A new operator for elliptic equations, and theP-compactification for ?u=Pu

Nakai, Mitsuru; Sario, Leo

doi:10.1007/bf01359704

Cited by 8 publications

(15 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our note we assume that s exists. The concept of a P-singular point was introduced in Nakai-Sario [4], and the term "P has a finite integral at x" in Glasner-Katz [2]. 4.…”

Section: -4 £0iw/ S(erp Is P-singular If and Only If Furir Pdv= Oo Fomentioning

confidence: 99%

“…The concept of a P-singular point was introduced in Nakai-Sario [4], and the term "P has a finite integral at x" in Glasner-Katz [2]. 4. We write/=BE-lim n /n on R if the sequence {f n } is uniformly bounded on P, converges to ƒ uniformly on compact subsets of P, and E R (f n -ƒ)-»0 as n-»oo.…”

Section: -4 £0iw/ S(erp Is P-singular If and Only If Furir Pdv= Oo Fomentioning

confidence: 99%

“…With Mp(R) one associates the Pcompactification Rp of R on which every ƒ G Mp(R) has a continuous extension (Nakai-Sario [4]). An interesting phenomenon is the occurrence of the P-singular point s£Rp defined by f(s)=0 for every fEM P (R).…”

mentioning

confidence: 99%

See 2 more Smart Citations

The 𝑃-singular point of the 𝑃-compactification for Δ𝑢=𝑝𝑢

Kwon¹,

Sario²

1971

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

ABSTRACT. By means of the P-algebra Mp(R) of bounded energy-finite Tonelli functions on a Riemannian manifold R, we construct the P-compactification R P of R as a quotient space of the Royden compactification. The P-singular point sp is explicitly characterized in terms of the density P. The dimension of the space PBE{R) of bounded energy-finite P-harmonic functions on R is shown to exceed exactly by one the cardinality of the P-harmonic boundary Ap if spGAp. If sp(£Ap one can replace the density P by another Q such that dim QBE(R) =dim PBE(R) and a Q-singular point does not exist.In the study of the equation Au=Pu, PâO, on a Riemannian manifold P, it is useful to consider the algebra MP(R) of bounded energy-finite Tonelli functions. With Mp(R) one associates the Pcompactification Rp of R on which every ƒ G Mp(R) has a continuous extension ). An interesting phenomenon is the occurrence of the P-singular point s£Rp defined by f(s)=0 for every fEM P (R).In the present note we construct Rp as a quotient space of the Royden compactification P*. Necessary and sufficient for the existence of an 5 is that Kj|Afp(P). If an s exists, it is unique. We shall give an explicit characterization of s in terms of P, thus establishing a link with a property considered by Glasner and Katz [2].We then show that if s lies on the P-harmonic boundary Ap, the cardinality of A P exceeds exactly by one the dimension of the space of bounded energy-finite P-harmonic functions on P.If 5 does not lie on A Pl it is removable in the sense that there exists a density Q on R without a ö-singular point such that dim QBE(R) = dim PBE(R) = the cardinality of A P .1. On a smooth Riemannian w-manifold P, n^2, consider P-A MS 1969 subject classifications. Primary 3045,3111.

show abstract

“…In our note we assume that s exists. The concept of a P-singular point was introduced in Nakai-Sario [4], and the term "P has a finite integral at x" in Glasner-Katz [2]. 4.…”

Section: -4 £0iw/ S(erp Is P-singular If and Only If Furir Pdv= Oo Fomentioning

confidence: 99%

Section: -4 £0iw/ S(erp Is P-singular If and Only If Furir Pdv= Oo Fomentioning

confidence: 99%

See 1 more Smart Citation

The 𝑃-singular point of the 𝑃-compactification for Δ𝑢=𝑝𝑢

Kwon¹,

Sario²

1971

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…it exists, and is unique, if and only if \ PdV -<*> (Nakai-Sario [7]). It is known that p<=R$ is P-singular if and only if I PdV=oo for every neighborhood U of φ in Rf (Kwon-Sario [4]).…”

Section: Space and Contains R As An Open Dense Subset; Every F^m P (R)mentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

The P-Harmonic Boundary and Energy-Finite Solutions of Δu = Pu

Kwon

Sario

Schiff

1971

Nagoya Mathematical Journal

View full text Add to dashboard Cite

The P-harmonic boundary Δ P and the P-singular point s of a Riemannian manifold R have been shown to play an important role in the study of bounded energy-finite solutions of Δu -Pu (Nakai-Sario [7], Kwon-Sario [4], Kwon-Sario-Schiff [5]). The objective of the present paper is to establish, in terms of Δ P and s, properties of unbounded energy-finite solutions {PE-functions) and of limits of decreasing sequences of positive Pis-functions (PEfunctions). Also, PE-and PZs-minimal functions will be discussed.For the convenience of the reader we shall briefly review, in 1, some preliminaries (for details see Kwon-Sario-Schiff [5]).1. On a connected, separable, oriented, smooth Riemanniari manifold of dimension N, consider the P-algebra M P {R) of bounded Tonelli functions / with finite energy integrals Here D R {f) = \ dfA*df is the Dirichlet integral of / over R, P(^0) a given nonnegative continuous function on. R 9 and dV = *1 the volume element of R. It is known that the P-algebra M P (R) is closed under the lattice operations fl)g= max(/,g) and fΠg= min(/,g) 9 and that it is complete in the ££-topology: if {f n } is a uniformly bounded sequence in M P {R), converges to / uniformly on compact subsets of R, and E R {f n -fj -> 0 as n, m ~> oo, then f<=M P (R).By means of the P-algebra M P {R) one constructs the P-compactification Rf of R, defined by the following properties: i?J is a compact Hausdorff

show abstract