Previous |  Up |  Next


subharmonic; surface integral
Let ${\Cal H}$ denote the class of positive harmonic functions on a bounded domain $\Omega$ in $\Bbb R^N$. Let $S$ be a sphere contained in $\overline{\Omega}$, and let $\sigma$ denote the $(N-1)$-dimensional measure. We give a condition on $\Omega$ which guarantees that there exists a constant $K$, depending only on $\Omega$ and $S$, such that $\int_Su\,d\sigma \le K\int_{\partial\Omega}u\,d\sigma$ for every $u\in {\Cal H}\cap C(\overline{\Omega})$. If this inequality holds for every such $u$, then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for $K$ are given. In particular the classical value $K=2$ for convex domains is slightly improved.
[1] Dahlberg B.: On estimates of harmonic measure. Arch. Rational Mech. Anal. 65 (1977), 275-288. MR 0466593
[2] Gabriel R.M.: A note upon functions positive and subharmonic inside and on a closed convex curve. J. London Math. Soc. 21 (1946), 87-90. MR 0019791 | Zbl 0061.23203
[3] Gardiner S.J.: Superharmonic extension from the boundary of a domain. Bull. London Math. Soc. 27 (1995), 347-352. MR 1335285 | Zbl 0835.31004
[4] Hayman W.K.: Integrals of subharmonic functions along two curves. Indag. Mathem. N.S. 4 (1993), 447-459. MR 1252989 | Zbl 0794.31003
[5] Helms L.L.: Introduction to Potential Theory. Wiley, New York, 1969. MR 0261018 | Zbl 0188.17203
[6] Kuran Ü.: Harmonic majorizations in half-balls and half-spaces. Proc. London Math. Soc. (3) 21 (1970), 614-636. MR 0315148 | Zbl 0207.41603
[7] Kuran Ü.: On NTA-conical domains. J. London Math. Soc. (2) 40 (1989), 467-475. MR 1053615 | Zbl 0726.31001
[8] Reuter G.E.H.: An inequality for integrals of subharmonic functions over convex surfaces. J. London Math. Soc. 23 (1948), 56-58. MR 0025642 | Zbl 0032.28202
[9] Widman K.-O.: Inequalities for the Green's function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21 (1967), 17-37. MR 0239264
Partner of
EuDML logo