A Caccioppoli estimate and fine continuity for superminimizers on metric spaces.
A lower bound for the gradient of -harmonic functions.
A note on p-subharmonic functions on complete manifolds.
A note on the Rellich formula in Lipschitz domains.
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.
A Radó type theorem for -harmonic functions in the plane.
A sharpening of a theorem of Bouligand. With an application to harmonic maps.
Analytic potential theory over the -adics
Over a non-archimedean local field the absolute value, raised to any positive power , is a negative definite function and generates (the analogue of) the symmetric stable process. For , 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.
Announcements of new results
Approximations by regular sets and Wiener solutions in metric spaces
Let be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of 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 -harmonic functions and to show that they coincide with three other notions of generalized solutions.
Arithmetic capacities on IP N.
Capacitary strong type estimates in semilinear problems
We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures for which the Sobolev space can be imbedded into . 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
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
Étant donnés et un arbre dont chaque sommet est de valence au moins , on étudie la constante de Sobolev d’exposant de , c’est-à-dire la plus petite constante telle que pour tout on ait . Notre motivation vient de la recherche de graphes finis avec des petites constantes de Poincaré d’exposant , en vue d’obtenir des exemples de groupes qui ont la propriété de point fixe sur les espaces .
Corrections to the article “The multivelocity Peierls potential in the problem of refining the classical fundamental acoustic potential near the source of sound in a homogeneous Maxwellian gas”.
Equivalence of weak and viscosity solutions to the -Laplace equation in the Heisenberg group.
Fine topology and -harmonic morphisms.
Fine topology of variable exponent energy superminimizers.
Growth of entire -subharmonic functions.
Harmonic morphisms and non-linear potential theory
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...
Inegalités de Kato et semi-groups sous-Markoviens.