On global homogeneous solutions to the Signorini problem

Problem's Description

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 $c_{m,n}^\pm>0$.

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

   • Alessio Figalli,
   • Joaquim Serra

   arXiv

1. Open Discreteness of possible homogeneities

2. Open On "small" homogeneities