Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks

We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed positive integer K there exist interior K–spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for any bounded and smooth domain there exist interior K–peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker. Received March 5, 1999 / Accepted June 11, 1999     

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

Remarks on an overdetermined boundary value problem

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006).     

Classical solutions of quasilinear parabolic systems on two dimensional domains

Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system.     

Boundary interface for the Allen–Cahn equation

We consider the Allen–Cahn equation

Local and asymptotic analysis of the flow generated by the Cahn–Hilliard–Gurtin equations

We consider the Cahn–Hilliard–Gurtin equation which corresponds, in the isotropic case, to the viscous Cahn–Hilliard equation. The convergence of its solutions toward some steady state is investigated by means of a proper generalization of the Lojasiewicz–Simon Theorem to nongradient-like flows. Furthermore, when the anisotropic coefficients are small, we prove that these steady states can be approximated by the corresponding stationary solutions of the viscous Cahn–Hilliard equation provided that the latter are local minimizers of the Ginzburg–Landau free energy. Received: April 26, 2004; revised: February 24, 2005     

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, centrifugal forcing as well as the quadratic potential with a fixed Ω > 0 being the rotational frequency. This model arises in the semiclassical limit of the Gross–Pitaevskii equation for Bose–Einstein condensates in a rotational frame. We investigate whether the action of dispersive rotational forcing complemented with the underlying potential prevents the generic finite time breakdown of the free nonlinear convection. We show that the rotating equations admit global smooth solutions for and only for a subset of generic initial configurations. Thus, the global regularity depends on whether the initial configuration crosses an intrinsic critical threshold, which is quantified in terms of the initial spectral gap associated with the 2 × 2 initial velocity gradient, λ ₂ (0) − λ ₁ (0), λ _j (0)=λ _j (∇ _x U₀) as well as the initial divergence, div_x (U₀). We also prove that for the case of isotropic trapping potential the smooth velocity field is periodic if and only if the ratio of the rotational frequency and the potential frequency is a rational number. The critical thresholds are also established for the case of repulsive potential. Finally the position density and the velocity field are explicitly recorded along the deformed flow map. Received: November 12, 2003; revised: May 4, 2004     

Decay rate of the range component of solutions to some semilinear evolution equations

We examine the rate of decay to 0, as t → +∞., of the projection on the range of A of the solutions of an equation of the form u′ + Au + |u| ^p−1 u = 0 or u′′ + u′ + Au + |u| ^p−1 u = 0 in a bounded domain of ^N , where A = −Δ with Neumann boundary conditions or A = −Δ − λ₁ I with Dirichlet boundary conditions. In general this decay is much faster than the decay of the projection on the kernel; it is often exponential, but apparently not always.     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

Boussinesq–Cerruti problems for the hexagonally symmetric system of elasticity in n–dimensional half-space

The Poisson matrices of the analoga to the Boussinesq–Cerruti boundary value problems for the operator of transversely isotropic elastostatics in n–dimensional half-space are computed by Fourier transformation and given in explicit form. (Received: May 4, 2004; revised: January 30, 2006)     

Boundary estimates for elliptic systems with L<Superscript>1</Superscript>–data

We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L ¹–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems.     

Mathematical foundation of the MFS for certain elliptic systems in linear elasticity

The method of fundamental solutions (MFS) is a Trefftz–type technique in which the solution of an elliptic boundary value problem is approximated by a linear combination of translates of fundamental solutions with singularities placed on a pseudo–boundary, i.e., a surface embracing the domain of the problem under consideration. In this work, we develop a mathematical framework for the numerical implementation of the MFS in elliptic systems. We obtain density results, with respect to the C ^ℓ-norms, which establish the applicability of the method in certain systems arising from the theory of elastostatics and thermo-elastostatics. The domains in our density results may possess holes and they satisfy the segment condition. This work was supported by a grant of the University of Cyprus.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The asymptotic behavior of viscosity solutions to the Cauchy–Dirichlet problem for the degenerate parabolic equation u _t  = Δ_∞ u in Ω × (0,∞), where Δ_∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of . Goro Akagi was supported by the Shibaura Institute of Technology grant for Project Research (no. 2006-211459, 2007-211455), and the grant-in-aid for young scientists (B) (no. 19740073), Ministry of Education, Culture, Sports, Science and Technology. Petri Juutinen was supported by the Academy of Finland project 108374. Ryuji Kajikiya was supported by the grant-in-aid for scientific research (C) (no. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe

We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved. Accepted 21 March 2001. Online publication 9 August 2001.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

Hilbert–Kunz multiplicity, McKay correspondence and good ideals¶in two-dimensional rational singularities

Hilbert–Kunz multiplicity is known to be a very mysterious invariant of a ring or an ideal. We will show a very beautiful formula on Hilbert–Kunz multiplicity for integrally closed ideals in two-dimensional Gorenstein rational singularities. In the proof, “McKay correspondence” and “Riemann–Roch formula” play essential roles. Also this formula gives a new significance to “good ideals”. Received: 25 October 2000     

BIFURCATION OF TWO-DIMENSIONAL KURAMOTO-SIVASHINSKY EQUATION

This paper deals with the steady state bifurcation of the K-S equation in two spatial dimensions with periodic boundary value condition and of zero mean. With the increase of parameter a, the steady state bifurcation behaviour can be very complicated. For convenience, only the cases a=2 and a=5 witl be discussed. The asymptotic expressions of the steady state solutions bifurcated from the trivial solution near a=2 and a=5 are given. And the stability of thenontriviat sotutions bifurcated from a=2 is studied. Of course, the cases a=n^2 m^2,n,m∈N(a≠2,5) can be similarly discussed by the same method which is used to discussing the cases a=2 and a= 5.