# Degree Theory: Old and New

• Haïm Brezis
Conference paper
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 27)

## Abstract

The theory of degree has a long history which is a cascade of successive generalizations. Presumably, the oldest notion is the degree of a (smooth) map u from S 1 into S 1 (S 1 = the unit circle). The degree of u, also called winding number, counts “how many times u covers its range taking into account the algebraic multiplicity.” More generally, a smooth (say C 1) map u from S 1 into ℂ, such that u ≠ 0 on S 1 has a degree which may be computed through the very classical integral formula
$$\deg u = \frac{1} {{2\pi i}}\int_{S^1 } {\frac{{\dot u}} {u}}$$
(1.1)
which measures the “algebraic change of phase” of u as the variable goes around S 1 once. Similarly, if Γ is a simple curve in ℝ2 and u is a smooth map from Γ into S 1, then its degree can be computed as
$$\deg (u,\Gamma ) = \frac{1} {{2\pi }}\int_\Gamma {u \times u_\tau ,}$$
(1.2)
where × denotes the cross product of vectors in ℝ2 (here S 1 is viewed as a subset of ℝ2, not ℂ)and u τ denotes the tangential derivative of u along Γ.

### Keywords

Vortex Manifold Convolution Stein

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### References

2. F. Bethuel [1], The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 153–206.
3. F. Bethuel, H. Brezis, and F. Hélein [1], Ginzburg—Landau Vortices, Birkäuser, 1994.
4. F. Bethuel and X. Zheng [1], Density of smooth functions between two manifolds in Sobolev spaces, J. Fund. Anal. 80 (1988), 60–75.
5. A. Boutet de Monvel-Berthier, V. Georgescu, and R. Purice [1], A boundary value problem related to the Ginzburg—Landau model, Comm. Math. Phys. 141 (1991), 1–23.
6. H. Brezis [1], “Large harmonic maps in two dimensions,” in: Nonlinear Variational Problems, A. Marino et al., eds., Pitman, 1985.Google Scholar
7. H. Brezis [2], “Metastable harmonic maps,” in: Metastability and Incompletely Posed Problems, S. Antman, J. Ericksen, D. Kinderlehrer and I. Miiller, eds., IMA Series, Springer, pp. 35–42, 1987.Google Scholar
8. H. Brezis [3], Lectures on the Ginzburg—Landau vortices, Scuola Normale Superiore, Pisa, 1997.Google Scholar
9. H. Brezis and J. M. Coron [1], Large solutions for harmonie maps in two dimensions, Comm. Math. Phys. 92 (1983), 203–215.
10. H. Brezis and L. Nirenberg [1], Degree theory and BMO; Part I: Compact manifolds without boundaries, Selecta Mathematica, New Series 1 (1995), 197–263.
11. H. Brezis and L. Nirenberg [2], Degree theory and BMO; Part II: Compact manifolds with boundaries, Selecta Mathematica, New Series 2 (1996), 1–60.
12. H. Brezis and L. Nirenberg [3], Nonlinear Functional Analysis and Applications, book in preparation.Google Scholar
13. R. R. Coifman and Y. Meyer [1], “Une généralisation du théorème de Calderón sur l’intégrale de Cauchy,” in: Fourier Analysis, Proc. Sem. at El Escorial, Asoc. Mat. Española, Madrid, 1980, 88–116.Google Scholar
14. R. G. Douglas [1], Banach Algebra Techniques in Operator Theory, Acad. Press, 1972.Google Scholar
15. C. Fefferman and E. Stein [1], H p spaces of several variables, Acta Math . 129 (1972), 137–193.
16. M. Giaquinta and S. Hildebrandt [1], A priori estimates for harmonic mappings, J. Reine Angew. Math. 336 (1982), 124–164.
17. F. John and L. Nirenberg [1], On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.
18. P. Jones [1], Extension Theorems for BMO, Indiana Univ. Math. J. 29 (1980), 41–66.
19. J. Jost [1], The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary value, J. Diff. Geom. 19 (1984), 393–401.
20. J. Leray and J. Schauder [1], Topologie et équations fonctionnelles, Ann. Sci. Ecole Norm. Sup. 51 (1934), 45–78.
21. L. Nirenberg [1], Topics in nonlinear functional analysis, Courant Institute Lecture Notes (1974).Google Scholar
22. D. Sarason [1], Trans. Amer. Math. Soc. 207 (1975), 391–405.
23. R. Schoen and K. Uhlenbeck [1], Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18 (1983), 253–268.