Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure (weight) $\gamma_n(x) dx$ and consequently $L^2(\mathbb{R}^n,d\gamma_n)$ and $H^k(\mathbb{R}^n,d\gamma_n)$, $k=1,2$, be the weighted versions of the Sobolev spaces with respect to the measure $d\gamma_n$.

We consider the following elliptic PDE on $\mathbb{R}^n$: $$ -a^{ij}(x)\partial_{ij}u+b^i\partial_iu+c(x)u=f,\tag{1} $$ where $(a^{i,j})$ satisfies the uniform elliptic condition on $\mathbb{R}^n$, and all coefficients $a^{i,j}$, $b^i$, $c$ are bounded and sufficiently smooth.

Then is there any reference saying whether the following well-posedness result holds or not: for all $f\in L^2(\mathbb{R}^n,d\gamma_n)$, (1) admits a unique solution $u\in H^2(\mathbb{R}^n,d\gamma_n)$?

I found a reference stating (without proof) that for $\gamma=\exp(-\mu\sqrt{1+|x|^2})$ with any $\mu>0$, the elliptic pde is well-posed in the corresponding weighted spaces.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.