TY - JOUR
TI - On a hardy type inequality and a singular Sturm-Liouville equation
DO - https://doi.org/doi:10.7282/T3R20ZGG
PY - 2014
AB - In this dissertation, we first prove a Hardy type inequality for $uin W^{m,1}_0(Omega)$, where $Omega$ is a bounded smooth domain in $mathbb{R}^N$ and $mgeq2$. For all $jgeq 0$, $1leq kleq m-1$, such that $1leq j+kleq m$, it holds that $frac{partial^ju(x)}{d(x)^{m-j-k}}in W^{k,1}_0(Omega)$, where $d$ is a smooth positive function which coincides with $dist(x,domega)$ near $domega$, and $partial^l$ denotes any partial differential operator of order $l$. We also study a singular Sturm-Liouville equation $-(x^{2alpha}u')'+u=f$ on $(0,1)$, with the boundary condition $u(1)=0$. Here $alpha>0$ and $fin L^2(0,1)$. We prescribe appropriate (weighted) homogeneous and non-homogeneous boundary conditions at 0 and prove the existence and uniqueness of $H^2_{loc}(0,1]$ solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator $mathcal{L}u:=-(x^{2alpha}u')'+u$ under homogeneous boundary conditions. Finally, we are interested in the equation $-(|x|^{2alpha}u')'+|u|^{p-1}u=mu$ on $(-1, 1)$ with boundary condition $u(-1)=u(1)=0$. Here $alpha>0$, $pgeq1$ and $mu$ is a bounded Radon measure on the interval $(-1,1)$. We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. We examine the limiting behavior of three approximation schemes. The isolated singularity at 0 is also investigated.
KW - Mathematics
KW - Sturm-Liouville equation--Numerical solutions
KW - Mathematics--Study and teaching
LA - eng
ER -