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Greenian potentials and concavity. (English) Zbl 0584.31003
Let D be a bounded domain in \({\mathbb{R}}^ n\), \(v\geq 0\) be a continuous function in D and \(G_ v\) be the Green operator relative to \(- (1/2)\Delta +v\). The author proves that the corresponding Green potentials satisfy a Brunn-Minkowski type inequality and obtains the following result:
If D is bounded convex and if \(f: D\to]0,\infty [\) and \(v^{-1/2}\) are concave (v\(\equiv 0\) or \(v>0)\), then \((G_ v f^ p)^{1/(2+p)}\) is concave, \(0\leq p\leq 1\).
Reviewer: V.Anandam

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31A10 Integral representations, integral operators, integral equations methods in two dimensions
31B99 Higher-dimensional potential theory
Full Text: DOI EuDML
[1] Borell, C.: Hitting probabilities of killed Brownian motion; a study on geometric regularity. Ann. Sci. Ec. Norm. Sup.17, 451-467 (1984) · Zbl 0573.60067
[2] Borell, C.: Capacitary inequalities of the Brunn-Minkowski type. Math. Ann.263, 179-184 (1983) · Zbl 0546.31001 · doi:10.1007/BF01456879
[3] Borell, C.: Convex set functions ind-space. Period. Math. Hungar.6, 111-136 (1975) · Zbl 0402.28007 · doi:10.1007/BF02018814
[4] Borell, C.: Convexity of measures in certain convex cones in vector space ?-algebras. Math. Scand.53, 125-144 (1983) · Zbl 0568.46007
[5] Borell, C.: Convex measures on locally convex spaces. Ark. Mat.12, 239-252 (1974) · Zbl 0297.60004 · doi:10.1007/BF02384761
[6] Brascamp, H.J., Lieb, E.H.: Some inequalities of Gaussian measures. In: Functional integral and its applications. Ed. A.M. Arthurs. Oxford: Clarendon Press 1975 · Zbl 0348.26011
[7] Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Pr?kopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal.22, 366-389 (1976) · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[8] Chorin, A.J., Marsden, J.E.: A mathematical introduction to fluid mechanics. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0417.76002
[9] Dynkin, E.B.: Markov Processes. Vol. I?II. Berlin, G?ttingen, Heidelberg: Springer 1965
[10] Korevaar, N.J.: Convex solutions to non-linear elliptic and parabolic boundary value problems. Indiana Univ. Math. J.32, 603-614 (1983) · Zbl 0528.35011 · doi:10.1512/iumj.1983.32.32042
[11] P?lya, G., Szeg?, G.: Isoperimetric inequalities in mathematical physics. Princeton: Princeton University Press 1951
[12] Port, S.C., Stone, C.J.: Brownian motion and classical potential theory. New York, San Francisco, London: Academic Press 1978 · Zbl 0413.60067
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