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On a planar conformal curvature problem

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Abstract

Given \(0<p<1, q>2\) and a positive \(L^{1}\)-function g on the unit circle, with \(g(\theta +T)=g(\theta )\) for some \(T\le 2\pi \), it is shown that the one dimensional conformal curvature problem \(-u''+u=\frac{g(\theta )}{u^{1-p}}+\uplambda u^{q-1}\;\text{ on }\;{\mathbb {S}}\) has two positive T-periodic solutions provided that \(\uplambda \) is smaller than an explicit constant.

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References

  1. Ai, J., Chou, K.S., Wei, J.: Self-similar solutions for the anisotropic affine curve shortening problem. Calc. Var. Partial Differ. Equ. 13, 311–337 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aleksandrov, A.D.: On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies. Mat.Sbornik N.S. 3, 27–46 (1938)

    Google Scholar 

  3. Andrews, B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138, 151–161 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews, B.: Classifications of limiting shapes for isotropic curve flows. J. Am. Math. Soc. 16, 443–459 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Böröczky, K., Lutwak, E., Yang, D., Zhang, G.: The logarithmic Minkowski problem. J. Am. Math. Soc. 26, 831–852 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for non-linear second order elliptic equations I. Monge–Ampère equations. Commun. Pure Appl. Math. 37, 369–402 (1984)

    Article  MATH  Google Scholar 

  7. Calabi, E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Mich. Math. J. 5, 105–126 (1958)

    Article  MATH  Google Scholar 

  8. Chen, W.: \(L_{p}\) Minkowski problem with not necessarily positive data. Adv. Math. 201, 77–89 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, S.Y., Yau, S.T.: On the regularity of the solution of the \(n\)-dimensional Minkowski problem. Commun. Pure Appl. Math. 29, 495–516 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chou, K.S., Wang, X.J.: The \(L_{p}\)-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dou, J., Zhu, M.: The two dimensional \(L_{p}\) Minkowski problem and nonlinear equations with negative exponents. Adv. Math. 230, 1209–1221 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper, Danske Vid. Selskab. Mat.-fys. Medd. 16, 1–31 (1938)

    MATH  Google Scholar 

  14. Gage, M., Li, Y.: Evolving planes curves by curvature in relative geometries I. Duke Math. J. 72, 441–466 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gage, M., Li, Y.: Evolving planes curves by curvature in relative geometries II. Duke Math. J. 75, 79–98 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gardner, R.: Geometric Tomography. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  17. Gardner, R., Hug, D., Weil, W.: The Orlicz—Brunn–Minkowski theory: a general framework, additions, and inequalities. J. Differ. Geom. 97, 427–476 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gardner, R., Hug, D., Weil, W., Ye, D.: The dual Orlicz–Brunn–Minkowski theory. J. Math. Anal. Appl. 430, 810–829 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gruber, P.: Convex and Discrete Geometry. Springer, Berlin (2007)

    MATH  Google Scholar 

  20. Guan, P., Lin, C.-S.: On equation \(det(u_{i,j}+\delta _{i,kJ}u)=u^{p}f\) on \(S^{n}\), preprint

  21. Haberl, C., Lutwak, E., Yang, D., Zhang, G.: The even Orlicz Minkowski problem. Adv. Math. 224, 2485–2510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Haberl, C., Schuster, F.: Asymmetric affine \(L_{p}\) Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Haberl, C., Schuster, F., Xiao, J.: An asymmetric affine Pólya–Szegö principle. Math. Ann. 352, 517–542 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Henk, M., Linke, E.: Cone-voluem measures of polytopes. Adv. Math. 253, 50–62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hu, C., Ma, X., Shen, C.: On the Christoffel–Minkowski problem of Firey’s \(p\)-sum. Calc. Var. Partial Differ. Equ. 21, 137–155 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang, Q., He, B.: On the Orlicz Minkowski problem for polytopes. Discrete Comput Geom 48, 281–297 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, Y., Liu, J., Xu, L.: On the uniqueness of the \(L_{p}\) Minkowski problems: the constant p-curvature case in \({\mathbb{R}}^{3}\). Adv. Math. 281, 906–927 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hug, D., Lutwak, E., Yang, D., Zhang, G.: On the \(L_{p}\) Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ivaki, M.N.: A flow approach to the \(L_{-2}\) Minkowski problem. Adv. Appl. Math. 50, 445–464 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jiang, M.: Remarks on the 2-dimensional \(L_{p}\)-Minkowski problem. Adv. Nonlinear Stud. 10, 297–313 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jian, H., Lu, J., Wang, X.J.: Nonuniqueness of solutions to the \(L_{p}\)-Minkowski problem. Adv. Math. 281, 845–856 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lewy, H.: On differential geometry in the large, I (Minkowski’s problem). Trans. Am. Math. Soc. 43, 258–270 (1938)

    MathSciNet  MATH  Google Scholar 

  33. Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. In: Contributions to Analysis, pp. 245–272. Academic Press, New York (1974)

