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

Annales Polonici Mathematici (2000)

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

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topWatson, 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

top- [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] 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] J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Grund- lehren Math. Wiss. 262, Springer, 1984. Zbl0549.31001
- [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] E. P. Smyrnélis, Sur les moyennes des fonctions paraboliques, Bull. Sci. Math. 93 (1969), 163-173. Zbl0203.09701
- [6] N. A. Watson, Uniqueness and representation theorems for parabolic equations, J. London Math. Soc. 8 (1974), 311-321. Zbl0285.35034
- [7] N. A. Watson, Boundary measures of solutions of partial differential equations, Mathematika 29 (1982), 67-82. Zbl0511.35020
- [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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