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Displaying similar documents to “On weighted spaces of functions harmonic in ℝ n ”

Fractional integro-differentiation in harmonic mixed norm spaces on a half-space

Karen L. Avetisyan (2001)

Commentationes Mathematicae Universitatis Carolinae

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In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces h ( p , q , α ) on the half-space are established. We prove that mixed norm is equivalent to a “fractional derivative norm” and that harmonic conjugation is bounded in h ( p , q , α ) for the range 0 < p ≤ ∞ , 0 < q ≤ ∞ . As an application of the above, we give a characterization of h ( p , q , α ) by means of an integral representation with the use of Besov spaces.

"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

A. Bonilla (2000)

Colloquium Mathematicae

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We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in ℝ N which is dense in the space of all harmonic functions in ℝ N and lim‖x‖→∞ x ∈ S ‖x‖μDαv(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other...

A note on the Cahn-Hilliard equation in H 1 ( ℝ N ) involving critical exponent

Jan W. Cholewa, Aníbal Rodríguez-Bernal (2014)

Mathematica Bohemica

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We consider the Cahn-Hilliard equation in H 1 ( ℝ N ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as | u | → ∞ and logistic type nonlinearities. In both situations we prove the H 2 ( ℝ N ) -bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012).