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

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.     

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.     

Compactness results for divergence type nonlinear elliptic equations

Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F _α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g _α and of coefficients a _α on M, we discuss the compactness problem of minimal energy type solutions of the equation

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.      

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).      

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

Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth

Let be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem in on ∂B_R when ɛ→⁺ for suitable positive numbers μ Mathematics Subject Classification (2000) 35J60, 35B33     

Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball

In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.     

Koksma–Hlawka type inequalities of fractional order

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order is in . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.      

Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups

We construct a family of completely regular semigroups with the property that each completely regular semigroup S with a finite number of -classes in each -class is non-cryptic if and only if S contains an isomorphic image of a member of . Each member F of is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to where is the natural partial order and a if and only if a ² = ab = ba. Hence the above result can be formulated in terms of and .      

Pseudo-conformal quaternionic CR structure on (4n + 3)-dimensional manifolds

We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}We study an integrable, nondegenerate codimension 3-subbundle on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an -valued 1-form ω locally on a neighborhood U such that and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic K?hler 4n-manifolds. From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure. The authors are grateful to ESI for financial support and hospitality during the preparation of this work. The first author acknowledge the support by Grant FWF Project P17108-N04 (Vienna) and Grant N MSM 0021622409 of the Ministry of Education, Youth and Sports (Brno).     

Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

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.      

$${\mathcal{R}}$$ -boundedness,pseudodifferential operators,and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class () and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with -bounded symbols, yielding by an iteration argument the -boundedness of λ(A−λ)⁻¹ in for some . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on with operator valued coefficients.     

Optimal Decompositions of Translations of L 2-Functions

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space . Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space , or equivalent the spectral theory of a unitary representation U of the rank-n lattice in . Starting with a non-zero vector , we look for relations among the vectors in the cyclic subspace in generated by ψ. Since these vectors involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L ²-independence. This refers to infinite linear combinations of integral translates of a fixed function with l ²-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals. Work supported in part by the U.S. National Science Foundation.     

Uniform Schauder estimates for regularized hypoelliptic equations

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: in , where Δ is the Laplace operator, m < n, and the limit operator is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter ϵ, of solution of the approximated equation L _ϵ u = f, using a modification of the lifting technique of Rothschild and Stein. These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.