynamical boundary condition,preprint
Stable standing waves of nonlinear Schrodinger equations
with a general nonlinear term
Masataka Shibata (Tokyo Institute of Technology)*1
In this talk, we consider the following minimizing problem under L2-constraint:
Eα = infu∈Mα
∫RN
1
2|∇u|2 − F (|u|) dx, Mα =
u ∈ H1(RN ); ∥u∥L2(RN ) = α
,
where N ≥ 1 and α > 0. The term F satisfies the following assumptions:
(F1) f ∈ C(C,C), f(0) = 0.(F2) f(r) ∈ R (r ∈ R), f(eiθz) = eiθf(z) (θ ∈ R, z ∈ C).(F3) lim|z|→0 f(z)/|z| = 0.
(F4) lim|z|→∞ f(z)/|z|l−1 = 0, where l = 2 + 4/N .
(F5) F (s) =∫ s
0f(τ)dτ (s ≥ 0), there exists s0 > 0 such that F (s0) > 0.
Under the assumptions (F1)–(F5), Eα > −∞ holds. Therefore, one would expect thatthere exists a global minimizer. If u is a global minimizer, there exists the Lagrangemultiplier µ ∈ R such that v(t, x) = eiµtu(x) is a standing wave of the nonlinearSchrodinger equation
ivt +∆v + f(v) = 0 in R× RN . (1)
To state orbital stability of (1), we put Sα = u ∈ Mα;u is a global minimizer. In[1], they showed that, if any minimizing sequence with respect to Eα isH1-precompactthen Sα is orbitally stable. Therefore, H1-precompactness for minimizing sequencesis important. Our main result is as follows:
Theorem ([2]). Assume (F1)–(F5). There exists α0 ≥ 0 such that there is no globalminimizer with respect to Eα if α < α0 and there exists a global minimizer withrespect to Eα if α > α0. Moreover, any minimizing sequence with respect to Eα isH1-precompact if α > α0.
In addition, we also discuss some almost critical conditions which determine α0 = 0or α0 > 0, and the existence results with respect to Eα0 under some conditions.
References
[1] T. Cazenave and P.-L. Lions. Comm. Math. Phys., 85(4):549–561, 1982.[2] M. Shibata. manuscripta mathematica, to appear
*1 Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku,Tokyo, 152-8551, JAPAN. e-mail: [email protected]
Vanishing diffusion limit of a parabolic equationwith a drift term
Hitoshi IshiiWaseda University, Japan
I present a pde approach to the vanishing viscosity limit of a par-abolic equation with a drift term, which is based on a joint work (inprogress) with Takis Souganidis of Chicago University. Let b be a vec-tor field on R
n, for which the origin is a (unique) globally asymptoticstable point. Let Ω ⊂ R
n be a domain which contains the origin andinvariant under the flow generated by b. Let ε > 0 and consider thesde dXε
t = b(Xεt ) d t +
√2ε dBt, where Bt denotes a n-dimensional
standard Brownian motion. The first exit time τ εx of the trajectory Xεt
from Ω is defined as
τ εx = inft ≥ 0 | Xεt ∈ Ω, Xε
0 = x.According to Freidlin and Wentzell (Random perturbations of dynami-cal systems, Springer-Verlag), there exists a constant m0 > 0 such thatτ εx ≈ em0/ε as ε → 0. This has an interpretation in terms of pde, wherethe pde is given by
uεt(x, t) = εΔuε(x, t) + b(x) · uε(x, t) in Ω × (0, ∞).
This interpretation and the related pde arguments are discussed.
Parabolic power concavity and
parabolic boundary value problems
Paolo SalaniUniversita di Firenze
Abstract. This talk is concerned with power concavity properties of the solution tothe parabolic boundary value problem
(P )
∂tu = ∆u + f(x, t, u,∇u) in Ω × (0,∞),
u(x, t) = 0 on ∂Ω × (0,∞),
u(x, 0) = 0 in Ω,
where Ω is a bounded convex domain in Rn and f is a nonnegative continuos functionon Ω × (0,∞) × R × Rn. We give a sufficient condition for the solution of (P ) to beparabolically power concave in Ω × [0,∞). This is a joint work with K. Ishige (TohokuUniversity).
References
[1] A. U. Kennington, Power concavity and boundary value problems, Indiana Univ.Math. J. 34 (1985), 687–704.
[2] K. Ishige and P. Salani, Parabolic quasi-concavity for solutions to parabolic prob-lems in convex rings, Math. Nachr. 283 (2010), 1526–1548.
The Phragmen-Lindelof theorem of fully nonlinear
elliptic PDEs in generalized unbounded domain
Kazushige NakagawaTohoku University
Abstract. A qualitative properties of viscosity solutions of fully nonlinear ellipticPDEs have been investigated as generalizations for classical elliptic PDE theory. Forinstance, the ABP maximum principle in unbounded domains, the Liouville property,the Hadamard principle, and the Phragmen-Lindelof theorem.
Our aim here is to extend the Phragmen-Lindelof theorem when PDEs have un-bounded coefficients in generalized weakG domaim.
