# Perturbation of harmonic structures and an index-zero theorem

Annales de l'institut Fourier (1970)

- Volume: 20, Issue: 1, page 317-359
- ISSN: 0373-0956

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topWalsh, Bertram. "Perturbation of harmonic structures and an index-zero theorem." Annales de l'institut Fourier 20.1 (1970): 317-359. <http://eudml.org/doc/74005>.

@article{Walsh1970,

abstract = {In the framework of an axiomatic theory of sheaves of “harmonic” functions, a notion of perturbation of these sheaves is introduced which corresponds to the replacement of the operator $\Delta $ by an operator $\Delta +f$, in the classical situation. The “harmonic” functions with which the paper is concerned are assumed to satisfy certain hypotheses (weaker than the axioms of Bauer); it is shown that the perturbed harmonic functions also satisfy these hypotheses. Moreover, the results obtained imply that the dimensions of the spaces $H^0(W,\{\bf H\})$ and $H^1(W,\{\bf H\})$ are (finite and) equal whenever the base space $W$ of a sheaf $\{\bf H\}$ satisfying these hypotheses is compact. That fact generalizes the classical theorem that the index of any second order elliptic operator on a (trivial bundle over $a$) compact manifold is zero. Further, it implies that whenever $\{\bf H\}$ satisfies the Brelot axioms and its adjoint sheaf $\{\bf H\}^*$ exists, the spaces $\{\bf H\}_W$ and $\{\bf H\}^*_W$ (where $W$ is again compact) have the same (finite) dimension.},

author = {Walsh, Bertram},

journal = {Annales de l'institut Fourier},

keywords = {partial differential equations},

language = {eng},

number = {1},

pages = {317-359},

publisher = {Association des Annales de l'Institut Fourier},

title = {Perturbation of harmonic structures and an index-zero theorem},

url = {http://eudml.org/doc/74005},

volume = {20},

year = {1970},

}

TY - JOUR

AU - Walsh, Bertram

TI - Perturbation of harmonic structures and an index-zero theorem

JO - Annales de l'institut Fourier

PY - 1970

PB - Association des Annales de l'Institut Fourier

VL - 20

IS - 1

SP - 317

EP - 359

AB - In the framework of an axiomatic theory of sheaves of “harmonic” functions, a notion of perturbation of these sheaves is introduced which corresponds to the replacement of the operator $\Delta $ by an operator $\Delta +f$, in the classical situation. The “harmonic” functions with which the paper is concerned are assumed to satisfy certain hypotheses (weaker than the axioms of Bauer); it is shown that the perturbed harmonic functions also satisfy these hypotheses. Moreover, the results obtained imply that the dimensions of the spaces $H^0(W,{\bf H})$ and $H^1(W,{\bf H})$ are (finite and) equal whenever the base space $W$ of a sheaf ${\bf H}$ satisfying these hypotheses is compact. That fact generalizes the classical theorem that the index of any second order elliptic operator on a (trivial bundle over $a$) compact manifold is zero. Further, it implies that whenever ${\bf H}$ satisfies the Brelot axioms and its adjoint sheaf ${\bf H}^*$ exists, the spaces ${\bf H}_W$ and ${\bf H}^*_W$ (where $W$ is again compact) have the same (finite) dimension.

LA - eng

KW - partial differential equations

UR - http://eudml.org/doc/74005

ER -

## References

top- [1] H. BAUER, Harmonische Räume und ihre Potentialtheorie, Springer Lecture Notes in Mathematics 22 (1966). Zbl0142.38402
- [2] N. BOBOC, C. CONSTANTINESCU and A. CORNEA, Axiomatic theory of harmonic functions. Nonnegative superharmonic functions, Ann. Inst. Fourier (Grenoble), 15 (1965), 283-312. Zbl0139.06604MR33 #1476
- [3] M. BRELOT, Lectures on Potential Theory, Tata Institute, Bombay, 1960. Zbl0098.06903MR22 #9749
- [4] C.H. DOWKER, Lectures on Sheaf Theory, Tata Institute, Bombay, 1957.
- [5] N. DUNFORD and J.T. SCHWARTZ, Linear Operators I, Interscience, New York, 1958. Zbl0084.10402MR22 #8302
- [6] R.C. GUNNING and H. ROSSI, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. Zbl0141.08601MR31 #4927
- [7] R.M. HERVÉ, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble), 12 (1962), 415-571. Zbl0101.08103MR25 #3186
- [8] T. KATO, Perturbation Theory for Linear Operators, Springer, Berlin-Göttingen-Heidelberg, 1966. Zbl0148.12601MR34 #3324
- [9] P.A. MEYER, Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Ann. Inst. Fourier (Grenoble), 13 (1963), 357-372. Zbl0116.30404MR29 #260
- [10] H. SCHAEFER, Topological Vector Spaces, Macmillan, New York, 1966. Zbl0141.30503MR33 #1689
- [11] B. WALSH, Flux in axiomatic potential theory. I : Cohomology, Inventiones Math. 8 (1969), 175-221. Zbl0179.15203MR42 #532
- [12] B. WALSH, Flux in axiomatic potential theory. II : Duality, Ann. Inst. Fourier, (Grenoble), 19 (1969). Zbl0181.11703MR42 #2023

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