Optimal regularity for the Signorini problem and its free boundary

Abstract

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\mathbf{u }=(u^1,u^2,\dots ,u^n)\in W^{1,2}(B_1^+:{\mathbb {R}}^n)$ minimizes

$$\begin{aligned} J(\mathbf{u })=\int _{B_1^+}|\nabla \mathbf{u }+\nabla ^\bot \mathbf{u }|^2+\lambda \hbox {div}(\mathbf{u })^2 \end{aligned}$$

in the convex set

$$\begin{aligned} K= & {} \left\{ \mathbf{u }=(u^1,u^2,\dots ,u^n)\in W^{1,2}(B_1^+:{\mathbb {R}}^n);\; u^n\ge 0 \quad \hbox {on}\quad \Pi ,\right. \\&\left. \mathbf{u }=f\in C^\infty (\partial B_1) \quad \hbox {on}\quad (\partial B_1)^+ \right\} , \end{aligned}$$

where $\lambda \ge 0$ say. Then $\mathbf{u }\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by

$$\begin{aligned} \Gamma _\mathbf{u }=\partial \{x;\;u^n(x)=0,\; x_n=0\}\cap B_{1}, \end{aligned}$$

will be a $C^{1,\alpha }$ graph close to points where $\mathbf{u }$ is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5, 6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.

Appendices

Appendix 1: Proof of Lemma 2.1

In this Appendix we will indicate how to prove $C^{1,\beta }$ regularity of the solutions to the Signorini problem. The proof follows the lines of the proof in the main body of the paper, but it is significantly simpler. We will therefore only briefly indicate some main points.

The idea of the proof of $C^{1,\beta }$-regularity is the same as the proof of $C^{1,1/2}$-regularity. In particular, if the regularity is worse than $C^{1,\beta }$ then $\sup _{B_r^+}|\mathbf{u }|$ grows faster than $r^{1+\beta }$ away from a free boundary point. Making a blow-up will give us a global solution that grows slower than $R^{1+\beta }$ as $R\rightarrow \infty$. But we know that the global solutions have growth $1, 3/2, 2, 5/2,\dots$ at infinity. So if $1+\beta <3/2$ then the blow-up can not be a solution—unless it grows like r which means that it is linear. We may thus argue as in the main body of the paper and assume that $\mathbf{u }\notin C^{1,\beta }$ for some $\beta <1/2$, this is the main intuition of (144), and make a blow-up that converges to a global solution that grows linearly.

Since we do not know that the solutions are $C^{1,\beta }$ yet we will no longer make the “standing assumption” we did in Sect. 2.

Lemma 14.1

Let $\mathbf{u }\ne 0$ be a global solution to the Signorini problem and assume that

$$\begin{aligned} \liminf _{r\rightarrow \infty }\frac{\ln \left( \Vert \mathbf{u }\Vert _{\tilde{L}^2(B_r^+)}\right) }{\ln (r)}< 3/2 \end{aligned}$$

(144)

then $\mathbf{u }$ is a linear function.

Proof

The proof is based on Proposition 7.1 and Lemma 6.2. In particular, arguing as in the first part of Proposition 7.1 it follows that $\mathbf{u }$ is two dimensional which by Lemma 6.2 implies that $\mathbf{u }$ is linear.

In order to repeat the argument in Proposition 7.1 we would have to repeat the arguments in Sect. 4 without knowing that the solutions are $C^{1,\beta }$. But the proofs are almost line for line the same so we will only indicate a few minor changes.

Following that proof of Lemma 4.1 we consider

$$\begin{aligned} w^{ij}(x)=\frac{\partial u^i}{\partial x_j}-\frac{\partial u^j}{\partial x_i}. \end{aligned}$$

It follows, exactly as in Lemma 4.1, that $w^{i,j}=$constant.

Noticing that

$$\begin{aligned} \sup _{B_R^+}|w^{ij}|\le C(1+R)^{\alpha -1} \end{aligned}$$

we may conclude that the constant is zero if $\alpha <1$. If $\alpha \ge 1$ we may without loss of generality subtract a linear function from $\mathbf{u }$ such $w^{ij}=0$. Equation (24) follows.

