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Regularity in the two-phase free boundary problems under non-standard growth conditions

Jun ZhengSubjects: Mathematics >> Mathematics （General）

In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla u)+(\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma})+gu\big)\text{d}x\rightarrow \text{min}$ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $\gamma=0$, chemical reaction problems with $0<\gamma<1$, and obstacle type problems with $\gamma=1$. Our results hold not only in the degenerate case of $p> 2$ for $p-$Laplace equations, but also in the singular case of $1 |

H\"{o}lder continuity of solutions to the $G$-Laplace equation

Jun Zheng; Yan ZhangSubjects: Mathematics >> Mathematics （General）

We establish regularity of solutions to the $G$-Laplace equation $-\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu$, where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g(t)$ is supposed to be nonnegative and $C^{1}$-continuous in $[0,+\infty)$, satisfying $g(0)=0$, and for some positive constants $\delta$ and $g_{0}$, $\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0$, that generalizes the structural conditions of Ladyzhenskaya-Ural'tseva for an elliptic operator. |

A Note on Elliptic Coordinates

Sun, CheSubjects: Mathematics >> Mathematics （General）

Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which an elliptic-coordinate function can be readily mapped on a uniform Cartesian mesh.Application to Kirchhoff vortex is provided. |

submitted time
2018-09-22
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