Regularity Near a Contact Point for Flame Propagation

In this paper we study a free boundary problem, arising from a model for the propagation of laminar flames. Consider a cylindrical region S in ? ⁿ , and the following free boundary problem with Dirichlet data on ? S: u _t  = Δ u in {u > 0} ∩ S, |? u|=1 on ? {u > 0} ∩ S and u = 0 on ? S. We show that if there is a contact point of the free boundary {u = 0, |? u|=1} with ? S, then the free boundary approaches ? S tangentially and it turns out to be a graph of C ^{1+α, α} function near the contact point. In particular, the space normal is Hölder continuous.     

On Some Finite Dimensional Representations of Symplectic Reflection Algebras Associated to Wreath Products

Let Γ _N : = S _N  ? Γ ^N be the wreath product of Γ, a finite subgroup of SL(2,C), by the symmetric group of degree N. In this article we classify all the irreducible representations of S _N  ? Γ ^N that can be extended to a representation of the associated symplectic reflection algebra H _1,k,c (Γ _N ) (where k is a complex number and c a class function on the nontrivial elements of Γ) for nonzero values of k.     

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ? ^N , N ≥ 2, possibly with boundary. Let Ω _R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω _R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.     

Multiplicative Lie Isomorphisms Between Prime Rings

Let ? be a prime ring with 1 containing a nontrivial idempotent E, and let ?′ be another prime ring. If Φ:? → ?′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ _T,S for all T, S ∈ ?, where Z′ _T,S is an element in the center 𝒵′ of ?′ depending on T and S.     

On Landis’ Conjecture in the Plane

In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W(z) be a measurable real vector-valued function and V(z) ≥0 be a real measurable scalar function, satisfying ‖W‖ _{L ^∞(R ²)} ≤ 1 and ‖V‖ _{L ^∞(R ²)} ≤ 1. Let u be a real solution of Δu ? ?(Wu) ? Vu = 0 in R ². Assume that u(0) = 1 and |u(z)| ≤exp (C ₀|z|). Then u satisfies inf _{|z 0| =R}  sup _{|z?z 0| <1}|u(z)| ≥exp (?CRlog R), where C depends on C ₀. In addition to the case of the whole plane, we also establish a quantitative form of Landis’ conjecture defined in an exterior domain.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Density results relative to the Dirichlet energy of mappings into a manifold

Let ?? be a smooth, compact, oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps u_k:Bⁿ → ?? with an equibounded Dirichlet integral give rise to elements of the space cart^2,1 (Bⁿ × ??). Assume that ?? is 1‐connected and that its 2‐homology group has no torsion. In any dimension n we prove that every element T in cart^2,1 (Bⁿ × ??) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u_k:Bⁿ → ?? with Dirichlet energies converging to the energy of T. © 2006 Wiley Periodicals, Inc.     

Many Solutions of Elliptic Problems on R n of Irrational Slope

ABSTRACT

We consider the problem ? Δ u + F _u (x, u) = 0 on R ⁿ , where F is a smooth function periodic of period 1 in all its variables. We show that, under suitable hypotheses on F, this problem has a family of non-self-intersecting solutions u _D , which are at finite distance from a plane of slope (ω,0,…,0) with ω irrational. These solutions depend on a real parameter D; if D ≠ D ^′, then the closures of the integer translates of u _D and of u _{D ^′} do not intersect.     

Some Extensions of PSL(2, p 2) are Uniquely Determined by Their Complex Group Algebras

In Tong-Viet's, 2012 work, the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras?

It is proved here that some simple groups of Lie type are determined by the structure of their complex group algebras. Let p be an odd prime number and S = PSL(2, p ²). In this paper, we prove that, if M is a finite group such that S < M < Aut(S), M = ?₂ × PSL(2, p ²) or M = SL(2, p ²), then M is uniquely determined by its order and some information about its character degrees. Let X ₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. As a consequence of our results, we prove that, if G is a finite group such that X ₁(G) = X ₁(M), then G ? M. This implies that M is uniquely determined by the structure of its complex group algebra.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

The Ambrosetti–Prodi Problem for the p-Laplace Operator

In this work we study the existence of a solution for the problem ? Δ _p u = f(u) + tΦ(x) + h(x), with homogeneous Dirichlet boundary conditions. Here the nonlinear term f(u) is a so-called jumping nonlinearity. In the proofs we use topological arguments and the sub-supersolutions method, together with comparison principles for the p-Laplacian.     

Finite Groups with Conjugacy Classes Number One Greater than Its Same Order Classes Number

ABSTRACT

Let k(G) be the number of conjugacy classes of finite groups G and π _e (G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |π _e (G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A ₅, L ₂(7), S ₅, Z ₃, Z ₄, S ₄, A ₄, D ₁₀, Hol(Z ₅), or Z ₃ ? Z ₄.     

An Equivalence of Categories for Lie Color Algebras

Abstract

Let k be a field with char k = p > 0 and G an abelian group with a bicharacter λ on G. For each p-(G,λ)-Lie color algebra L over k the p-universal enveloping algebra u(L) is a G-graded Hopf algebra,i.e.,a Hopf algebra in the category ^kG ? of kG-comodules. In this paper we describe a subcategory of ^kG ? which is equivalent to the category of the finite dimensional p-(G,λ)-Lie color algebras over k.     

Power Values of Derivations with Annihilator Conditions on Lie Ideals in Prime Rings

Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If au ^{n ₁} d(u) ^{n ₂} u ^{n ₃} d(u) ^{n ₄} u ^{n ₅} … d(u) ^{n _k?1} u ^{n _k}  = 0 for all u ∈ U, where n ₁, n ₂,…,n _k are fixed non-negative integers not all zero, then a = 0 and if a(u ^s d(u)u ^t ) ⁿ  ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S ₄, the standard identity in four variables.     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.