It is also easy to see that

$$\begin{aligned} 2\frac{\partial \xi }{\partial x_n}+\tau \in W^{2,2}_{loc}({\mathbb {R}}^n_+) \end{aligned}$$

which by the trace Theorem implies that

$$\begin{aligned} 2\frac{\partial \xi }{\partial x_n}+\tau \in W^{3/2,2}_{loc}(\Pi ). \end{aligned}$$

(145)

In particular we may deduce, as in (32), that

$$\begin{aligned} 2\frac{\partial \xi }{\partial x_n}+\tau =c. \end{aligned}$$

(146)

The rest of Lemma 4.1 follows as in the proof of that lemma.

The rest of Sect. 4 follows as before with some minor changes.

This means that we may argue as in Proposition 7.1 to conclude that $\mathbf{u }(x)=\mathbf{u }(x_1,x_n)$. By Lemma 6.2 it follows that $\mathbf{u }$ may be written as a series of eigenfunctions. But our assumption (144) assures that no eigenfunction in the series representation has homogeneity greater than 1 which implies proves the lemma. $\square$

The following Corollary is an easy consequence of Lemma 14.1.

Corollary 14.1

Let $\mathbf{u }\ne 0$ be a global solution to the Signorini problem and assume that

$$\begin{aligned} \liminf _{r\rightarrow \infty }\frac{\ln \left( \Vert \mathbf{u }\Vert _{\tilde{L}^2(B_r^+)}\right) }{\ln (r)}\le 1 \end{aligned}$$

then

$$\begin{aligned} \liminf _{r\rightarrow \infty }\frac{\ln \left( \Vert \mathbf{u }\Vert _{\tilde{L}^2(B_r^+)}\right) }{\ln (r)}= 1. \end{aligned}$$

Definition 14.1

We will denote the $L^2$-projection of $\mathbf{u }\in L^2(B_{r}^+(x^0);{\mathbb {R}}^n)$ onto the space $\mathcal {P}$ by $\mathbf {Pr}(\mathbf{u },r,x^0)$. The space $\mathcal {P}$ we is the space of affine functions l satisfying

• (i) l(x) is affine of the following form;

  $$\begin{aligned} l(x)=\left[ \begin{array}{l} b_{1} \\ b_{2} \\ \vdots \\ b_{n} \end{array}\right] +\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} a_{11} &{} a_{12} &{} \dots &{} 0 \\ a_{21} &{} a_{22} &{} \dots &{} 0 \\ \vdots &{} &{} \ddots &{} 0 \\ 0 &{} 0 &{} \dots &{} a_{33} \end{array}\right] \left[ \begin{array}{l} x_1 \\ x_2 \\ \vdots \\ x_n \end{array}\right] , \end{aligned}$$

• (ii) $a_{nn}+\frac{\lambda }{4}(a_{11}+a_{22}+\dots +a_{nn})$=0.

That is $\mathbf {Pr}(\mathbf{u },r,x^0)$ is the element in $\mathcal {P}$ that satisfies

$$\begin{aligned} \left\| \mathbf{u }-\mathbf {Pr}(\mathbf{u },r,x^0)\right\| _{L^2(B_r^+(x^0))}=\inf _{\mathbf{p }\in \mathcal {P}}\Vert \mathbf{u }-\mathbf{p }\Vert _{L^2(B_r^+(x^0))}. \end{aligned}$$

When $x^0=0$ we will just write $\mathbf {Pr}(\mathbf{u },r)$ for $\mathbf {Pr}(\mathbf{u },r,0)$.

Remark

Notice that the condition $a_{1k}=a_{k1}=0$ for $k=1,2,\dots ,n-1$ and (ii) just implies that l(x) satisfies the same boundary data as $\mathbf{u }$ in $\Omega _\mathbf{u }$.

We are now ready to sketch a proof of Lemma 2.1. The idea is to make a blow-up as in Lemma 14.1—but subtract the closest linear solution. Then we show that unless the solution grows like $R^{3/2}$ at infinity then the blow-up must be linear. But since we have subtracted the linear solution this means that the blow-up must be zero.

