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 .      

Renormalized solutions of nonlinear parabolic equations with general measure data

Let a bounded open set, N ≥  2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model is

On the number of moduli of plane sextics with six cusps

Let be the variety of irreducible sextics with six cusps as singularities. Let be one of irreducible components of . Denoting by the space of moduli of smooth curves of genus 4, we consider the rational map sending the general point [Γ] of Σ, corresponding to a plane curve , to the point of parametrizing the normalization curve of Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We know that , where ρ(2, 4, 6) is the Brill–Noether number of linear series of dimension 2 and degree 6 on a curve of genus 4. We prove that both irreducible components of have number of moduli equal to seven.      

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Variational integrals of splitting-type: higher integrability under general growth conditions

Besides other things we prove that if , , locally minimizes the energy

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term. Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated. Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.     

Characterization of balls by Riesz-potentials

For a bounded convex domain and consider the unit- density Riesz-potential . We show in this paper that u  =  const. on ∂G if and only if G is a ball. This result corresponds to a theorem of L.E. Fraenkel, where the ball is characterized by the Newtonian-potential (α = 2) of unit density being constant on ∂G. In the case α = N the kernel |x − y| ^α-N is replaced by  − log|x − y| and a similar characterization of balls is given. The proof relies on a recent variant of the moving plane method which is suitable for Green-function representations of solutions of (pseudo-)differential equations of higher-order.      

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ _∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U _s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P _s and units C _s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).     

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

Dy namic of Abelia n Subgroups of GL( n, $$\mathbb{C}$$): A Structure Theorem

In this paper, we characterize the dynamic of every Abelian subgroups of , or . We show that there exists a -invariant, dense open set U in saturated by minimal orbits with a union of at most n -invariant vector subspaces of of dimension n−1 or n−2 over . As a consequence, has height at most n and in particular it admits a minimal set in . This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15     

Mean-value theorems and extensions of the Elliott–Daboussi theorem on Beurling’s generalized integers (I)

Let be the generalized integers n _j associated with a set of generalized primes p _i in Beurling’s sense. On the basis of the general mean-value theorems, established in our previous work, for multiplicative function f(n _j ) defined on , we prove extensions, in functional form and in mean-value form, of the Elliott–Daboussi theorem to high order mean-values. For the main result, let α,ρ, and τ be positive real constants such that α > 1,ρ≥1 and . Then a multiplicative function f satisfies the following conditions, with some constant , (1) All four series

Multilevel frames for sparse tensor product spaces

For Au = f with an elliptic differential operator and stochastic data f, the m-point correlation function of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A ^(m) of A. Sparse tensor products of hierarchic FE-spaces in are known to allow for approximations to which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here how to achieve the fast computation of sparse approximations of for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method.     

Rectifiable oscillations in second-order half-linear differential equations

Second-order half-linear differential equation (H): on the finite interval I = (0,1] will be studied, where , p > 1 and the coefficient f(x) > 0 on I, , and . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when , λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L ^p nonintegrability of the derivative of all solutions of the equation (H).      

Cyclicity of bicyclic operators and completeness of translates

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on .     

Optimal quadrature problem on classes defined by kernels satisfying certain oscillation properties

We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula

Some remarks on solutions to an overdetermined elliptic problem in divergence form in a ball

We prove the existence of a continuum of non-radial pairs (k,u) solutions to the following overposed problem div in B _r , u = 0 and on ∂B _r , where B _r is the Euclidean ball centered at zero of radius r in . Dedicato a Marco e Andrea Provera.     

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation