MathSciDoc: An Archive for Mathematician ∫

The continuity of weak solutions of elliptic partial differential equations $$div \mathcal{A}(x,\nabla u) = 0$$ is considered under minimal structure assumptions. The main result guarantees the continuity at the point$x$_{0}for weakly monotone weak solutions if the structure of$A$is controlled in a sequence of annuli $$B(x_0 ,R_j )\backslash \bar B(x_0 ,r_j )$$ with uniformly bounded ratio$R$_{$j$}$/r$_{$j$}such that lim_{$j→∞$}$R$_{$j$}=0. As a consequence, we obtain a sufficient condition for the continuity of mappings of finite distortion.

We study a semilinear elliptic equation of the form $$ - \Delta u + u = f(x,u), u \in H_0^1 (\Omega ),$$ where$f$is continuous, odd in$u$and satisfies some (subcritical) growth conditions. The domain Ω⊂R^{N}is supposed to be an unbounded domain ($N$≥3). We introduce a class of domains, called strongly asymptotically contractive, and show that for such domains Ω, the equation has infinitely many solutions.

We prove that the linearized Riesz transforms and the imaginary powers of the Laplacian are$H$^{1}-bounded on complete Riemannian manifolds satisfying the doubling property and the Poincaré inequality, where$H$^{1}denotes the Hardy space on$M$.

Let$D$be a relatively compact domain in$C$^{2}with smooth connected boundary ∂$D$. A compact set$K⊂∂D$is called removable if any continuous CR function defined on ∂$D/K$admits a holomorphic extension to$D$. If$D$is strictly pseudoconvex, a theorem of B. Jöricke states that any compact$K$contained in a smooth totally real disc$S⊂∂D$is removable. In the present article we show that this theorem is true without any assumption on pseudoconvexity.

We prove the following Hartogs-Bochner type theorem: Let M be a connected C^{2}hypersurface of P_{n}(C) (n≥2) which divides P_{n}(C) in two connected open sets Ω_{1}and Ω_{2}. Suppose that M has at most one open CR orbit. Then there exists i∈{1,2} such that C^{1}CR functions defined on M extends holomorphically to Ω_{$i$}.

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We prove that intermediate Banach spaces $$\mathcal{A}$$ and $$\mathcal{B}$$ with respect to arbitrary Hilbert couples $$\bar {H}$$ and $$\bar {K}$$ are exact interpolation if and onlyif they are exact$K$-monotonic, i.e. the condition $$f^0 \in \mathcal{A}$$ and the inequality $$K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$$ , imply$g$^{0}∈$B$and ‖$g$^{0}‖$B$≤‖$f$^{0}‖_{$A$}($K$is Peetre’s$K$-functional). It is well known that this property is implied by the following: for each ϱ>1 there exists an operator $$T:\bar {H} \to \bar {K}$$ such that$Tf$^{0}=$g$^{0}, and $$K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0$$ . Verifying the latter property, it suffices to consider the “diagonal case” where $$\bar {H} = \bar {K}$$ is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ϱ=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.

We prove that if$T$is a strictly singular one-to-one operator defined on an infinite dimensional Banach space$X$, then for every infinite dimensional subspace$Y$of$X$there exists an infinite dimensional subspace$Z$of$X$such that$Z∩Y$is infinite dimensional,$Z$contains orbits of$T$of every finite length and the restriction of$T$to$Z$is a compact operator.

Completely invariant components of the Fatou sets of meromorphic maps are discussed. Positive answers are given to Baker’s and Bergweiler’s problems that such components are the only Fatou components for certain classes of meromorphic maps.

In this paper we study the spectral counting function for the weighted$p$-laplacian in one dimension. First, we prove that all the eigenvalues can be obtained by a minimax characterization and then we show the existence of a Weyl-type leading term. Finally we find estimates for the remainder term.