Proof of Lemma 2.4

We start by proving that if $\mathbf{u }$ is a solution in $B_1^+$ such that

$$\begin{aligned} \Vert \mathbf{u }\Vert _{\tilde{L}^2(B_1^+)}=1 \end{aligned}$$

(147)

then there exist an $r_0$ such that

$$\begin{aligned} \Vert \mathbf{u }(r_j x)-\mathbf {Pr}(u,r_j)\Vert _{\tilde{L}^2(B_{r_j}^+)}\le C r^{1+\beta } \end{aligned}$$

for $r\le r_0$.

In particular we will show that if $\mathbf{u }^j$ is a sequence of solutions satisfying (147) and $0\in \Gamma$ then

$$\begin{aligned} \liminf _{j\rightarrow \infty ,r_j\rightarrow 0}\frac{|\ln ( \Vert \mathbf{u }^j(r_j x)-\mathbf {Pr}(u^j,r_j))|\Vert _{\tilde{L}^2(B_{r}^+)}}{|\ln (r_j)|}\ge 1+2\beta \end{aligned}$$

(148)

for some $\beta >0$. Once we have shown (148) the Lemma follows as Corollary 8.1.

We will assume the contrary that we have sequences $\mathbf{u }^j$ satisfying (147) and $r_j\rightarrow 0$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty }\frac{\ln \left( \left\| \mathbf{u }^j(r_j x)-\mathbf {Pr}(u^j,r_j)\right\| _{\tilde{L}^2(B_1^+)}\right) }{|\ln (r_j)|}=\alpha \le 1. \end{aligned}$$

The proof will progress in several steps. $\square$

Step 1

Arguing as we did in Lemma 11.2 we may find a sub-sequence of $\mathbf{u }^j$ and a sequence $r_j\rightarrow 0$ such that

$$\begin{aligned} \mathbf{v }^j(x)=\frac{u^j(r_jx)-\mathbf {Pr}(u^j,r_j)}{\left\| u^j(r_jx)-\mathbf {Pr}(u^j,r_j)\right\| _{\tilde{L}^2(B_1^+)}}\rightarrow \mathbf{u }^0 \end{aligned}$$

where

$$\begin{aligned} \sup _{B_R^+}\Vert \mathbf{u }^0\Vert _{\tilde{L}^2(B_R^+)}\le C(1+R)^{1+\epsilon } \end{aligned}$$

for some small $\epsilon$. In particular from Corollary 14.1 it follows that $\mathbf{u }^0$ is linear.

Step 2

Let $\mathbf{u }^j$ be as in Step 1 and $\mu$ some small constant. Then there exist a sequence $x^j\in B_{1/2}\cap \Pi$ and a sequence of real numbers $s_j$ such that

$$\begin{aligned} \inf _{\gamma \in {\mathbb {R}}}\frac{\left\| \mathbf{u }^j(s_jr_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j,r_js_j)-\gamma \mathbf{f }(x)\right\| _{\tilde{L}^2(B_1^+)}}{\left\| \mathbf{u }^j(s_jr_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j,r_js_j)\right\| _{\tilde{L}^2(B_1^+)}}=\mu . \end{aligned}$$

(149)

Proof of Step 2

Notice that since $0\in \Gamma$ we have $\Lambda \cap B_{1/2}\ne 0$ so we may find a small ball $B_{\delta }(x^j)$ such that $e_n\cdot \mathbf{u }^j(r_jx)>0$ in $B_{\delta }\cap \Pi$.

