Almost minimizers for the thin obstacle problem

Abstract

We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,\beta }$ regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the $C^{1,\gamma }$ regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperimetric inequalities for the solutions of the thin obstacle problem.

Appendix A: Some examples of almost minimizers

Example A.1

If u is a minimizer of the functional

$$\begin{aligned} \int _D a(x)|\nabla u|^2 \end{aligned}$$

over the set with strictly positive $a\in C^{0,\alpha }({\overline{D}})$, $0<\alpha \le 1$, then u is an almost minimizer for the Signorini problem with a gauge function $\omega (r)=C r^\alpha$.

Proof

This is rather immediate. $\square$

Example A.2

Let u be a solution of the Signorini problem for the Laplacian with drift with the velocity field $b\in L^p(B_1)$, $p>n$:

$$\begin{aligned}&-\Delta u + b(x)\nabla u=0\quad \text {in }B_1^\pm \\&-\partial _{x_n}u\ge 0,\quad u\ge 0,\quad u\partial _{x_n}u=0\quad \text {on }B_1', \end{aligned}$$

even in $x_n$-variable. We understand this in the weak sense that u satisfies the variational inequality

$$\begin{aligned} \int _{B_1}\nabla u\nabla (w-u)+(b(x)\nabla u) (w-u)\ge 0, \end{aligned}$$

for any competitor $w\in {\mathfrak {K}}_{0,u}(B_1,B_1')$, i.e. $w\in u+W^{1,2}_0(B_1)$ such that $w\ge 0$ on $B_1'$ in the sense of traces. Then u is an almost minimizer for the Signorini problem with $\psi =0$ on and a gauge function $\omega (r)=Cr^{1-n/p}$.

Proof

Let $B_r(x_0)\Subset B_1$ and $w\in {\mathfrak {K}}_{0,u}(B_r(x_0),B_1')$. Extending w as equal to u in $B_1\setminus B_r(x_0)$, and applying the variational inequality for u, we obtain

$$\begin{aligned} \int _{B_r(x_0)}\nabla u\nabla (w-u)+b(x)\nabla u(w-u)\ge 0. \end{aligned}$$

(A.1)

Let v be the Signorini replacement of u on $B_r(x_0)$. Then v satisfies the variational inequality

$$\begin{aligned} \int _{B_r(x_0)} \nabla v\nabla (w-v)\ge 0, \end{aligned}$$

(A.2)

for all w as above. Now, taking $w=u\pm (u-v)^+$ in (A.1) we will have

$$\begin{aligned} \int _{B_r(x_0)}\nabla u\nabla (u-v)^+ +(b(x)\nabla u)(u-v)^+=0. \end{aligned}$$

Next, taking $w=v+(u-v)^+$ in (A.2), we have

$$\begin{aligned} \int _{B_r(x_0)}\nabla v\nabla (u-v)^+\ge 0. \end{aligned}$$

Taking the difference, we then obtain

$$\begin{aligned} \int _{B_r(x_0)}|\nabla (u-v)^+|^2\le -\int _{B_r(x_0)}b(x)\nabla u(u-v)^+. \end{aligned}$$

Similarly, taking $w=v\pm (v-u)^+$ in (A.2) and $w=u+(v-u)^+$ in (A.1) and subtracting the resulting inequalities, we obtain

$$\begin{aligned} \int _{B_r(x_0)}|\nabla (v-u)^+|^2\le \int _{B_r(x_0)}b(x)\nabla u(v-u)^+. \end{aligned}$$

Hence, combining the inequalities above, we arrive at

$$\begin{aligned} \int _{B_r(x_0)}|\nabla (v-u)|^2\le \int _{B_r(x_0)}|b(x)| |\nabla u| |v-u|. \end{aligned}$$

Then, applying Hölder’s inequality, we have for $p>n$

$$\begin{aligned} \int _{B_r(x_0)}|\nabla (v-u)|^2\le \Vert b\Vert _{L^p(B_r(x_0))}\Vert \nabla u\Vert _{L^2(B_r(x_0))}\Vert v-u\Vert _{L^{p^*}(B_r(x_0))}, \end{aligned}$$

with $p^*=2p/(p-2)$. Next, since $v-u\in W^{1,2}_0(B_r(x_0))$, from the Sobolev’s inequality we have

$$\begin{aligned} \Vert v-u\Vert _{L^{p^*}(B_r(x_0))}\le C_{n,p} r^{1-n/p}\Vert \nabla (v-u)\Vert _{L^2(B_r(x_0))} \end{aligned}$$

and hence we can conclude that

$$\begin{aligned} \int _{B_r(x_0)}|\nabla (v-u)|^2\le Cr^{2(1-n/p)}\int _{B_r(x_0)}|\nabla u|^2 \end{aligned}$$

with $C=C_{n,p}\Vert b\Vert _{L^p(B_1)}^2$. This implies

$$\begin{aligned} \int _{B_r(x_0)}|\nabla u|^2-\int _{B_r(x_0)}|\nabla v|^2&=\int _{B_r(x_0)} (\nabla u+\nabla v)(\nabla u- \nabla v)\\&\le Cr^\gamma \int _{B_r(x_0)}( |\nabla u|^2+|\nabla v|^2)+Cr^{-\gamma }\int _{B_r(x_0)}|\nabla (v-u)|^2\\&\le Cr^\gamma \int _{B_r(x_0)}(|\nabla u|^2+|\nabla v|^2)+Cr^{2(1-n/p)-\gamma }\int _{B_r(x_0)}|\nabla u|^2, \end{aligned}$$

where we have used Young’s inequality in the second line. Choosing $\gamma =1-n/p$ we then deduce that for small enough $0<r<r_0(n,p,\Vert b\Vert _{L^p(B_1)})$

$$\begin{aligned} \int _{B_r(x_0)}|\nabla u|^2\le (1+Cr^{1-n/p})\int _{B_r(x_0)}|\nabla v|^2 \end{aligned}$$

with $C=C_{n,p}\Vert b\Vert _{L^p(B_1)}^2$. $\square$