-Capacity and -Hyperbolicity of Submanifolds.
Parabolic Harnack inequality and estimates of Markov chains on graphs.
On a graph, we give a characterization of a parabolic Harnack inequality and Gaussian estimates for reversible Markov chains by geometric properties (volume regularity and Poincaré inequality).
Parabolic measure on domains of class Lip
Periodic harmonic functions on lattices and points count in positive characteristic
This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.
Perturbation by differences of unbounded potentials.
Perturbation theory for nonlinear Dirichlet problems.
Perturbations and nonlinear harmonic spaces. (Perturbations et espaces harmoniques non linéares.)
p-harmonic functions on graphs and manifolds.
Plateau-Stein manifolds
We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.
Plurifine potential theory
We give an overview of the recent developments in plurifine pluripotential theory, i.e. the theory of plurifinely plurisubharmonic functions.
Pluriharmonic extension in proper image domains
Let be a bounded hyperconvex domain in and set , j=1,...,s, s ≥ 3. Also let be the image of D under the proper holomorphic map π. We characterize those continuous functions that can be extended to a real-valued pluriharmonic function in .
Pluriharmonic interpolation and hulls of C1 curves in the unit sphere.
Pluriharmonically dentable complex Banach spaces.
Pluriharmonicity in terms of harmonic slices.
Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains
Plurisubharmonic extension in non-degenerate analytic polyhedra.
Plurisubharmonic functions on compact sets
Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.
Plurisubharmonic functions with logarithmic singularities
To a plurisubharmonic function on with logarithmic growth at infinity, we may associate the Robin functiondefined on , the hyperplane at infinity. We study the classes , and (respectively) of plurisubharmonic functions which have the form and (respectively) for which the function is not identically . We obtain an integral formula which connects the Monge-Ampère measure on the space with the Robin function on . As an application we obtain a criterion on the convergence of the Monge-Ampère...
Plurisubharmonic saddles
A certain linear growth of the pluricomplex Green function of a bounded convex domain of at a given boundary point is related to the existence of a certain plurisubharmonic function called a “plurisubharmonic saddle”. In view of classical results on the existence of angular derivatives of conformal mappings, for the case of a single complex variable, this allows us to deduce a criterion for the existence of subharmonic saddles.
Poisson kernel and Green function of balls for complex hyperbolic Brownian motion
The aim of this paper is to give a description of the Poisson kernel and the Green function of balls in the complex hyperbolic space. The description is in terms of the hypergeometric function and unitary spherical harmonics in ℂⁿ.