zbMATH — the first resource for mathematics

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

MSC:
 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:
References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.