  34. Ludwig, M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119, 159–188 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ludwig, M.: General affine surface areas. Adv. Math. 224, 2346–2360 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ludwig, M., Reitzner, M.: A classification of \(SL(n)\) invariant valuations. Ann. Math. 172, 1219–1267 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differential Geom 38(1), 131–150 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lutwak, E.: The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lutwak, E., Oliker, V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227–246 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lutwak, E., Yang, D., Zhang, G.: On the \(L_{p}\)-Minkowski problem. Trans. Am. Math. Soc. 356, 4359–4370 (2004)

    Article  MATH  Google Scholar 

  41. Lutwak, E., Yang, D., Zhang, G.: A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375–390 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lutwak, E., Yang, D., Zhang, G., Sobolev: Optimal, norms and the \(L_{p}\)-Minkowski problem, Int. Math. Res. Not. IMRN,: 2006, p. 62987. Art, ID (2006)

  43. Lutwak, E., Yang, D., Zhang, G.: Orlicz projection bodies. Adv. Math. 223, 220–242 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lutwak, E., Yang, D., Zhang, G.: Orlicz centroid bodies. J. Differ. Geom. 84, 365–387 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Meyer, M., Werner, E.: On the \(p\)-affine surface area. Adv. Math. 152, 288–313 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  46. Minkowski, H.: Volumen und oberfläche. Math. Ann. 57, 447–495 (1903)

    Article  MathSciNet  MATH  Google Scholar 

  47. Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Commun. Pure Appl. Math. 6, 337–394 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  48. Pogorelov, A.V.: The Minkowski Multidimensional Problem. V.H Winston Sons, Washington D.C (1978)

    MATH  Google Scholar 

  49. Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  50. Schütt, C., Werner, E.: Surface bodies and \(p\)-affine surface area. Adv. Math. 187, 98–145 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  51. Stancu, A.: The discrete plannar \(L_{0}\)-Minkowski problem. Adv. Math. 167, 160–174 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  52. Stancu, A.: On the number of solutions to the discrete two-dimensional \(L_{0}\)-Minkowski problem. Adv. Math. 180, 290–323 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. Thompson, A.C.: Minkowski Geometry. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  54. Tzitzéica, G.: Sur une nouvelle classe de surfaces. Rend. Circ. Mat. Palermo 25, 180–187 (1908)

    Article  MATH  Google Scholar 

  55. Tzitzéica, G.: Sur une nouvelle classe de surfaces. Rend. Circ. Mat. Palermo 28, 210–216 (1909)

    Article  MATH  Google Scholar 

  56. Umanskiy, V.: On the solvability of the two dimensional \(L_{p}\)-Minkowski problem. Adv. Math. 225, 3214–3228 (2010)

    Article  MathSciNet  Google Scholar 

  57. Werner, E.: On \(L_{p}\) affine surface areas. Indiana Univ. Math. J. 56, 2305–2323 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. Xi, D., Jin, H., Leng, G.: The Orlicz Brunn–Minkowski inequality. Adv. Math. 260, 350–374 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. Ye, D.: Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz \(L_{\varphi }\) affine and geominimal surface areas. J. Math. Anal. Appl. 443, 352–371 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Yijing, S.: Compatibility phenomena in singular problems. Proc. R. Soc. Edinb. Sect. A 143, 1321–1330 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  61. Yijing, S., Duanzhi, Z.: The role of the power 3 for elliptic equations with negative exponents. Calc. Var. Partial Differ. Equ. 49, 909–922 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  62. Yijing, S., Shaoping, W.: An exact estimate result for a class of singular equations with critical exponents. J. Funct. Anal. 260, 1257–1284 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  63. Yijing, S., Xiaoqiang, C.: Remarks on a planar conformal curvature problem. Monatsh Math. 176, 623–636 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  64. Yijing, S., Yiming, L.: The planar Orlicz Minkowski problme in the \(L^{1}\)-sense. Adv. Math. 281, 1364–1383 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  65. Zhu, B., Zhou, J., Xue, W.: Dual Orlicz–Brunn–Minkowski theory. Adv. Math. 264, 700–725 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhu, G.: The logarithmic Minkowski problem for polytopes. Adv. Math. 262, 909–931 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  67. Zhu, G.: The centro-affine Minkowski problem for polytopes. J. Differ. Geom. 101, 159–174 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  68. Zhu, G.: The \(L_{p}\) Minkowski problem for polytopes for \(0<p<1\). J. Funct. Anal. 269, 1070–1094 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Science Foundation of China (Grants 11971027 and 11771468).

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Authors and Affiliations

  1. Department of Mathematics, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Sun Yijing & Shuang Zhen

  2. School of Computer Science and Mathematics, FuJian University of Technology, Fuzhou, 350118, People’s Republic of China

    Tan Yuxin

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  1. Sun Yijing
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Yijing, S., Yuxin, T. & Zhen, S. On a planar conformal curvature problem. Math. Z. 299, 1565–1585 (2021). https://doi.org/10.1007/s00209-021-02728-4

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