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Mean values of subtemperatures over level surfaces of Green functions. (English) Zbl 0784.35039
Earlier, the author presented a result for subtemperatures in which the mean values are taken over level surfaces of the Green function. Thus, he established an analogue to the classical result of F. Riesz that the mean values of subharmonic functions over concentric spheres of radius \(r\), form convex functions of the form \(\log r\) or \(r^{2-n}\), depending on the dimension of the space. In this paper, he provides a more elementary proof of the theorem for subtemperatures and establishes two new results on thermic majorization, one developing its properties, and the second characterizing, in terms of the mean values, subtemperatures which have thermic majorization.

MSC:
35K05 Heat equation
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[1] Bauer, H.,Probability theory and elements of measure theory (2nd ed.), Academic Press, London, 1981. · Zbl 0466.60001
[2] Bauer, H., Heat balls and Fulks measures,Ann. Acad. Sci. Fennicae Ser. A. I. Math.,10 (1985), 67–82. · Zbl 0592.35057
[3] Boboc, N., Cornea, A., Comportement des balayées des mesures ponctuelles. Comportement des solutions du problème de Dirichlet aux points irréguliers,C. R. Acad. Sci. Paris Sér. A,264 (1967), 995–997. · Zbl 0156.12303
[4] Brelot, M., A new proof of the fundamental theorem of Kellogg-Evans on the set of irregular points in the Dirichlet problem,Rend. Circ. Math. Palermo,4 (1955), 112–122. · Zbl 0065.33602 · doi:10.1007/BF02846030
[5] Brelot, M., Choquet, G., Espaces et lignes de Green,Ann. Inst. Fourier,3 (1951), 199–263. · Zbl 0046.32701
[6] Doob, J. L.,Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, 1984. · Zbl 0549.31001
[7] Federer, H.,Geometric measure theory, Springer-Verlag, New York, 1969. · Zbl 0176.00801
[8] Lukeš, J., Malý, J., On the boundary behaviour of the Perron generalized solution,Math. Ann.,257 (1981), 355–366. · Zbl 0461.31003 · doi:10.1007/BF01456505
[9] Netuka, I., Fine behaviour of solutions of the Dirichlet problem near an irregular point,Bull. Sci. Math. (2),114 (1990), 1–22. · Zbl 0699.31015
[10] Protter, M. H., Weinberger, H. F.,Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, 1967. · Zbl 0153.13602
[11] Sternberg, S.,Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, 1964. · Zbl 0129.13102
[12] Watson, N. A., A theory of subtemperatures in several variables,Proc. London Math Soc. (3),26 (1973), 385–417. · Zbl 0253.35045 · doi:10.1112/plms/s3-26.3.385
[13] Watson, N. A., Green functions, potentials, and the Dirichlet problem for the heat equation,Proc. London Math. Soc. (3),33 (1976), 251–298. · Zbl 0336.35046 · doi:10.1112/plms/s3-33.2.251
[14] Watson, N. A., A convexity theorem for local mean values of subtemperatures,Bull. London Math. Soc.,22 (1990), 245–252. · Zbl 0722.35019 · doi:10.1112/blms/22.3.245
[15] Watson, N. A., Mean values and thermic majorization of subtemperatures,Ann. Acad. Sci. Fennicae Ser. A. I. Math.,16 (1991), 113–124. · Zbl 0784.35038
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