Let $\Omega\subset\mathbb{R}^{n}$ be a smoothly bounded domain. Is the following statement true? There exists a positive constant $C$ such that $$ \intop_{\Omega}\dfrac{\left|f\left(x\right)\right|^{2}}{\delta\left(x\right)}dx\leq C\left\Vert f\right\Vert^{2} _{W^{1,2}}, $$ for any $f\in C_{c}...

I came across another interesting integral equations problem: Consider the integral equation (inspired by non-relativistic scattering of an electron): $\phi(x)=exp(ikx)+\int_{\mathbb{R}}\frac{exp(ik|x-t|)}{2ik}V(t)\phi(t)$, where $\phi(x)$ represents the wave function of the single particle. I...