7 papers
Physical derivation of the coarea formula and an elementary proof via gradient flow
Shibo Liu
In this note, we derive an elementary version of the coarea formula by considering the mass of a solid body with density . Then we present an rigorous proof using the changi…
On the integral formula of the Jacobian determinant
Shibo Liu
It is known that the integral of the Jacobian determinant of a smooth map depends only on and this result leads to an anal…
Schrodinger-Poisson-Slater equations with nonlinearity subscaled near zero
Shibo Liu, Kanishka Perera
We study the following zero-mass Schr{ö}dinger-Poisson-Slater equation \[ - Îu + \left( \frac{1}{4 Ï| x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1…
Multiple solutions for Schr{ö}dinger-Poisson-Slater equations with critical growth
Shibo Liu
We obtain multiple solutions for the zero mass Schr{ö}dinger-Poisson-Slater equation \[ - Îu + \left( \frac{1}{4 Ï| x |} \ast u^2 \right) u = λg (x) | u |^{p - 2} u + | u |^{6…
Polynomial 2D Green Coordinates for High-order Cages
Shibo Liu, Ligang Liu, Xiao-Ming Fu
We propose conformal polynomial coordinates for 2D closed high-order cages, which consist of polynomial curves of any order. The coordinates enable the transformation of the input…
Polynomial 2D Biharmonic Coordinates for High-order Cages
Shibo Liu, Ligang Liu, Xiao-Ming Fu
We derive closed-form expressions of biharmonic coordinates for 2D high-order cages, enabling the transformation of the input polynomial curves into polynomial curves of any order.…