On the elliptic equation L u - k + K exp [ 2 u ] = 0

Carlos E. Kenig; Wei-Ming Ni

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1985)

  • Volume: 12, Issue: 2, page 191-224
  • ISSN: 0391-173X

How to cite

top

Kenig, Carlos E., and Ni, Wei-Ming. "On the elliptic equation $Lu - k + K \exp [2u] = 0$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 12.2 (1985): 191-224. <http://eudml.org/doc/83956>.

@article{Kenig1985,
author = {Kenig, Carlos E., Ni, Wei-Ming},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {elliptic equation; degenerate elliptic operator},
language = {eng},
number = {2},
pages = {191-224},
publisher = {Scuola normale superiore},
title = {On the elliptic equation $Lu - k + K \exp [2u] = 0$},
url = {http://eudml.org/doc/83956},
volume = {12},
year = {1985},
}

TY - JOUR
AU - Kenig, Carlos E.
AU - Ni, Wei-Ming
TI - On the elliptic equation $Lu - k + K \exp [2u] = 0$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1985
PB - Scuola normale superiore
VL - 12
IS - 2
SP - 191
EP - 224
LA - eng
KW - elliptic equation; degenerate elliptic operator
UR - http://eudml.org/doc/83956
ER -

References

top
  1. [A] L.V. Ahlfors, An extension of Schwarz's Lemma, Trans. Amer. Math. Soc., 43 (1938), pp. 359-364. Zbl0018.41002MR1501949JFM64.0315.04
  2. [AB] L.V. Ahlfors - L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math., 72 (1960), pp. 385-404. Zbl0104.29902MR115006
  3. [Av] P. Aviles, Prescribing conformal complete metrics with given positive Gaussian curvature in R2, preprint. 
  4. [FJK] E.B. Fabes - D.S. Jerison - C.E. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble), 32 (1982), pp. 151-182. Zbl0488.35034MR688024
  5. [FKS] E.B. Fabes - C.E. Kenig - R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Communications in Partial Differential Equations, 7 (1982), pp. 77-116. Zbl0498.35042MR643158
  6. [F] A. Friedman, Bounded entire solutions of elliptic equations, Pacific J. Math., 44, (1973), pp. 497-507. Zbl0256.35024MR320506
  7. [Kp] L. Karp, The exterior Dirichlet problem for nonuniformly elliptic equations and applications, Workshop on Nonlinear Elliptic P.D.E., Jan. 1983MSRI, Berkeley. 
  8. [K] J. Kazdan, Gaussian and scalar curvature, an update, Ann. of Math. Studies, 102 (S.-T. Yau, Ed.), pp. 185-192. Zbl0481.53035
  9. [KW] J. Kazdan - F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), pp. 14-47. Zbl0273.53034MR343205
  10. [Kr] J.B. Keller, On solutions of Δu = f(u), Comm. Pure Appl. Math., 10 (1957), pp. 503-510. Zbl0090.31801
  11. [KN1] C.E. Kenig - W.-M. NI, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math., 106 (1984), pp. 689-702. Zbl0559.35025MR745147
  12. [KN2] C.E. Kenig - W.-M. NI, On the elliptic equation Lu - k + K exp [2u] = 0 and conformal metrics with prescribed curvatures, Abstracts Amer. Math. Soc., 3, no. 5 (Aug. 1982), 796-35-81, p. 343. 
  13. [KN3] C.E. Kenig - W.-M. Ni, Conformal metrics with prescribed curvatures, Abstracts Amer. Math. Soc., 4, no. 3 (April 1983), 803-35-84, p. 254. 
  14. [LSW] W. Littman - G. Stampacchia - H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, Ser. III, 17 (1963), pp. 45-79. Zbl0116.30302MR161019
  15. [M1] R. McOwen, On the elliptic equation Δu + K exp [2u] = f and prescribed negative curvature in R2, preprint. 
  16. [M 2] R. McOwen, Conformal metrics in R2 with prescribed Gaussian and positive total curvature, preprint. 
  17. [N1] W.-M. Ni, On the elliptic equation Δu + K(x) exp [2u] = 0 and conformal metrics with prescribed Gaussian curvatures, Invent. Math., 66 (1982), pp. 343-352. Zbl0487.35042
  18. [N2] W.-M. Ni, On the elliptic equation Δu + K(x)u (n+2)/(n-2) = 0, its generalization, and applications in geometry, Indiana Univ. Math. J., 31 (1982), pp. 493-529. Zbl0496.35036
  19. [O] O.A. Oleinik, On the equation Δu + k(x) exp [u] = 0, Russian Math. Surveys, 33 (1978), pp. 243-244. Zbl0401.35051
  20. [Os] R. Osserman, On the inequality Δu ≽ f(u), Pacific J. Math., 7 (1957), pp. 1641-1647. Zbl0083.09402
  21. [S] D.H. Sattinger, Conformal metrics in R2 with prescribed curvature, Indiana Univ. Math. J., 22 (1972), pp. 1-4. Zbl0236.53009MR305307
  22. [T] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup., Ser. IV, 3 (1976), pp. 697-718. Zbl0341.35031MR601601
  23. [W] H. Wittich, Ganze Lösungen der Differentialgleichung Δu = exp [u], Math. Z., 49 (1944), pp. 579-582. Zbl0028.41001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.