Dini Condition Research Articles

Morrey Regularity of Solutions of Nonlinear Elliptic Systems of Arbitrary Order with Restrictions on the Modulus of Ellipticity

We investigate the dependence of the regularity of generalized solutions of nonlinear elliptic systems on the modulus of ellipticity and regularity of the right-hand side. We establish Morrey regularity with limit exponent determined by the modulus of ellipticity in the case where the right-hand side belongs to a space with a norm stronger than the Dini function. These conditions are exact for second-order systems, namely, for any violation of the Dini condition, we construct a solution that does not belong to the Morrey space with limit exponent.

On Ruelle-Perron-Frobenius Operators.¶II. Convergence Speeds

We study Ruelle operators on expanding and mixing dynamical systems with potential function satisfying the Dini condition. We give an estimate for the convergence speed of the iterates of a Ruelle operator. Our proof avoids Markov partitions. This is the second part of our research on Ruelle operators.

On Ruelle-Perron-Frobenius Operators.¶I. Ruelle Theorem

We study Ruelle–Perron–Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper, we give a proof of the Ruelle Theorem on Gibbs measures. It is the first part of our research on the subject. The rate of convergence of powers of the operator will be presented in a forthcoming paper.

Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition

This paper is concerned with systems of nonlinear partial differential equations –D_{\alpha}a^{\alpha}_i (x, u, \nabla u) = b_i (x, u, \nabla u) (i = 1,..., N) where the coefficients a^{\alpha}_i are assumed to satisfy the condition a^{\alpha}_i (x, u, \xi) – a^{\alpha}_i (y, v, \xi) ≤ \omega |x–y| + |u–v| (1+|\xi|) for all {x, u}, {y, v} \in \Omega \times \mathbb R^N and all \xi \in \mathbb R^{nN} , and where \int^1_0 \frac {\omega (t)}{t} dt < + \infty while the functions \frac {\partial a_i^{\alpha}}{\partial \xi ^j_{\beta}} satisfy the standard boundedness and ellipticity conditions and the function \xi \mapsto b_i (x, u, \xi) may have quadratic growth. With these assumptions we prove partial Hölder continuity of bounded weak solutions u to the above system provided the usual smallness condition on \|u \|_{L^\infty ({\Omega})} is fulfilled.

Boundary asymptotics for orthogonal rational functions on the unit circle

Let w(θ) be a positive weight function on the unit circle of the complex plane. For a sequence of points { αk }k = 1∞ included in a compact subset of the unit disk, we consider the orthogonal rational functions φn that are obtained by orthogonalization of the sequence { 1, z / π1, z2 / π2, ... } where \( {\pi }_k \left( {\rm Z} \right) = \prod\nolimits_{j^{ = 1} }^k {\left( {1 - \overline {\alpha } j{\rm Z}} \right)} \), with respect to the inner product \(\left\langle {f,g} \right\rangle = \frac{1}{{2{\pi }}}\int_{{ - \pi }}^{\pi } {f\left( {{e}^{i\theta } } \right)} \overline {g\left( {{e}^{i\theta } } \right)} w\left( {\theta } \right){d\theta }\) In this paper we discuss the behaviour of φn(t) for ∣ t ∣ = 1 and n → ∞ under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szegő in the polynomial case, that is when all αk = 0.

Iterated Function System and Ruelle Operator

We consider a generalized iterated function system where the weights are variable functions. By using the Ruelle operator and a dynamical system consideration we prove that if the system is contractive and the weights are strictly positive functions and satisfy the Dini condition, then there exists a unique eigenmeasure (corresponding to the Ruelle operator) on the attractor. If in addition the maps are conformal and satisfy the open set condition, then we prove that they satisfy the strong open set condition, and by using this we can give a description of theLp-scaling spectrum and the multifractal structure of the eigenmeasure. The work extends some results of [Proc. London Math. Soc.73(1996), 105–154;Adv. Appl. Math.19(1997), 486–513;J. Statist. Phys.86(1997), 233–275;Indiana Univ. Math. J.42(1993), 367–411].