This work is joint work with S. Koike (Tohoku University) and A. Vitolo (Universityof Salerno, Italy).
FABER-KRAHN INEQUALITIES
IN SHARP QUANTITATIVE FORM I & II
LORENZO BRASCO AND GUIDO DE PHILIPPIS
In this talk we present a sharp quantitative improvement of the celebrated Faber-Krahn inequality. The latter asserts that balls uniquely minimize the first eigenvalueof the Dirichlet-Laplacian, among sets with given volume. We prove that indeedmore can be said: the difference between the first eigenvalue λ(Ω) of a set Ω andthat of a ball of the same volume controls the deviation from spherical symmetry ofΩ. Moreover, such a control is the sharpest possible, in a sense that we will makeprecise. This settles a conjecture by Bhattacharya, Nadirashvili and Weitsman.The result is valid for more general geometric quantities, like
λ2,q(Ω) = minu∈W 1,2
0 (Ω)
∫Ω
|∇u|2 : ‖u‖Lq(Ω) = 1
.
The proof is based on various reduction steps: among these, a central role is playedby a Selection Principle for the torsional rigidity functional, which essentially per-mits to reduce the task to prove the desired result for small smooth deformationsof a ball.
The result here presented is contained in a recent joint paper with BozhidarVelichkov (SNS, Pisa).
L. B. Laboratoire d’Analyse, Topologie, Probabilites, Aix-Marseille Universite, 39Rue Frederic Joliot Curie, 13453 Marseille Cedex 13, France
E-mail address: [email protected]
G. D. P. Hausdorff Center for Mathematics, Endenicher Allee 62, D-53115 Bonn,
Germany
E-mail address: [email protected]
Turing patterns in prey-predator systems
with dormancy of predators
Masataka KuwamuraKobe University
Abstract. In this talk, we consider a prey-predator reaction-diffusion system incorpo-rating the effect of predator dormancy as follows:
ut = du∆u + s(u) − f(u)v
vt = dv∆v + kµ(u)f(u)v + α(u)w − m(v)v
wt = k(1 − µ(u))f(u)v − α(u)w − n(w)w,
(1)
where u and v denote the prey and predator densities, respectively, and w denotesthe density of predators with dormant state. The function µ(u) with 0 < µ(u) < 1determines the distribution of reproduction energy of predators between active anddormant states; α(u) determines the hatching of dormant predators (average dormancyperiod). These functions control predator dormancy dependent on the current preydensity. It should be noted that (1) can be reduced to a typical 2-component prey-predator reaction-diffusion system if µ ≡ 1 and α ≡ 0.
When n(w) ≡ 0, we show that periodic traveling waves as well as spatially peri-odic steady states can bifurcate from a coexisting equilibrium (spatially homogeneouspositive steady state) as du decreases, and give a simple and useful criterion for classi-fying these bifurcating patterns. Moreover, when n is sufficiently small, we numericallyshow transient spatio-temporal complex patterns which are the mixture of spatiallyperiodic steady states and periodic traveling waves. These results suggests that dor-mancy of predators is not a generator but an enhancer of spatio-temporal patterns inprey-predator systems.
References
[1] Kuwamura, M., Nakazawa, T., Ogawa, T., A minimum model of prey-predatorsystem with dormancy of predators and the paradox of enrichment, J. Math. Biol.58 (2009), 459-479.
[2] Kuwamura, M., Turing patterns in prey-predator systems with dormancy of preda-tors, submitted.
Blow-up set for a superlinear heat equation
and pointedness of the initial data
Yohei FujishimaOsaka University
Abstract. We study the location of the blow-up set of the solution for a superlinearheat equation
∂tu = ε∆u + f(u), x ∈ Ω, t > 0,
u(x, t) = 0, x ∈ ∂Ω, t > 0,
u(x, 0) = ϕ(x) ≥ 0 ( 6≡ 0), x ∈ Ω,
(P)
where ∂t = ∂/∂t, ε > 0, N ≥ 1, Ω ⊂ RN is a domain, ϕ ∈ C2(Ω)∩C(Ω) is a nonnegativebounded function, and f is a positive convex function in (0,∞). In [1], the location ofthe blow-up set for problem (P) with f(u) = up (p > 1) was studied for the casewhere ε > 0 is sufficiently small, and the invariance of the equation under some scaletransformation for the solution played an important role to characterize the location ofthe blow-up set. However, (P) does not possess such scale invariance in general, and wecan not directly apply the argument of [1] to problem (P).
In this talk we introduce a new transformation for problem (P), and generalize theresults of [1]. In particular, we show the relationship between the location the blow-upset for problem (P) and pointedness of the initial function ϕ under suitable assumptionson f for the case where ε > 0 is sufficiently small.
References
[1] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and point-edness of the initial data, Indiana Univ. Math. J. 61 (2012), 627–663.
