Let $\rho$ be the standard Gaussian measure and let $h\in H^1(\mathbb{R},\rho)$ be such that $\exp h \in H^1(\mathbb{R},\rho)$. By Sobolev embeddings, we have $H^1(\mathbb{R},\rho)\hookrightarrow C^0(\mathbb{R})$. Let $X$ be the linear subspace of $H^1(\mathbb{R},\rho)$ given by $$ X = \{ u \in H^1(\mathbb{R},\rho) \ |\ u(0) = 0\}. $$ Now consider the ODE $$ u' - u\, h' = 0. $$ All possible solutions to this differential equation form the one-dimensional linear space $E = \{c\,\exp h \ | \ c\in\mathbb{R}\}$. Denote by $S(X) = \{u\in X \ |\ \Vert u\Vert_{H^1} = 1 \}$ the unit sphere in $X$.
My questions are the following:
- Does a sequence $\{u_n\}\subset H^1(\mathbb{R},\rho)$ with $\Vert u'_n - u_n\, h'\Vert_{H^1} \to 0$ imply that there exists some $c\in\mathbb{R}$, such that $\Vert u_n - c\, \exp h\Vert_{H^1} \to 0$.
- Is there a way to show, that there exists some positive constant $C_d>0$, such that $$ \operatorname{dist}_{H^1}(S(X),E) = \inf_{u\in S(X)}\inf_{\eta\in E} \Vert u-\eta \Vert_{H^1} \geq C_d > 0. $$
Note: There is a related question here but the assumptions are a little different. Here, $E$ is closed, convex and finite dimensional while $S(X)$ is closed and bounded but not convex and infinite dimensional.
Here is what I tried so far: Due to the Sobolev embedding, we have for any $c \in \mathbb{R}$ $$ \inf_{u\in S(X)} \Vert u- c\,\exp h \Vert_{H^1} \geq C_\mathrm{Sob} \inf_{u\in S(X)} \Vert u- c\,\exp h \Vert_{C^0} \geq C_\mathrm{Sob} \inf_{u\in S(X)} \vert u(0) - c\,\exp h(0) \vert = C_\mathrm{Sob} \exp h(0) \vert c\vert. $$ In other words $\operatorname{dist}(E,S(X)) \gtrsim \inf_{c\in\mathbb{R}} \vert c\vert$. On the other hand, we have for $\{c_n\, \exp h\}_n \subset E$ with $c_n\to 0$ $$ \inf_{u\in S(X)} \Vert u- c_n\,\exp h \Vert_{H^1} \geq \inf_{u\in S(X)} \vert \Vert u \Vert_{H^1} - \vert c_n\vert \Vert \exp h \Vert_{H^1} \vert = \vert 1 - \vert c_n\vert \Vert \exp h \Vert_{H^1} \vert \to 1 \qquad\mbox{as }n\to \infty. $$ Since not both these lower bounds can go to zero simultaneously, this yields some equilibrium value $0 < c^* < \Vert \exp h\Vert_{H^1}^{-1}$ such that e.g. $\operatorname{dist}(E,S(X)) \geq C_\mathrm{Sob} \exp h(0) c^* > 0$.
