Extreme harmonic functions and boundary value problems

Gowrisankaran, Kohur

doi:10.5802/aif.148

Cited by 28 publications

(20 citation statements)

References 11 publications

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“…Definition 1.2 (cf. [13,26]). (i) For any nonnegative superharmonic function u on X and E cz X, the reduced function ofuonE is defined on X by REu(x, t) = inf{w(;ï, t): wis nonnegative, superharmonic on Xand w > uon £}.…”

Section: 2])mentioning

confidence: 99%

Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab

Mair¹

1984

Trans. Amer. Math. Soc.

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Abstract. This paper investigates the boundary behaviour of positive solutions of the equation Lu = 0, where L is a uniformly parabolic second-order differential operator in divergence form having Holder-continuous coefficients on X= R" X (0, 7"), where 0 < T < oo. In particular, the notion of semithinness for the potential theory on X associated with L is introduced, and the relationships between fine, semifine and parabolic convergence at points of R" X {0} are studied.The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of Lu = 0 on X has parabolic limits Lebesgue-almost-everywhere on R" X (0). Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions. 0. Introduction. Let £ be a second-order linear parabolic differential operator having divergence structure on X = R" X (0, T) where 0 < T < oo. The coefficients of L are assumed to be such that the classical fundamental solution exists, the results in [2] for classical solutions hold, and the solutions of Lu = 0 form a strong harmonic space in the sense of Bauer [3].In the particular case of the heat operator Ax -9/3i, Doob [11] proved the almost everywhere convergence through parabolic regions of quotients of positive solutions on X. Hattemer [16] showed that if £ c R", « is a solution of the heat equation on X, and for each b e £, u is bounded on a parabolic region with vertex b, then u has finite parabolic limits almost everywhere on £ (cf. For certain parabolic operators with divergence structure, Johnson [19] proved the Lebesgue-almost-everywhere convergence through parabolic regions of positive solutions on X.In this paper, Johnson's result for £ is deduced from the abstract Fatou-NaimDoob theorem. Also, a Carleson-type result is established for solutions of Lu = 0 defined on a union of parabolic regions. The methods employed in this paper were inspired mainly by those in [6,21 and 22].

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Section: 2])mentioning

confidence: 99%

Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab

Mair¹

1984

Trans. Amer. Math. Soc.

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“…See Gowrisankaran [8] for a discussion of the possibilities. Cartan pointed out in .1940 that the topology on R w^hich is the smallest topology (fewest open sets) making superharmonic functions continuous is precisely the fine topology on R, defined above in terms of thinness.…”

Section: Itmentioning

confidence: 99%

Some classical function theory theorems and their modern versions

Doob¹

1965

Annales De L’institut Fourier

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“…However, a generalisation of Doob's theorem to functions of several variables can be proved, but the limits are those following the "fine" filters [5] canonical to the structure of the multiply harmonic functions. More precisely, we have to consider "fine" filters corresponding to the minimal positive multiply harmonic functions; and these filters are finer than the product of the fine filters corresponding to the minimal positive harmonic functions.…”

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confidence: 99%