Fully nonlinear elliptic equations and the dini condition

Fully nonlinear elliptic equations and the dini condition

Vector-Valued Inequalities, Factorization, and Extrapolation for a Family of Rough Operators

We prove very general weighted norm inequalities for rough maximal and singular integral operators whose kernels satisfy some common Fourier transform decay estimates. Examples include homogeneous singular integral operators with kernels which do not necessarily satisfy a Dini condition as well as several operators whose measures are supported on lower dimensional sets, such as the discrete spherical maximal operator and the Hilbert transform along a homogeneous curve. The weights are seen to satisfy analogues of Jones′ factorization theorem and Rubio de Francia′s extrapolation theorem, and so are complete with respect to these important properties. We also obtain the corresponding weighted vector-valued inequalities for these operators and for the maximal operator associated with a starlike set.

ON THE DIRICHLET PROBLEM FOR A SECOND-ORDER ELLIPTIC EQUATION

The function space , , where is a bounded domain in , consists of elements that on sets of positive -dimensional Hausdorff measure have traces with a property analogous to joint continuity. For the set of traces of the functions in on coincides with , and the imbedding is valid.Solutions of the Dirichlet problem in are considered for the elliptic equation Under the assumption that the normal to and the coefficients of the equation satisfy the Dini condition on , it is established that for all and there is a unique solution that depends continuously on and . It is proved that in this situation the solution in coincides with the concept of a solution in introduced by Mikhailov.Bibliography: 39 titles.

An extension of the theory of orthogonal polynomials and Gaussian quadrature to trigonometric and hyperbolic polynomials

A class of functions was introduced by P. E. Koch (Research Report No. 72, Institute of Informatics, University of Oslo, 1982) that includes the algebraic, trigonometric, and hyperbolic polynomials, but retains sufficient similarities to the algebraic polynomials so that it was possible to generalize the Jackson theorems. This class need not satisfy the Haar condition. Some of the properties of orthogonal polynomials are extended here to this class. In particular, the recurrence relation, the Erdös-Turan theorem, the Dini condition for pointwise convergence of orthogonal expansions, and some of the basic properties of Gaussian quadrature are generalized.

A note on singular integrals with weights

We prove two results concerning the behavior on weighted ${L^p}$ spaces of Calderón-Zygmund singular integrals formed with kernels whose ${L^1}$ moduli of continuity satisfy the Dini condition. We also prove a result about the behavior of multiplier operators.

The Dini condition and regularity of weak solutions of elliptic equations

Regularity results in domains of Euclidean n-space are established for generalized solutions of second order elliptic equations for which the coefficients of the differential operator and the nonhomogeneous term satisfy a Dini criterion. Generalized solutions are shown to be essentially classical solutions and a bound for the modulus of continuity of second order partial derivatives of the solution is established which yields Weyl's lemma as a corollary. A differentiability theorem is also established for the case the terms of the equation have further differentiability properties.

Regularity of boundary points for linear equations of parabolic type

The paper considers a second-order linear parabolic equation whose coefficients satisfy a Dini condition. It is proven that the conditions for regularity of the boundary points for such an equation and for the heat-conduction equation coincide.

Modulus of continuity of the solution to the Dirichlet problem for a second-order parabolic equation

The modulus of continuity of the solution to the Dirichlet problem is investigated for a second-order parabolic equation at a regular boundary point. A bound for the modulus of continuity is obtained in terms of the capacity. The coefficients of the equation are required to satisfy a Dini condition (uniformly).

The maximum principle and local Lipschitz estimates near the lateral boundary for solutions of second-order parabolic equations

