Abstract
We consider the non-local Liouville equation
corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up, when hε approaches a function h as ε → 0, at a critical point of the harmonic extension of h provided some generic assumptions are satisfied.
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L. Battaglia, M. Medina and A. Pistoia, Large conformal metrics with prescribed gaussian and geodesic curvatures, Calc. Var. Partial Differential Equations 60 (2021), Article no. 39.
K. C. Chang and J. Q. Liu, A prescribing geodesic curvature problem, Math. Z. 223 (1996), 343–365.
F. Da Lio, L. Martinazzi and T. Rivière, Blow-up analysis of a nonlocal Liouville-type equation, Anal. PDE 8 (2015), 1757–1805.
Y.-X. Guo and J.-Q. Liu, Blow-up analysis for solutions of the Laplacian equation with exponential Neumann boundary condition in dimension two, Commun. Contemp. Math. 8 (2006), 737–761.
A. Jevnikar, R. López-Soriano, M. Medina and D. Ruiz, Blow-up analysis of conformal metrics of the disk with prescribed gaussian and geodesic curvatures, Analysis and PDE, to appear, arXiv:2004.14680 [math.AP]
P. Liu and W. Huang, On prescribing geodesic curvature on D2, Nonlinear Anal. 60 (2005), 465–473.
B. Ou, A uniqueness theorem for harmonic functions on the upper-half plane, Conform. Geom. Dyn. 4 (2000), 120–125.
H. Zhang, Prescribing the boundary geodesic curvature on a compact scalar-flat Riemann surface via a flow method, Pacific J. Math. 273 (2015), 307–330.
L. Zhang, Classification of conformal metrics on R 2+ with constant Gauss curvature and geodesic curvature on the boundary under various integral finiteness assumptions, Calc. Var. Partial Differential Equations 16 (2003), 405–430.
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M. Medina was partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement N 754446 and UGR Research and Knowledge Transfer Fund — Athenea3i.
M. Medina also acknowledges the hospitality of Università La Sapienza di Roma, where this work was carried out during a long visit in the academic year 2019–2020.
A. Pistoia was partially supported by Fondi di Ateneo “Sapienza” Universita di Roma (Italy).
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Battaglia, L., Medina, M. & Pistoia, A. A blow-up phenomenon for a non-local Liouville-type equation. JAMA 149, 343–367 (2023). https://doi.org/10.1007/s11854-022-0260-1
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DOI: https://doi.org/10.1007/s11854-022-0260-1