On "small" homogeneities

General Description View the group

Let $(x,y)\in \mathbb R^n\times \mathbb R$, and let $u(x,y)=u(x,-y)$ solve $$ \left\{ \begin{array}{ll} \Delta u=0 &\text{for $y>0$}\\ \partial_{y}u\leq 0&\text{on $y=0$}\\ u\geq 0&\text{on $y=0$}\\ u\partial_{y}u= 0&\text{on $y=0$}, \end{array} \right. $$ where $\partial_yu(x,0):=\lim_{\epsilon\to 0^+}\frac{u(x,\epsilon)-u(x,0)}{\epsilon}$. We recall that, by the regularity theory in http://www.mathnet.ru/links/83200b51be5ed58c12caff3fc76afe89/znsl805.pdf, the function $u$ is locally Lipschitz in $\mathbb R^n\times \mathbb R$, and it is of class $C^{1,1/2}$ in the set $\mathbb R^n\times [0,\infty)$ (hence, also in the set $\mathbb R^n\times (-\infty,0]$ by symmetry).

Assume that $u$ is $\lambda$-homogeneous for some $\lambda>0$, namely $$ u(r x,r y)=r^\lambda u(x,y)\qquad \forall\,r>0. $$ Let $\mathcal O_n$ denote the set of possible homogeneities in dimension $n$.

It is well-known that, for $n=1$, the set of possible values of $\lambda$ is given by $$ \mathcal O_1:=\bigl\{1,2,3,4,\ldots\bigr\}\cup\biggl\{\frac{3}{2},\frac72,\frac{11}2,\ldots\biggr\} $$ (see for instance Appendix A.1 in https://arxiv.org/pdf/1703.00678.pdf, specified to the particular case $s=1/2$).

For instance, possible solutions are given by $$ u(x,y)=-|y| \quad \leftrightsquigarrow \quad \lambda=1, $$ $$ u(x,y)=x^2-y^2 \quad \leftrightsquigarrow \quad \lambda=2, $$ $$ u(x,y)=\frac{|y|^3}{3} -x^2|y|\quad \leftrightsquigarrow \quad \lambda=3. $$ Also, if we use polar coordinates $(\rho,\theta)$ with $\theta=0$ corresponding to the positive $x$ axis, then $$ u(\rho,\theta)=-\rho^{(2k+1)/2}\sin\biggl(\frac{(2k+1)\theta}{2}\biggr) \quad \leftrightsquigarrow \quad \lambda=\frac{2k+1}2,\qquad k\in \mathbb N. $$

In the case $n\geq 2$, much less is known. By the results in https://arxiv.org/pdf/math/0609031.pdf and https://arxiv.org/pdf/1709.03120.pdf, we know that the set $\mathcal O_n$ is disjoint from $$ \bigl(0,1\bigr)\cup \biggl(1,\frac32\biggr) \cup \biggl(\frac32,2\biggr) \cup\bigcup_{m\in \mathbb N}\bigl((2m-c_{m,n}^-,2m)\cup(2m,2m+c_{m,n}^+)\bigr), $$ where $c_{m,n}^\pm>0$.

Problem's Description

Let $n\geq 2$, and let $\mathcal O_n$ denote the set of possible homogeneities in dimension $n$ for global solutions to the Signorini problem.

Let $n_0$ be the largest integer such that the set $\mathcal O_{n_0}$ is disjoint from the interval $(2,3)$. Find the best possible lower bound on $n_0$.

Note that, at the moment, we only know that $n_0\geq 1$. By the results in https://arxiv.org/pdf/1709.04002.pdf, any improvement on the value of $n_0$ would give a stronger bound on the dimension of the set of anomalous free boundary points in the classical obstacle problem.

1. Article On the fine structure of the free boundary for the classical obstacle problem

   • Alessio Figalli,
   • Joaquim Serra

   arXiv