ellipticity constants and the maximum moduli of the coefficients of the equation. In this paper we initiate the study of a similar problem for linear second-order parabolic equations. We shall show that a regular solution of a second-order parabolic equation satisfies a Lipschitz condition (in a parabolic metric) near the lateral part of the parabolic boundary of a region if the boundary surface has the property of strict (with a Dini condition) p-paraboloidness from without with respect to the region, and if natural conditions are satisfied for the behavior of the boundary values of the solution in the neighborhood under consideration. For n = 1, the class of surfaces satisfying a strict (with a Dini condition) Pz paraboloidness condition from without will include Gevrey curves and even curves of the type F ~ [7], where the modulus of continuity ~ satisfies a Dini condition; for n -> 2, it includes surfaces with the following two properties: 1) Their cuts by hyperplanes orthogonal to the Ot axis have a continuously varying outer normal with a modulus of continuity that satisfies a Dini condition, and they lie in a half-space specified by the outer normal; 2) the cuts by hyperplanes parallel to the Ot axis satisfy with respect to t a Dini-- H6lder condition with an Windex" l/2 + ~2, where the modulus of continuity ~2 likewise satisfies a Dini condition. The obtained conditions of local regularity (in the Lipschitz sense) are exact. We shall Show that there exist surfaces near which a regular solution (that reaches a strict extremum at a boundary point) of a parabolic equation cannot satisfy a Lipschitz condition whatever the behavior of the solution at the boundary. The surfaces have the following properties: 1) Their cuts by a hyperplane orthogonal to the Ot ~is (and passing through the boundary point of an extremum) have a continuously varying outer normal with a modulus of continuity that does not satisfy a Dini condition; 2) these surfaces are located in a half-space specified by the outer normal in such a way that the region near these surfaces has a very "narrow" completion. Let us note that Dini's condition occurs also in the requirement towards the behavior of the solution at the boundary, and we show that Dini's condition is also an exact (i. e., necessary) condition for local regularity of the solution in the Lipschitz sense near the lateral part of a parabolic boundary.

On the modulus of continuity of the solution to the Dirichlet problem at a regular boundary point

A linear elliptic equation of second order with coefficients satisfying a Dini condition is considered in the paper. The modulus of continuity of a solution at a regular boundary point is investigated. An estimate for the modulus of continuity in terms of the Wiener capacity is obtained.

On Differentiability of Minimal Surfaces at a Boundary Point

Let $F(z) = \{ u(z),v(z),w(z)\} ,|z| < 1$, represent a minimal surface spanning the curve $\Gamma :\{ U(s),V(s),W(s)\} ,s$ being the arc length. Suppose $\Gamma$ has a tangent at a point $P$. Then $F(z)$ is differentiable at this point if $U’(s),V’(s),W’(s)$ satisfy a Dini condition at $P$.

On differentiability of minimal surfaces at a boundary point

Let F ( z ) = { u ( z ) , v ( z ) , w ( z ) } , | z | &gt; 1 F(z) = \{ u(z),v(z),w(z)\} ,|z| &gt; 1 , represent a minimal surface spanning the curve Γ : { U ( s ) , V ( s ) , W ( s ) } , s \Gamma :\{ U(s),V(s),W(s)\} ,s being the arc length. Suppose Γ \Gamma has a tangent at a point P P . Then F ( z ) F(z) is differentiable at this point if U ′ ( s ) , V ′ ( s ) , W ′ ( s ) U’(s),V’(s),W’(s) satisfy a Dini condition at P P .

On the conditions that the net method for the Laplace equation converges with speed h2

The paper gives a uniform estimate of order h2 of the error in the net method of solving the Dirichlet problem for the Laplace equation under the assumptions that the modulus of continuity of the second derivatives of the boundary values and the modulus of continuity of the curvature of the region's boundary do not exceed a function satisfying the Dini condition. It is shown that with the removal of the Collatz boundary values the constraints on the boundary values cannot be significantly relaxed in terms of the moduli of continuity of the second derivatives, while the constraints on the moduli of continuity of the curvature of the boundary cannot be completely lifted.

Estimates for the highest derivatives of solutions of elliptic and parabolic equations with continuous coefficients

We show that, in a certain sense, the Dini condition on the modulus of continuity of the coefficients of elliptic and parabolic equations of arbitrary order in a domain Ω is necessary both for the existence of a classical solution of the first boundary problem under arbitrarily smooth data, and for deriving an a priori estimate for the maximum of the moduli of the highest derivatives in a do man Ω' ⊂Ω in terms of max¦u¦in\(\bar \Omega\). The bibliography contains seven references.