On parabolic measures and subparabolic functions

Wu, Jang Mei G.

doi:10.1090/S0002-9947-1979-0531974-X

On parabolic measures and subparabolic functions
HTML articles powered by AMS MathViewer

by Jang Mei G. Wu

Trans. Amer. Math. Soc. 251 (1979), 171-185

DOI: https://doi.org/10.1090/S0002-9947-1979-0531974-X

PDF | Request permission

Erratum: Trans. Amer. Math. Soc. 259 (1980), 636-636.

Abstract:

Let D be a domain in $R_x^n \times R_t^1$ and ${\partial _p}D$ be the parabolic boundary of D. Suppose ${\partial _p}D$ is composed of two parts B and S: B is given locally by $t = \tau$ and S is given locally by the graph of ${x_n} = f({x_1},{x_2}, \cdots ,{x_{n - 1}},t)$ where f is Lip 1 with respect to the local space variables and Lip $\tfrac {1} {2}$ with respect to the universal time variable. Let $\sigma$ be the n-dimensional Hausdorff measure in ${R^{n + 1}}$ and $\sigma ’$ be the $(n - 1)$-dimensional Hausdorff measure in ${\textbf {R}^n}$. And let $dm(E) = d\sigma (E \cap B) + d{\sigma ’} \times dt(E \cap S)$ for $E \subseteq {\partial _p}D$. We study (i) the relation between the parabolic measure on ${\partial _p}D$ and the measure dm on ${\partial _p}D$ and (ii) the boundary behavior of subparabolic functions on D.

References

A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II, Proc. Cambridge Philos. Soc. 42 (1946), 1–10. MR 14414, DOI 10.1017/s0305004100022660
Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
Björn E. J. Dahlberg, On the existence of radial boundary values for functions subharmonic in a Lipschitz domain, Indiana Univ. Math. J. 27 (1978), no. 3, 515–526. MR 486569, DOI 10.1512/iumj.1978.27.27035
J. L. Doob, A probability approach to the heat equation, Trans. Amer. Math. Soc. 80 (1955), 216–280. MR 79376, DOI 10.1090/S0002-9947-1955-0079376-0

Linear operators

Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Richard A. Hunt and Richard L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307–322. MR 226044, DOI 10.1090/S0002-9947-1968-0226044-7
John T. Kemper, Kernel functions and parabolic limits for the heat equation, Bull. Amer. Math. Soc. 76 (1970), 1319–1320. MR 264246, DOI 10.1090/S0002-9904-1970-12658-1
John T. Kemper, Temperatures in several variables: Kernel functions, representations, and parabolic boundary values, Trans. Amer. Math. Soc. 167 (1972), 243–262. MR 294903, DOI 10.1090/S0002-9947-1972-0294903-6

On functions subharmonic in a circle

28

Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
I. Petrowsky, Zur ersten Randwertaufgabe der Wärmeleitungsgleichung, Compositio Math. 1 (1935), 383–419 (German). MR 1556900
P. J. Rippon, On the boundary behaviour of Green potentials, Proc. London Math. Soc. (3) 38 (1979), no. 3, 461–480. MR 532982, DOI 10.1112/plms/s3-38.3.461
N. A. Watson, Green functions, potentials, and the Dirichlet problem for the heat equation, Proc. London Math. Soc. (3) 33 (1976), no. 2, 251–298. MR 425145, DOI 10.1112/plms/s3-33.2.251

Boundary limits of Green’s potentials along curves

60

Jang-Mei Gloria Wu, On functions subharmonic in a Lipschitz domain, Proc. Amer. Math. Soc. 68 (1978), no. 3, 309–316. MR 470234, DOI 10.1090/S0002-9939-1978-0470234-7