It is not hard to show that for some sequence $t_k\rightarrow 0$

$$\begin{aligned} \lim _{t_k\rightarrow 0}\frac{\mathbf{u }^j(t_kr_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j, t_kr_j, x^j)}{\left\| \mathbf{u }^j(t_kr_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j, t_kr_j, x^j)\right\| _{\tilde{L}^2(B_1^+)}}=\tilde{\mathbf{u }}^j \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} \Delta \tilde{\mathbf{u }}^j+\frac{\lambda +2}{2} \nabla \hbox {div}(\tilde{\mathbf{u }}^j)= 0 &{} \quad \hbox {in } B_1^+\\ \frac{\partial e_n\cdot \tilde{\mathbf{u }}^j}{\partial x_i}+\frac{\partial e_i\cdot \tilde{\mathbf{u }}^j}{\partial x_n}=0 &{} \quad \hbox {on } \Pi \quad \hbox {for } i=1,2,\dots ,n-1 \\ \frac{\partial e_n\cdot \tilde{\mathbf{u }}^j}{\partial x_n}+\frac{\lambda }{4}\hbox {div}(\tilde{\mathbf{u }}^j)=0 &{} \quad \hbox {on } \Pi \\ \mathbf {Pr}(\tilde{\mathbf{u }}^j,1,0)=0. &{} \end{array} \end{aligned}$$

Standard regularity theory implies that $\tilde{\mathbf{u }}^j$ can be written as a sum of polynomials and $\mathbf {Pr}(\tilde{\mathbf{u }}^j,1,0)=0$ implies that the zeroth and first order polynomial are identically zero.

It follows that,

$$\begin{aligned} \inf _{\gamma \in {\mathbb {R}}}\frac{\left\| \mathbf{u }^j(t_kr_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j,t_kr_j)-\gamma \mathbf{u }^0(x)\right\| _{\tilde{L}^2(B_1^+)}}{\left\| \mathbf{u }^j(t_kr_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j,t_kr_j)\right\| _{\tilde{L}^2(B_1^+)}}\rightarrow 1 \end{aligned}$$

as $k\rightarrow \infty$. But by Step 1 we also have that

$$\begin{aligned} \inf _{\gamma \in {\mathbb {R}}}\frac{\left\| \mathbf{u }^j(r_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j,r_j)-\gamma \mathbf{u }^0(x)\right\| _{\tilde{L}^2(B_1^+)}}{\left\| \mathbf{u }^j(r_jx+x^j)-\mathbf {Pr}(\mathbf{u }^j,r_j)\right\| _{\tilde{L}^2(B_1^+)}}\rightarrow 0 \end{aligned}$$

as $j\rightarrow \infty$. An argument of continuity shows that we may chose the $s_j$ as claimed in the step. $\square$

Step 3

Let

$$\begin{aligned} \mathbf{w }^j=\frac{\mathbf{u }^j(r_js_j x+x^j)-\mathbf {Pr}(\mathbf{u }^j,r_js_j,x^j)}{\Vert \mathbf{u }^j(r_js_j x+x^j)-\mathbf {Pr}(\mathbf{u }^j,r_js_j,x^j)\Vert _{\tilde{L}^2(B_1^+)}}, \end{aligned}$$

with $x^j$ and $s_j$ as in Step 2.

Then there exists a sub-sequence such that $\mathbf{w }^j\rightarrow \mathbf{w }^0$ strongly in $W^{1,2}$ and weakly in $W^{2,2}$. Moreover if we chose the sequences appropriately $\mathbf{w }^0$ will be a linear function.

Proof of Step 3

It it follows from strong convergence and Step 2 that

$$\begin{aligned} \inf _{\gamma \in {\mathbb {R}}}\frac{\left\| \mathbf{w }^0-\gamma \mathbf{u }^0(x)\right\| _{\tilde{L}^2(B_1^+)}}{\left\| \mathbf{w }^0\right\| _{\tilde{L}^2(B_1^+)}}=\mu \end{aligned}$$

(150)

and, if we chose the $s_j$ as the largest $s\in (0,1)$ such that (149) holds,

$$\begin{aligned} \inf _{\gamma \in {\mathbb {R}}}\frac{\left\| \mathbf{w }^0-\mathbf {Pr}(\mathbf{w }^0,R)-\gamma \mathbf{u }^0(x)\right\| _{\tilde{L}^2(B_R^+)}}{\left\| \mathbf{w }^0\right\| _{\tilde{L}^2(B_R^+)}}\le \mu , \end{aligned}$$

(151)

for each $R\ge 1$.

