Under suitable assumptions on , we prove that generates a positive -semigroup on and, hence, many previous (linear or nonlinear) results are extended substantially.
The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation where and are polynomials of degree and , is under discussion. We concentrate on the case when has only real zeros and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients in the partial fraction decomposition , we allow the presence of both positive and negative coefficients . The corresponding electrostatic interpretation of the zeros of the solution as points of equilibrium in an electrostatic field generated by charges at is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.
We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,
exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.
We prove for many self-similar, and some more general, sets that if is the Hausdorff dimension of and is Hölder continuous with exponent , then the -dimensional Hausdorff measure of is .
Let be the disk algebra. In this paper we address the following question: Under what conditions on the points do there exist operators such that
and , , for every ? Here the convergence is understood in the sense of norm in . Our first result shows that if satisfy Carleson condition, then there exists a function such that , . This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if are required to be projections, then for any choice of the operators do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.
We study maps and give detailed estimates on in terms of and . These estimates are used to prove a lemma by D. Henry for the case . Here is an open subset and and are Banach spaces.
On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions. 相似文献