Precise asymptotic formulas for variational eigencurves of semilinear two-parameter elliptic eigenvalue problems

We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u ^p + f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort of criticality for two-parameter problems seems to be new. Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003     

Anisotropic branching random walks on homogeneous trees

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff dimension of Λ∩Ω_μ, where Ω_μ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ω_μ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ω_μ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point. Received: 30 June 1998 / Revised version: 10 March 1999     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

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).     

Harmonic maps from manifolds of <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-Riemannian metrics

For a bounded domain Ω ⊂ R ⁿ endowed with L ^∞-metric g, and a C ⁵-Riemannian manifold (N, h) ⊂ R ^k without boundary, let u ∈ W ^1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C ^α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H ^n-2(Σ) = 0, such that for some α ∈ (0, 1). C. Y. Wang Partially supported by NSF.     

Solutions of elliptic equations involving critical Sobolev exponents with neumann boundary conditions

We consider the problem −Δu=|u| ^p−1u+λu in Ω with on δΩ, where Ω is a bounded domain inR ^N ,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR ^N , then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR ³, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial solution. This work was supported by the Paris VI-Leiden exchange program Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016.     

Extreme Jensen measures

Let Ω be an open subset of R ^d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u  dμ≤u(x) for every superharmonic function u on Ω. Denote by J _x (Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J _x (Ω)), the set of extreme elements of J _x (Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53. As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α _n } _n=1 ^∞ and a continuous function , there exists an entire function f≢0 satisfying f(α _n )=0 for all n, and |f(z)|≤M(z) for all z∈C.     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.     

Proprietes generiques de processus croises

LetS _φ be the skew product transformation(x, g)↦(Sx, gφ(x)) defined on Ω×G, where Ω is a compact metric space,G a compact metric group with its Haar measureh. IfS is a μ-continuous transformation where μ is a Borel measure on Ω, ergodic with respect toS, we study the setE ₀ of μ-continuous applications φ:Ω→G such that μ⩀h is ergodic (with respect toS _φ). For example,E ₀ is residual in the group of μ-continuous applications from Ω toG with the uniform convergence topology. We also study the weakly mixing case. Some arithmetic applications are given.     

Quasiconformal and harmonic mappings between Jordan domains

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C ^1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C ^1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C ^1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ^∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).      

Continuity of Marcinkiewicz integrals on homogeneous Morrey-Herz spaces

Let μmΩ,b be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO(Rn) function b(x). In this paper, we will study the continuity ofμΩ and μmΩ,b on homogeneous Morrey-Herz spaces.     

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ _Ω ^ρ , μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ on BMO(ℝ ⁿ ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog⁺ L(S ⁿ⁻¹). Here μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g _λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ ⁿ ) or to a certain Campanato space, then [μ _Ω,λ ^*,ρ (f)]², [μ _Ω,S ^ρ (f)]² and [μ _Ω ^ρ (f)]² are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.     

Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals

 If u is a minimizer of ∫_Ω F(|∇u|)dx−∫_Ω udμ, then the pointwise estimate

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 ⁻ⁿ where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ω_μ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ⁻¹ (Ω_μ) in the metric d is

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

L p -Boundedness of Marcinkiewicz Integrals with Hardy Space Function Kernels

We give the L ^p -boundedness for a class of Marcinkiewicz integral operators μ_Ω, μ^ast; _{Ω, λ} and μ_Ω, _s related to the Littlewood-Paley g-function, g ^* _λ-function and the area integral S, respectively. These operators have the kernel functions Ω∈H ¹ (S ⁿ⁻¹), the Hardy space on S ⁿ⁻¹. These results in this paper substantially improve and extend the known results. Received August 25, 1998, Accepted July 6, 1999     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

Convexes divisibles IV : Structure du bord en dimension 3

Divisible convex sets IV: Boundary structure in dimension 3 Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex: The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M. Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these triangles is dense in the boundary of Ω (see Figs. 1 to 4). Moreover, we construct examples of such divisible convex open sets Ω.      

Boundedness of generalized marcinkiewicz integral on the <Emphasis Type="Italic">Hp</Emphasis>-sobolev spaces

In this paper, the general Marcinkiewicz integral operator μΩ,α on the Hp Sobolev spaces under the proper condition of kernel Ω(x′) is considered. It is obtained that μΩ,α is bounded from H{ } to Lp for some 0<p≤1. Jiang and Jia were supported in part by Education Department of Zhejiang province.