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- W3022154815 abstract "Abstract We consider an evolution problem associated to the Kazdan–Warner equation on a closed Riemann surface $$(Sigma ,g)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Σ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> $$begin{aligned} -Delta _{g}u=8pi left( dfrac{he^{u}}{int _{Sigma }he^{u}mathop {}mathrm {d}mu _{g}}-dfrac{1}{int _{Sigma }mathop {}mathrm {d}mu _{g}}right) end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> <mml:mfenced> <mml:mstyle> <mml:mfrac> <mml:mrow> <mml:mi>h</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>u</mml:mi> </mml:msup> </mml:mrow> <mml:mrow> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mi>h</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>u</mml:mi> </mml:msup> <mml:mrow /> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mo>-</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mrow /> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:mfenced> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where the prescribed function $$hge 0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$max _{Sigma }h>0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mo>max</mml:mo> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We prove the global existence and convergence under additional assumptions such as $$begin{aligned} Delta _{g}ln h(p_0)+8pi -2K(p_0)>0 end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>ln</mml:mo> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>K</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> for any maximum point $$p_0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> of the sum of $$2ln h$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>ln</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> and the regular part of the Green function, where K is the Gaussian curvature of $$Sigma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Σ</mml:mi> </mml:math> . In particular, this gives a new proof of the existence result by Yang and Zhu (Pro Am Math Soc 145:3953–3959, 2017) which generalizes existence result of Ding et al. (Asian J Math 1:230–248, 1997) to the non-negative prescribed function case." @default.
- W3022154815 created "2020-05-13" @default.
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- W3022154815 date "2021-01-24" @default.
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- W3022154815 title "Global existence and convergence of a flow to Kazdan–Warner equation with non-negative prescribed function" @default.
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