Sobolev integrability implies weak flatness?

Question

Let $B_1\subset \mathbb R^n$ be the unit ball, and suppose $u\in W^{1,2}(B_1)$, i.e. $\int_{B_1} u^2+|Du|^2\,dx\le K<\infty$. Moreover, suppose $u$ is nonnegative, with $ess\inf u=0$.

Let us also define the distribution function $\rho(t)=|\{u\le t\}|$. The nonnegativity and infimum condition simply mean $\rho(t)=0$ precisely when $t< 0$.

Question 1 (answered by supinf): can we deduce from the above that $\rho(t)\ge C t^\alpha$ for some $C,\alpha>0$ depending on $K,n$?

Question 2 (supinf): Replace $\rho(t)\ge Ct^\alpha$ with $\rho(t)\ge \rho(0)+Ct^\alpha$? If $\rho(0)=0$?

Question 3: Can we deduce $\rho(t)\ge Ct^{\alpha}$ for small $t\ge 0$, and some $C,\alpha>0$ depending on $K$?

Here, $\rho(0)\neq 0$ is allowed, and this smallness should remove counterexamples related to the putative ``power growth" of $\rho(t)$.

Heuristics:

1. If $u$ is smooth, then this is certainly true, with $\alpha=n$. So it is reasonable that $\int |Du|^2<\infty$ would have a similar effect.

2. If $n=1$, one would initially guess that the family $u_k(x)=|x|^{1/k}$ would comprise a counterexample, since the distribution function decays with $k$, but in fact, this family is not in $W^{1,2}$.

3. Note that the distribution function also vanishes for the $W^{1,2}$ family $u_k(x)=U(|x|)^{1/k}$, where $U(x)=x-x\ln x$. But it can be checked that the Poincare constant $\int |Du|^2/\int u^2$ blows up, so $K=K(k)$ is unbounded.

4. If $u$ is Lipschitz, the distribution function is somewhat related to the $W^{1,2}$ norm using the co-area formula:

$$ \int_{B_1} |Du|=\int_{0}^\infty |\{u=t\}|dt. $$

Answer 1

No this is not true.

Consider the example $u=0$.

Then $\rho(t) = |B_1|$ if $t\geq 0$ according to the definition of $\rho$.

Then for $C,\alpha>0$ we have $$ \lim_{t\to\infty}\rho(t)= | B_1 | < \infty = \lim_{t\to\infty} C t^\alpha. $$ Thus $\rho(t)\geq Ct^\alpha$ cannot hold for every $t\geq 0$.

question 2 and general remarks:

For question 2, the function $u=1$ is a similar counterexample (here we have $\rho(0)=0$).

Fundamentally, the problem is that $$ \rho(t) \leq |B_1| < \infty \qquad \forall t\in\mathbb R $$ is true for all functions $u \in W^{1,2}(B_1)$, whereas $$ \lim_{t\to\infty} C t^\alpha = \infty $$ holds for all $C>0,\alpha>0$.

So slight modifications will probably not save your conjecture.