A Remark on the Linearized Stability of Positive Solutions for Systems Involving the p-Laplacian

The aim of this paper is to prove some stability result for nonlinear elliptic systems of the form where Δ_p denotes the p-Laplacian operator defined by Δ_pz = div(|∇ z|^p-2∇ z); p > 2, Ω is a bounded domain in R^N (N > 1) with smooth boundary where with h = 1 when α = 1, λ is a positive parameter and f,g are C² functin on [0,∞) × [0,∞). We prove stability and instability results of positive stationary solutions under various choices of f and g.     

The divergence of fluctuations for shape in first passage percolation

We consider the first passage percolation model on Z ^d for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R ^d and It is well known that if p < p _c , there exists a compact shape B _d ⊂ R ^d such that for all > 0, t B _d (1 − ) ⊂ B(t) ⊂ tB _d (1 + ) and G(t)(1 − ) ⊂ B(t) ⊂ G(t)(1 + ) eventually w.p.1. We denote the fluctuations of B(t) from tB _d and G(t) by In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB _d ) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting. Research supported by NSF grant DMS-0405150     

Commutativity of Almost <Emphasis Type="Italic">F</Emphasis>-algebras

Let A be an Archimedean vector lattice, let be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ ₀:A × A→ B is a positive orthosymmetric bimorphism, then there exists a positive bimorphism extension Ψ of Ψ ₀ to × in B which is orthosymmetric. This leads to a new and short proof of the commutativity of the almost f-algebras multiplications.     

Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation

The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space H^s(ℝ²). For critical case (α=), existence of global (large) solutions in H^s is proved for s≥ when is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu [4] for s = 1, 2 and the result of A. Cordoba and D. Cordoba [8] for s=. For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to is somewhat surprising and unexpected. For super-critical case (α ∈ (0,)), existence and uniqueness of global (large) solution in H^s is proved when the product is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].     

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Positive Projective Space of a <Emphasis Type="Italic">C</Emphasis>*-Algebra with a Tracial Conditional Expectation

Let be a C*-algebra, a subalgebra of its center and Φ: → a tracial faithful conditional expectation. We define the positive projective space as the quotient where G⁺ is the space of positive invertible elements of , and if there exists g invertible in such that a′ = |g|²a. When is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow ℙ⁺ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in ℙ⁺, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space ℙ⁺ with the given geodesic distance is non positively curved.     

Odd Degree Polynomials on Real Banach Spaces

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = expⁿ⁺¹τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ . Some of the work on this paper was done while the first author was a visitor to the Departamento de Análisis Matemático of the Universidad Complutense de Madrid, to which great thanks are given. The research of the second author was supported by grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

Approximating a Feller Semigroup by Using the Yosida Approximation of the Symbol of its Generator

We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (T_t)_t≥0 then the Feller semigroups (T_t^(v))_t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (T_t)_t≥0 as ν →∞.     

On the disconnection of a discrete cylinder by a random walk

We investigate the large N behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N ^{2 d} and comparable to the cover time of the slice , but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time disconnection occurs, a massive ``clogging' typically takes place in the truncated cylinders of height . These mechanisms are in contrast with what happens when d=1.     

The Singularity Property of Banach Function Spaces and Unconditional Convergence in L 1[0, 1]

Let X be a Banach function space, L^∞ [0, 1] ⊂ X ⊂ L¹[0, 1]. It is proved that if dual space of X has singularity property in closed set E ⊂ [0, 1] then: 1) there exists no orthonormal basis in C[0, 1], which forms an unconditional basis in X in metric of L¹[0, 1] space, 2) for the Hardy-Littlewood maximal operator M we have      

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

The Optimal Order for the p-th Moment of Sums of Independent Random Variables with Respect to Symmetric Norms and Related Combinatorial Estimates

For n independent random variables f₁, . . . ,f_n and a symmetric norm || ||_X on ℝⁿ, we show that for 1≤ p < ∞ Here is the disjoint sum of the f_i's and h* is the non-increasing rearrangement. Similar results (where L_p is replaced by a more general rearrangement invariant function space) were obtained first by Litvak, Gordon, Schütt and Werner for Orlicz spaces X and independently by S. Montgomery-Smith [22] for general X but without an explicit analysis of the order of growth for the constant in the upper estimate. The order is optimal and obtained from combinatorial estimates for doubly stochastic matrices. The result extends to Lorentz-norms l_{f, q} on ℝⁿ under mild assumptions on f. We give applications to the theory of noncommutative L_p spaces.     

A schlichtness theorem for envelopes of holomorphy

Let Ω be a domain in . We prove the following theorem. If the envelope of holomorphy of Ω is schlicht over Ω, then the envelope is in fact schlicht. We provide examples showing that the conclusion of the theorem does not hold in , n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in whose envelopes are multiply sheeted.     

Gaussian Estimates of Order α and L p -Spectral Independence of Generators of C 0-Semigroups

We prove L^p-spectral independence for generators of C₀-semigroups estimated by the positive C₀-semigroup . In the preliminary process of the proof, we obtain the asymptotic expansion formula for the integral kernel of the C₀-semigroup .     

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λ_G is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of . Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST “Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the hospitality.     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.     

Matrix Representations for Positive Noncommutative Polynomials

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x<sub>1</sub>, . . . , } and {h<sub>1</sub>, . . . , } for which q(X,H) is positive semidefinite whenever are tuples of selfadjoint matrices with ||X<sub>j</sub>|| ≤ 1 but H<sub>j</sub> unconstrained. The representation we obtain is a Gram representation in the variables h where P<sub>q</sub> is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose P_q such that the matrix P_q(X) is positive semidefinite for all ||X_j|| ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when g_x = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically to appear IOET July 2003] when restricted to x-domains of the type ||X_j|| ≤ 1. Partially supported by NSF, DARPA and Ford Motor Co. Partially supported by NSF grant DMS-0140112 Partially supported by NSF grant DMS-0100367