# The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation

• Volume: 75, Issue: 3, page 281-287
• ISSN: 0066-2216

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## Abstract

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Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip $X={ℝ}^{n}×\right]0,a\left[$. We prove a representation theorem for an arbitrary ${C}^{2,1}$ function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with ${C}^{\infty }$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of Lu=0, are also presented.

## How to cite

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Watson, Neil. "The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation." Annales Polonici Mathematici 75.3 (2000): 281-287. <http://eudml.org/doc/208401>.

@article{Watson2000,
abstract = {Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip $X = ℝ^\{n\} × ]0,a[$. We prove a representation theorem for an arbitrary $C^\{2,1\}$ function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with $C^\{∞\}$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of Lu=0, are also presented.},
author = {Watson, Neil},
journal = {Annales Polonici Mathematici},
keywords = {fundamental solution; parabolic equation; representation theorem; bounded Hölder continuous coefficients},
language = {eng},
number = {3},
pages = {281-287},
title = {The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation},
url = {http://eudml.org/doc/208401},
volume = {75},
year = {2000},
}

TY - JOUR
AU - Watson, Neil
TI - The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation
JO - Annales Polonici Mathematici
PY - 2000
VL - 75
IS - 3
SP - 281
EP - 287
AB - Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip $X = ℝ^{n} × ]0,a[$. We prove a representation theorem for an arbitrary $C^{2,1}$ function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with $C^{∞}$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of Lu=0, are also presented.
LA - eng
KW - fundamental solution; parabolic equation; representation theorem; bounded Hölder continuous coefficients
UR - http://eudml.org/doc/208401
ER -

## References

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1. [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968), 607-694. Zbl0182.13802
2. [2] J. Chabrowski and N. A. Watson, Properties of solutions of weakly coupled parabolic systems, J. London Math. Soc. 23 (1981), 475-495. Zbl0479.35049
3. [3] J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Grund- lehren Math. Wiss. 262, Springer, 1984. Zbl0549.31001
4. [4] A. M. Il'in, A. S. Kalashnikov and O. A. Oleĭnik, Linear equations of the second order of parabolic type, Uspekhi Mat. Nauk 17 (1962), no. 3, 3-146 (in Russian); English transl.: Russian Math. Surveys 17 (1962), no. 3, 1-143.
5. [5] E. P. Smyrnélis, Sur les moyennes des fonctions paraboliques, Bull. Sci. Math. 93 (1969), 163-173. Zbl0203.09701
6. [6] N. A. Watson, Uniqueness and representation theorems for parabolic equations, J. London Math. Soc. 8 (1974), 311-321. Zbl0285.35034
7. [7] N. A. Watson, Boundary measures of solutions of partial differential equations, Mathematika 29 (1982), 67-82. Zbl0511.35020
8. [8] N. A. Watson, A decomposition theorem for solutions of parabolic equations, Ann. Acad. Sci. Fenn. Math. 25 (2000), 151-160. Zbl0939.35080

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