Bubbling solutions for an anisotropic Emden–Fowler equation

We consider the following anisotropic Emden–Fowler equation where is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity can approach to as . These results show a striking difference with the isotropic case [ Constant].     

Rank one solvable <Emphasis Type="Italic">p</Emphasis>-adic differential equations and finite Abelian characters via Lubin–Tate groups

We introduce a new class of exponentials of Artin–Hasse type, called π-exponentials. These exponentials depend on the choice of a generator π of the Tate module of a Lubin–Tate group over . They arise naturally as solutions of solvable differential modules over the Robba ring. If is isomorphic to over , we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin–Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of , whose residue field is a given Artin–Schreier extension of , where k is the residue field of K. We then compute explicitly the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one φ-module over attached to a rank one representation of , defined by an Artin–Schreier character.     

Higher-order energy expansions and spike locations

We consider the following singularly perturbed semilinear elliptic problem: where is a bounded domain in R ^N with smooth boundary , is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional defined by where . Ni and Takagi ([29, 30]) proved that for a single boundary spike solution , the following asymptotic expansion holds: where c ₁ > 0 is a generic constant, is the unique local maximum point of and is the boundary mean curvature function at . In this paper, we obtain a higher-order expansion of where c ₂, c ₃ are generic constants and is the scalar curvature at . In particular c ₃ > 0. Some applications of this expansion are given.Received: 14 January 2003, Accepted: 28 July 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): Primary 35B40, 35B45; Secondary 35J25     

Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs

Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C^*-algebra C^*(E) of E which is generated by a Cuntz-Krieger E-family {s_e, p_v}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C^*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Let be such that each is a signed measure on R ^d belonging to the Kato class K _{d, 1}. A Brownian motion in R ^d with drift is a diffusion process in R ^d whose generator can be informally written as . When each is given by U ⁱ (x)dx for some function U ⁱ , a Brownian motion with drift is a diffusion in R ^d with generator . In Kim and Song (Ill J Math 50(3):635–688, 2006), some properties of Brownian motions with measure-value drifts in bounded smooth domains were discussed. In this paper we prove a scale invariant boundary Harnack principle for the positive harmonic functions of Brownian motions with measure-value drifts in bounded Lipschitz domains. We also show that the Martin boundary and the minimal Martin boundary with respect to Brownian motions with measure-valued drifts coincide with the Euclidean boundary for bounded Lipschitz domains. The results of this paper are also true for diffusions with measure-valued drifts, that is, when is replaced by a uniformly elliptic divergence form operator with C ¹ coefficients or a uniformly elliptic non-divergence form operator with C ¹ coefficients. The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. The research of P. Kim is supported by Research Settlement Fund for the new faculty of Seoul National University.     

Large solutions for biharmonic maps in four dimensions

We seek critical points of the Hessian energy functional , where or Ω is the unit disk in and u : Ω → S ⁴. We show that has a critical point which is not homotopic to the constant map. Moreover, we prove that, for certain prescribed boundary data on ∂B, E _B achieves its infimum in at least two distinct homotopy classes of maps from B into S ⁴. The author was partially supported by SNF 200021-101930/1.     

A formula for the Bloch norm of a C 1-function on the unit ball of ℂ n

For a C ¹-function f on the unit ball ⊂ ℂ ⁿ we define the Bloch norm by , where is the invariant derivative of f, and then show that . Supported by MNZŽS Serbia, Project No. 144010.     

Minimizing total completion time for UET tasks with release time and outtree precedence constraints

Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n ²)-time algorithm for the problems , outtree and , outtree . In this note, we show that their algorithm admits an O(n log n)-time implementation.     

The <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> sphere of a 4-manifold

Let denote the set of even integers . We prove that when H ≥ X ^0.33, almost all integers can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.      

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with ${\nabla=D + S}In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with induced by the one-parameter family of connections given by and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-K?hler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt ^*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt ^*-bundles generalised para-pluriharmonic maps into , respectively, into SO ₀(n,n)/U ^π(C ⁿ ), where U ^π(C ⁿ ) is the para-complex analogue of the unitary group.      

Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X E and C F, two multifunctions Γ : X → 2 ^X and Ф : X → 2 ^C , and a single-valued map ψ : , find a pair such that , Ф and for all . We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Ф. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213–228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E ^* and ).     

Characterization of order 3 algebraic immersed minimal surfaces of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

In this paper we prove that if is a minimal immersion of a compact surface and , for some homogeneous polynomial f of degree 3 on R ⁴, then, M is a torus and is one of the examples given by Lawson (1970, Complete minimal surfaces in S ³. Ann. Math. 92(2), 335–374).      

Asymptotic behaviour of renormalized solutions to elliptic equations with measure data and G-converging operators

We study the limit as n goes to +∞ of the renormalized solutions u ⁿ to the nonlinear elliptic problems

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007     

Blow-up criterion for 2-D Boussinesq equations in bounded domain

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U _t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control and by and . Furthermore, can control by using vorticity transportation equations. At last, can control . Thus, we can find a blow-up criterion in the form of .      

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

A Multiplicity Result for the p-Laplacian Involving a Parameter

We study existence and multiplicity of positive solutions for the following problem

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation