Nonlinear balayage on metric spaces

Abstract

Abstract. We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and pharmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

“…The first definition for metric spaces was given by Kinnunen-Martio [25]. We will follow Björn-Björn-Mäkäläinen-Parviainen [6], and we also follow the custom of not making the dependence on p explicit in the notation. holds for all nonnegative ϕ ∈ N 1,p 0 (V ).…”

“…The obstacle problem has been studied for bounded sets in (weighted) R n (see, e.g., Heinonen-Kilpeläinen-Martio [20] and the references therein), and later also for bounded sets in more general metric spaces (see, e.g., Björn-Björn [3], [4], [5], Björn-Björn-Mäkäläinen-Parviainen [6], Björn-Björn-Shanmugalingam [8], Kinnunen-Martio [25], Kinnunen-Shanmugalingam [26], and Shanmugalingam [31]). The double obstacle problem has also been studied (see, e.g., Farnana [13,14,15,16] and Eleuteri-Farnana-Kansanen-Korte [12]).…”

Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces

“…The advantage with our formulation is that it allows for connecting balayage and obstacle problems without requiring an unnatural (and unnecessary) condition on which obstacle problems are under consideration. This connection is established in Björn-Björn-Mäkäläinen-Parviainen [5]. In [5] the fundamental convergence theorem is used as a starting point for the development of the theory of balayage, this is in contrast to earlier developments of the theory of balayage where the fundamental convergence theorem is obtained as a consequence of the theory.…”

“…The fundamental convergence theorem is a basic tool in the theory of balayage: it implies several fundamental properties of the balayage in a straightforward manner, see Björn-Björn-Mäkäläinen-Parviainen [5]. The theory of balayage in turn plays an essential role in the study of regular boundary points, capacity and polar sets, see [5] for some of these applications on metric spaces.…”

“…The theory of balayage in turn plays an essential role in the study of regular boundary points, capacity and polar sets, see [5] for some of these applications on metric spaces.…”