Also from the convergence it follows that

$$\begin{aligned} \mathbf {Pr}(\mathbf{w }^0,1)=0. \end{aligned}$$

Arguing as in Lemma 9.2 it follows from (151) that

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{\ln \left( \Vert \mathbf{w }^0\Vert _{\tilde{L}^2(B_R^+)}\right) }{\ln \left( R\right) } \le 1+\epsilon . \end{aligned}$$

(152)

Using (152) and Corollary 14.1 we may conclude that $\mathbf{w }^0$ is a linear solution to the Signorini problem, as Step 3 claims.

We have thus constructed a solution $\mathbf{w }^0$ such that $\mathbf {Pr}(\mathbf{w }^0,1)=0$ and such that (150) holds. It follows from $\mathbf {Pr}(\mathbf{w }^0,1)=0$ and linearity that $\mathbf{w }^0= \mathbf{f }$ for some $\mathbf{f }$ in the complement of the range of $\mathbf {Pr}$, but that contradicts (150). Our argument of contradiction is therefore complete and we have shown (148).

From (148) and similar argument as in Corollary 8.1 we may conclude that

$$\begin{aligned} \frac{1}{d(x^0)^{n+2+2\beta }}\int _{B_{d(x^0)}(x^0)\cap {\mathbb {R}}^n_+}\left| \mathbf{u }- \mathbf{u }(x^0)-(\nabla \mathbf{u })_{r,x^0}\cdot (x-x^0)\right| ^2\le C \end{aligned}$$

for each ball $B_{d(x^0)}(x^0)$ with $|x^0|<1/2$, $r\le 1/4$ and $d(x^0)=\hbox {dist}(x^0,\Gamma )$. It is now fairly standard, using the interior regularity for the Lame system, to show that this implies that $\mathbf{u }^0\in C^{1,\beta }$. $\square$

Appendix 2: Sketch of the proof of Lemma 6.2

In this Appendix we will briefly indicate how to prove the eigenfunction expansion in Lemma 6.2.

Lemma 6.2

Let $\mathbf{w }=(w^1,w^2,\dots ,w^{n})\in W^{1,2}(B_1^+)$ be a solution to the following linear problem

$$\begin{aligned} \begin{array}{ll} \Delta \mathbf{w }+\frac{2+\lambda }{2}\nabla \hbox {div}(\mathbf{w })=0 &{} \quad \hbox {in } B_1^+ \\ w^n(x_1,x_2,\dots ,0)=0 &{} \quad \hbox {on } \{x_1>0\} \cap \Pi \\ \frac{\partial w^i}{\partial x_n}+\frac{\partial w^n}{\partial x_i}=0 &{} \quad \hbox {on } \Pi \quad \hbox {for } i=1,2,\dots ,n-1 \\ \frac{\partial w^n}{\partial x_n}+\frac{\lambda }{4}\hbox {div}(\mathbf{w })=0 &{} \quad \hbox {on } \{x_1<0\}\cap \Pi \\ \Vert \mathbf{w }\Vert _{L^\infty (B_1^+)}\le 1 \end{array} \end{aligned}$$

(153)

then $\mathbf{w }=\sum _{i=1}^\infty a_i \mathbf{q }_i$ where $\mathbf{q }_i$ is a homogeneous solution of order i / 2 to the same problem.

Furthermore if $\mathbf{w }\in W^{2,2}(B_1^+)$ then

$$\begin{aligned} \mathbf{w }=a\mathbf{p }_{3/2}+\sum _{2=0}^\infty a_i \mathbf{q }_i. \end{aligned}$$

Proof

We will prove the Lemma in four steps. In the first three steps we will show that $w^n$ can be represented as a series of i / 2 homogeneous solutions. We will do that by showing that it is enough to show that a harmonic function with homogeneous Dirichlet and Neumann boundary data on $\{x_n=0,\; \pm x_1>0\}$ can be represented by i / 2 homogeneous functions. In Step 2 we will show that it is enough to show to prove this in ${\mathbb {R}}^2$. And in Step 3 we will show that it is indeed true n ${\mathbb {R}}^2$. In Step 4 we show that $w^i$, $i=1,2,\dots ,n-1$, can also be represented in such way. $\square$