SYMMETRY BREAKING IN CONSTRAINED CHEEGER TYPE
ISOPERIMETRIC INEQUALITIES
FRANCESCO DELLA PIETRA (UNIVERSITA DEGLI STUDI DEL MOLISE)
Joint work with B. Brandolini, C. Nitsch and C. Trombetti (Universita degli studi diNapoli “Federico II”)
Abstract. The aim of the talk is to present some results on the optimal constant C(Ω)in the Sobolev inequality
‖u‖Lq(Ω) ≤ C(Ω)‖Du‖(Rn)
1 ≤ q < 1∗, for BV function which are zero outside Ω and with zero mean value inside Ω.The study of C(Ω) leads to the definition of a Cheeger type constant. We are interested infinding the best possible embedding constant in terms of the measure of Ω alone. We setup an optimal shape problem and we completely characterize, on varying the exponentq, the behavior of optimal domains. Among other things we establish the existence ofa threshold value 1 ≤ q < 1∗ above which the symmetry of optimal domains is broken.Significant differences between the cases n = 2 and n ≥ 3 are emphasized.
Date: June 24, 2013.
The dynamics of a parabolic quasilinear
boundary value problem
associated to the mean curvature operator
Kazuhiro TakimotoHiroshima University
Abstract. In this talk, we are concerned with the dynamics of the parabolic quasilinearmodel
∂u
∂t−
(u′√
1 + κ(u′)2
)′
= λV (x)u, 0 < x < 1, t > 0
u(0, t) = 0, u(1, t) = 0,
u(x, 0) = u0 > 0,
where λ ∈ R, κ ∈ (0, 1], u0 ∈ C[0, 1], V ∈ C1[0, 1] satisfies V > 0 (V ≥ 0 but V 6= 0),and ′ stands for the spatial derivative. This is a quasilinear perturbation, through themean curavture operator, of the classical linear heat equation. Among our results forthis model, we can say that the mean curvature has the effect of maintaining boundedall classical positive steady-states of the model, though their gradient may blow-upsomewhere. The dynamics of the positive solutions of the model is fully described.
This is a joint work with S. Cano-Casanova (Comillas University) and J. Lopez-Gomez (Complutense University).
New mathematical models for suspension bridges
Filippo GazzolaPolitecnico di Milano
Abstract. We suggest two new mathematical models in order to describe oscillationsin suspension bridges. A Hamiltonian system is introduced as a model for a discretizedversion of a bridge; this model enables us to explain some unexpected phenomena inbridges and to numerically replicate their oscillations. Then we introduce a continuousmodel, based on a variational problem related to a fourth order PDE. The combinationof these two models gives a detailed description of the statics and dynamics of suspensionbridges.
The scale-invariant log-Hardy inequalities and
related variational problems
Michinori IshiwataFukushima University
Abstract. The standard Hardy inequality states that(N − p
p
)p ∫RN
|u(x)|p
|x|pdx ≤
∫RN
|∇u|pdx (1)
holds for every u ∈ W 1,p(RN), where N ≥ 2 and 1 ≤ p < N . This type of inequalityarises in vast area of mathematical sciences, e.g, the verification of the stability of ahydrogen atom. In this talk, we focus on the critical case p = N in the unit ballB1 := x ∈ RN : |x| < 1. In this case, it is known that the inequality(
N − 1
N
)N ∫B1
|u(x)|N
|x|N(1 + log 1
|x|
)N≤∫
B1
|∇u(x)|Ndx, u ∈ W 1,N0 (B1) (2)
holds. Indeed, the critical inequality of the type (2) in a bounded domain which containsthe origin is known (see [1]).
In spite of the natural similarity between the critical inequality (2) and the originalinequality (1), there exists a crucial difference between them. Namely, there seem to existno natural scale invariance properties for the critical inequality (2) and its generalizationwith remainder terms, while the original inequality (1) is invariant under the scaling
uλ(x) = λN−p
p u(λx). In this talk, we introduce a generalization of the critical inequality(2) which is invariant under a certain scaling and discuss related variational problems.This is a joint work with Prof. Ioku in Ehime university.
References
[1] D. E. Edmunds and H. Triebel: Sharp Sobolev embeddings and related Hardy inequali-ties: the critical case. Math. Nachr. 207 (1999), 79-92.
[2] N. Ioku and M. Ishiwata: Scale invariant form of critical Hardy’s inequality. preprint.
Open problems in the study of critical points of
solutions of elliptic PDE’s
Rolando MagnaniniUniversita di Firenze
Abstract. Anybody who has taken a class in Calculus knows the importance of the criticalpoints of a differentiable function. In mathematical research, information such as the number,nature and location of critical points has often proven decisive in fields such as DifferentialGeometry, Dynamical Systems, Partial Differential Equations or Inverse Problems. In my talk,I shall focus on methods that allow to infer information on the occurrence and/or number ofcritical points of solutions of elliptic PDE’s from their (qualitative) behavior on the boundaryof a domain. Most of the results I will present concern solutions on domains of the Euclideanplane. Obtaining similar results in general dimension is a much needed and challenging task.With the aim of stimulating further research, I shall present a selection of open problems inthis field .
