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A note on the Rellich formula in Lipschitz domains.

Alano Ancona (1998)

Publicacions Matemàtiques

Let L be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain ­Ω of RN and having Lipschitz coefficients in Ω­. It is shown that the Rellich formula with respect to Ω­ and L extends to all functions in the domain D = {u ∈ H01(Ω­); L(u) ∈ L2(­Ω)} of L. This answers a question of A. Chaïra and G. Lebeau.

Analytic potential theory over the p -adics

Shai Haran (1993)

Annales de l'institut Fourier

Over a non-archimedean local field the absolute value, raised to any positive power α > 0 , is a negative definite function and generates (the analogue of) the symmetric stable process. For α ( 0 , 1 ) , this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

Approximations by regular sets and Wiener solutions in metric spaces

Anders Björn, Jana Björn (2007)

Commentationes Mathematicae Universitatis Carolinae

Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of X can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for p -harmonic functions and to show that they coincide with three other notions of generalized solutions.

Capacitary strong type estimates in semilinear problems

D. Adams, Michel Pierre (1991)

Annales de l'institut Fourier

We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures μ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures μ for which the Sobolev space W 2 , p can be imbedded into L p ( μ ) . The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...

Characterizations of p-superharmonic functions on metric spaces

Anders Björn (2005)

Studia Mathematica

We show the equivalence of some different definitions of p-superharmonic functions given in the literature. We also provide several other characterizations of p-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our results also...

Constantes de Sobolev des arbres

Marc Bourdon (2007)

Bulletin de la Société Mathématique de France

Étant donnés p [ 1 , + [ et un arbre T dont chaque sommet est de valence au moins  3 , on étudie la constante de Sobolev d’exposant p de T , c’est-à-dire la plus petite constante σ p telle que pour tout u p ( T 0 ) on ait u p p σ p d u p p . Notre motivation vient de la recherche de graphes finis avec des petites constantes de Poincaré d’exposant  p , en vue d’obtenir des exemples de groupes qui ont la propriété de point fixe sur les espaces L p .

Harmonic morphisms and non-linear potential theory

Ilpo Laine (1992)

Banach Center Publications

Originally, harmonic morphisms were defined as continuous mappings φ:X → X' between harmonic spaces such that h'∘φ remains harmonic whenever h' is harmonic, see [1], p. 20. In general linear axiomatic potential theory, one has to replace harmonic functions h' by hyperharmonic functions u' in this definition, in order to obtain an interesting class of mappings, see [3], Remark 2.3. The modified definition appears to be equivalent with the original one, provided X' is a Bauer space, i.e., a harmonic...

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