A critical growth rate for harmonic and subharmonic functions in the open ball in
Krzysztof Samotij (1987)
Colloquium Mathematicae
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Krzysztof Samotij (1987)
Colloquium Mathematicae
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Peter Sjögren (1975)
Annales de l'institut Fourier
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Let be an elliptic linear operator in a domain in . We imposse only weak regularity conditions on the coefficients. Then the adjoint exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type given distribution. We then apply to R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of are also studied. The results generalize earlier work of the author.
H. Hueber, M. Sieveking (1982)
Annales de l'institut Fourier
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Let , be elliptic operators with Hölder continuous coefficients on a bounded domain of class . There is a constant depending only on the Hölder norms of the coefficients of and its constant of ellipticity such that where (resp. ) are the Green functions of (resp. ) on .
Stephen M. Buckley, Pekka Koskela (1999)
Commentationes Mathematicae Universitatis Carolinae
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Let be a relatively closed subset of a Euclidean domain . We investigate when solutions to certain elliptic equations on are restrictions of solutions on all of . Specifically, we show that if is not too large, and has a suitable decay rate near , then can be so extended.
Piotr Nayar (2009)
Bulletin of the Polish Academy of Sciences. Mathematics
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We prove that if is harmonic and there exists a polynomial such that f + W is nonnegative, then f is a polynomial.
V. Anandam, M. Damlakhi (2003)
Annales Polonici Mathematici
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Some properties of the functions of the form in ℝⁿ, n ≥ 2, where each is a harmonic function defined outside a compact set, are obtained using the harmonic measures.
Sh. Chen, S. Ponnusamy, X. Wang (2012)
Annales Polonici Mathematici
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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function is said to be p-harmonic in Ω if each component function (i∈ 1,...,m) of is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...
Orazio Arena (2009)
Bollettino dell'Unione Matematica Italiana
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A "matching" method for planar harmonic functions is exhibited, using an elliptic adjoint equation .
Fanghua Lin, Tristan Rivière (1999)
Journal of the European Mathematical Society
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There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a -valued function defined on the boundary of a bounded regular domain of . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within...
Jiayu Li, Meng Wang (2009)
Journal of the European Mathematical Society
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Let be a compact Riemannian manifold. A quasi-harmonic sphere on is a harmonic map from to () with finite energy ([LnW]). Here is the Euclidean metric in . Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target . We also derive gradient estimates and Liouville theorems...
Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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We study local reflections with respect to a curve in a Riemannian manifold and prove that is a geodesic if is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if is harmonic for all geodesies .
Xi Fu, Bowen Lu (2016)
Czechoslovak Mathematical Journal
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The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic --Bloch space and characterize it in terms of and where is a majorant. Similar results are extended to harmonic little --Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).
R. P. Gilbert (1964)
Annales Polonici Mathematici
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Daoud Bshouty, Abdallah Lyzzaik (2012)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form , where is a continuously non-decreasing function that satisfies , assume every value finitely many times in the disc?
Luigi D'Onofrio, Tadeusz Iwaniec (2003)
Studia Mathematica
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We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation . In this example the p-harmonic transform is essentially inverse to . To every vector field our operator assigns the gradient of the solution, . The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our...
Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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We study local reflections with respect to a curve in a Riemannian manifold and prove that is a geodesic if is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if is harmonic for all geodesies .
A. Sadullaev (2011)
Annales de la faculté des sciences de Toulouse Mathématiques
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The main result of the present paper is : every separately-subharmonic function , which is harmonic in , can be represented locally as a sum two functions, , where is subharmonic and is harmonic in , subharmonic in and harmonic in outside of some nowhere dense set .