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
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 unit disk. We show that for some relatively closed set there is a function that can be uniformly approximated on by functions of , but such that cannot be written as , with and uniformly continuous on . This answers a question of Stray.
Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .
For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.
Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents. 相似文献
We show that finite dimensional injective operator spaces are corners of finite dimensional -algebras .
We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,
exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.
For each integer we construct a compact, geodesic metric space which has topological dimension , is Ahlfors -regular, satisfies the Poincaré inequality, possesses as a unique tangent cone at almost every point, but has no manifold points.
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial -complex into for , and it was recently shown to be incomplete for . We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of -complexes in .
For a group let be the number of subgroups of index and let be the number of normal subgroups of index . We show that for 2$">. If and does not divide or if and or , we show that for all sufficiently large . On the other hand if and divides , then is not even bounded as a function of .
It is proved in this paper that