A uniqueness theorem for nonnegative superharmonic functions in planar domains

Kuran, Ü.; Schiff, Joel L.

doi:10.1016/0022-247x(83)90225-1

Cited by 9 publications

(16 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, in this paper, we demonstrate that s(x) cannot tend to zero "too rapidly" as x approaches a point on dD, unless s = 0. Similar considerations in other contexts may be found in Brelot [1, Appendix, §26d], Kuran [2], Kuran and Schiff [3], and Nairn [4]. 2.…”

supporting

confidence: 72%

“…We first consider the case of the open n-ball, B" = B"(y; r), centered at y of radius r > 0. The proof is a generalization of that given in [3] for n = 2. for all x E B". Taking lim infx^iJteB», yields /9B» dp(z) = 0, implying u = 0 and hence h = 0 in B".…”

mentioning

confidence: 78%

“…Now, the Green's function is given by In [3], for n = 2, an analog of this result was established under the more restrictive boundary condition that dD be Dini-smooth in a neighborhood of £, but under the more general condition that * G D could tend to £ via any approach in D. That version, however, requires the Riemann mapping theorem which is no longer at our disposal for « > 3.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A uniqueness theorem for superharmonic functions in 𝑅ⁿ

Schiff¹

1982

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let s(x) be a nonnegative superharmonic function defined on the /i-ball B"(y; r) in R", n > 3. If s(x) tends to zero "too rapidly" as x tends to a single point £ on the boundary of B"(y; r), then we prove that s = 0. The same result can then be extended to domains D Ç R", whose boundary dD is locally C ' at £ G 3D. These results generalize some earlier work of the author and U. Kuran for n -2.

show abstract

supporting

confidence: 72%

mentioning

confidence: 78%

mentioning

confidence: 99%

See 1 more Smart Citation

A uniqueness theorem for superharmonic functions in 𝑅ⁿ

Schiff¹

1982

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Analogous results on Euclidean domains for the case p = 2 (that is, for potential theory of the Laplacian operator) can be found in [25], Theorem 7'-16 from page 226 of [21,23,24], and [20]. and hence, as…”

Section: On P-superharmonic Functionsmentioning

confidence: 86%

Boundary Harnack Principle for p-harmonic Functions in Smooth Euclidean Domains

et al. 2007

View full text Add to dashboard Cite

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C 1,1 -domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain.

show abstract

“…Similar considerations in other contexts may be found in Brelot [1, Appendix, ?26d], Kuran [2], Kuran and Schiff [3], and Naim [4]. However, in this paper, we demonstrate that s(x) cannot tend to zero "too rapidly" as x approaches a point on aD, unless s _ 0.…”

mentioning

confidence: 54%