On the planar 𝐿_{𝑝} Minkowski problem with sign-changing data

Torres, Pedro; Zamora, Manuel

doi:10.1090/proc/15378

On the planar $L_p$ Minkowski problem with sign-changing data
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by Pedro J. Torres and Manuel Zamora PDF

Proc. Amer. Math. Soc. 149 (2021), 3077-3088 Request permission

Abstract:

We provide novel sufficient conditions for the existence of $T-$periodic solutions to the second-order singular equation \begin{equation*} u''+u=\frac {h(t)}{u^{\mu }}, \end{equation*} where $\mu \geq 1$ and $h$ is a general sign-changing function defined on $[0,T]$. Our main result has a direct reading on the two-dimensional $L_p$ Minkowski problem with sign-changing data.

References

Jun Ai, Kai-Seng Chou, and Juncheng Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 13 (2001), no. 3, 311–337. MR 1865001, DOI 10.1007/s005260000075
Károly J. Böröczky and Hai T. Trinh, The planar $L_p$-Minkowski problem for $0<p<1$, Adv. in Appl. Math. 87 (2017), 58–81. MR 3629263, DOI 10.1016/j.aam.2016.12.007
Anna Capietto, Jean Mawhin, and Fabio Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992), no. 1, 41–72. MR 1042285, DOI 10.1090/S0002-9947-1992-1042285-7
Wenxiong Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math. 201 (2006), no. 1, 77–89. MR 2204749, DOI 10.1016/j.aim.2004.11.007
Claus Dohmen and Yoshikazu Giga, Selfsimilar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 7, 252–255. MR 1303574
Claus Dohmen, Yoshikazu Giga, and Noriko Mizoguchi, Existence of selfsimilar shrinking curves for anisotropic curvature flow equations, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 103–119. MR 1379195, DOI 10.1007/BF01189949
Jingbo Dou and Meijun Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents, Adv. Math. 230 (2012), no. 3, 1209–1221. MR 2921178, DOI 10.1016/j.aim.2012.02.027
José Godoy and Manuel Zamora, A general result to the existence of a periodic solution to an indefinite equation with a weak singularity, J. Dynam. Differential Equations 31 (2019), no. 1, 451–468. MR 3935151, DOI 10.1007/s10884-018-9704-9
Robert Hakl and Pedro J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations 248 (2010), no. 1, 111–126. MR 2557897, DOI 10.1016/j.jde.2009.07.008
Robert Hakl, Pedro J. Torres, and Manuel Zamora, Periodic solutions to singular second order differential equations: the repulsive case, Topol. Methods Nonlinear Anal. 39 (2012), no. 2, 199–220. MR 2985878
Robert Hakl and Manuel Zamora, Periodic solutions to second-order indefinite singular equations, J. Differential Equations 263 (2017), no. 1, 451–469. MR 3631312, DOI 10.1016/j.jde.2017.02.044
Mohammad N. Ivaki, A flow approach to the $L_{-2}$ Minkowski problem, Adv. in Appl. Math. 50 (2013), no. 3, 445–464. MR 3011439, DOI 10.1016/j.aam.2012.09.003
Meiyue Jiang, Liping Wang, and Juncheng Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 41 (2011), no. 3-4, 535–565. MR 2796243, DOI 10.1007/s00526-010-0375-6
Meiyue Jiang and Juncheng Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem II, Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 785–803. MR 3392904, DOI 10.3934/dcds.2016.36.785
Mei-Yue Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem, Adv. Nonlinear Stud. 10 (2010), no. 2, 297–313. MR 2656684, DOI 10.1515/ans-2010-0204
Rafael Ortega and David Rojas, Periodic oscillators, isochronous centers and resonance, Nonlinearity 32 (2019), no. 3, 800–832. MR 3909067, DOI 10.1088/1361-6544/aaee9a
Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
Pedro J. Torres, Mathematical models with singularities, Atlantis Briefs in Differential Equations, vol. 1, Atlantis Press, Paris, 2015. A zoo of singular creatures. MR 3328358, DOI 10.2991/978-94-6239-106-2
Vladimir Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem, Adv. Math. 180 (2003), no. 1, 176–186. MR 2019221, DOI 10.1016/S0001-8708(02)00101-9