$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations
Der-Chen Chang Jie Xiao
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1905-1920 doi: 10.3934/dcds.2015.35.1905
Based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in the upper-half Euclidean space, this note characterizes a nonnegative Randon measure $\mu$ on $\mathbb R^{1+n}_+$ such that the extension $R_\alpha L^p(\mathbb R^n)\subseteq L^q(\mathbb R^{1+n}_+,\mu)$ holds for $(\alpha,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$ where $u=R_\alpha f$ is the weak solution of the fractional diffusion equation $(\partial_t + (-\Delta_x)^\alpha)u(t, x) = 0$ in $\mathbb R^{1+n}_+$ subject to $u(0,x)=f(x)$ in $\mathbb R^n$.
keywords: $L^p$-type capacities fractional diffusion equations heat kernel. $L^q$-extensions
On the variational $p$-capacity problem in the plane
Jie Xiao
Communications on Pure & Applied Analysis 2015, 14(3): 959-968 doi: 10.3934/cpaa.2015.14.959
Under $1 < p \leq 2$, this note presents some new optimal isoperimetric type properties of the variational $p$-capacitary potentials on convex plane rings.
keywords: Isoperimetric inequalities $p$-capacitary potentials convex plane rings.

Year of publication

Related Authors

Related Keywords

[Back to Top]