A Fully Nonlinear Nonhomogeneous Neumann Problem

Let Ω be an open bounded set in ℝ^N, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|^p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂ν_a related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝ^N, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, f∈L¹(ℝ^N), g∈L¹(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,

Regularity of the Extremal Solution of Some Nonlinear Elliptic Problems Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian

We consider the equation on a smooth bounded domain of with zero Dirichlet boundary conditions where p ≥ 2, λ > 0 and f satisfies typical assumptions in the subject of extremal solutions. We prove that, for such general nonlinearities f, the extremal solution u ^* belongs to L ^∞(Ω) if N < p + p/(p − 1) and if N < p(1 + p/(p − 1)). This work was partially supported by MCyT BMF 2002-04613-CO3-02.     

Second-Order Optimality Conditions for Constrained Domain Optimization

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫ _Ω g(x) dx=1, a domain is sought which maximizes either , fixed x ₀∈Ω, or ℱ(Ω)=∫ _Ω F(x,u(x)) dx, where u solves the Dirichlet problem Δu(x)=−f(x), x∈Ω, u(x)=0, x∈∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals and ℱ. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.     

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

On sum-product bases

We investigate additive-multiplicative bases in . Let , s>2, and . It is proved that , provided min {|B| ^s/2|A|^(s−2)/2,|A| ^s/2|B|^(s−2)/2}>p ^s/2. This note is supported by “Balaton Program Project” and OTKA grants K 61908, K 67676.     

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

Weighted a priori estimates for solution of (−Δ)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis> = <Emphasis Type="Italic">f</Emphasis> with homogeneous dirichlet conditions

Let u be a weak solution of (-△)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω  Rn. Then, the main goal of this paper is to prove the following a priori estimate:‖u‖ Wω2 m,p(Ω) ≤ C ‖f‖ Lωp (Ω),where ω is a weight in the Muckenhoupt class Ap.     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

On the Existence of Multiple Periodic Solutions for the Vector <Emphasis Type="Italic">p</Emphasis>-Laplacian via Critical Point Theory

We study the vector p-Laplacian

Some Remarks on a Class of Nonuniformly Elliptic Equations of p-Laplacian Type

This paper deals with the existence of weak solutions in W ₀¹(Ω) to a class of elliptic problems of the form

Greedy expansions in Banach spaces

We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary . For convenience we assume that is symmetric: implies . The primary goal of this paper is to study representations of an element f∈X by a series

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂_ν u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L ^∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s| ^p ) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) =  s ^p then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential equation is of the form h(x, u) with h satisfying suitable growth conditions.     

Distances Between <Emphasis Type="Italic">σ</Emphasis>-Fields on a Probability Space

For a probability space (Ω,ℱ,P) and two sub-σ-fields we consider two natural distances: and . We investigate basic properties of these distances. In particular we show that if a distance (ρ or ) from ℬ to is small then there exists Z∈ℱ with small P(Z), such that for every B∈ℬ there exists such that B∖Z and A∖Z differ by a set of probability zero. This improves results of Neveu (Ann. Math. Stat. 43(4):1369–1371, [1972]), Jajte and Paszkiewicz (Probab. Math. Stat. 19(1):181–201, [1999]).      

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

A Conditional 0–1 Law for the Symmetric <Emphasis Type="Italic">σ</Emphasis>-field

Let (Ω,ℬ,P) be a probability space, a sub-σ-field, and μ a regular conditional distribution for P given . For various, classically interesting, choices of (including tail and symmetric), we prove the following 0–1 law: There is a set such that P(A ₀)=1 and μ(ω)(A)∈{0,1} for all and ω∈A ₀. If ℬ is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.      

Iterated Logarithm Law for Anticipating Stochastic Differential Equations

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.