Step 1

In order to show that $w^n$ is representable by a series of i / 2 homogeneous functions it is enough to show that a solution to

$$\begin{aligned} \begin{array}{ll} \Delta h=0 &{} \quad \mathrm{in }\,B_1^+\\ h=0 &{} \quad \mathrm{on } \,B_1\cap \{x_n=0, x_1>0\} \\ \frac{\partial h}{\partial x_n}=0 &{} \quad \mathrm{on }\, B_1\cap \{x_n=0, x_1<0\}, \end{array} \end{aligned}$$

(154)

may be represented by a series of 1 / 2-homogeneous harmonic functions.

Proof Step 1

Since $\mathbf{w }\in W^{1,2}(B_1^+)$ it follows that $g(x)=\hbox {div}(\mathbf{w })\in L^2(B_1^+)$ and $\Delta g=0$. We may thus, by Fubini’s theorem extend g so that $g\in L^2(B_1\cap \{x_n=0\})$ and $\lim _{x_n\rightarrow 0^+}g(x',x_n)=g(x',0)$ non-tangentially for almost every $x'$. In particular, we may approximate g(x) by a polynomial q(x) so that

$$\begin{aligned} \Vert g-q\Vert _{L^2(B_1^+)}+\Vert g-q\Vert _{L^2(B_1\cap \{x_n=0\})}\le \epsilon . \end{aligned}$$

We let $v^n$ be the solution to

$$\begin{aligned} \begin{array}{ll} \Delta v^n=-\frac{\lambda +2}{2}\frac{\partial q(x)}{\partial x_n} &{} \quad \hbox {in } B_1^+\\ v^n=u^n &{} \quad \hbox {on } \partial B_1\cap \{x_n>0\} \\ v^n=0 &{} \quad \hbox {on } B_1\cap \{x_n=0, x_1>0\} \\ \frac{\partial v^n}{\partial x_n}=-\frac{\lambda }{4}q &{} \quad \hbox {on } B_1\cap \{x_n=0, x_1<0\}. \end{array} \end{aligned}$$

Then

$$\begin{aligned}&\int _{B_1^+}|\nabla (v^n-w^n)|^2=-\int _{B_1^+}\Delta (v^n-w^n)(v^n-w^n)\\&\qquad + \int _{\partial B_1\cap \{x_n>0\}}(v^n-w^n)\frac{\partial (v^n-w^n)}{\partial \nu }\\&\qquad - \int _{\partial B_1\cap \{x_n=0, x_1>0\}}(v^n-w^n)\frac{\partial (v^n-w^n)}{\partial x_n}\\&\qquad -\int _{\partial B_1\cap \{x_n=0, x_1<0\}}(v^n-w^n)\frac{\partial (v^n-w^n)}{\partial x_n}\\&\quad =\frac{2+\lambda }{2}\int _{B_1^+}\frac{\partial (q-g)}{\partial x_n}(v^n-w^n)-\frac{\lambda }{4}\int _{\partial B_1\cap \{x_n=0, x_1<0\}}(v^n-w^n)(p-g)\\&\quad =-\frac{2+\lambda }{2}\int _{B_1^+}\frac{\partial (v^n-w^n)}{\partial x_n}(g-q)\\&\qquad -\frac{2+\lambda }{2}\int _{\partial B_1\cap \{x_n=0, x_1<0\}}(v^n-w^n)(p-g)\\&\qquad -\frac{\lambda }{4}\int _{\partial B_1\cap \{x_n=0, x_1<0\}}(v^n-w^n)(p-g)\\&\quad =-\frac{2+\lambda }{2}\int _{B_1^+}\frac{\partial (v^n-w^n)}{\partial x_n}(g-q)\\&\qquad -\frac{4+3\lambda }{4}\int _{\partial B_1\cap \{x_n=0, x_1<0\}}(v^n-w^n)(p-g)\\&\quad \le \frac{1}{C}\int _{B_1^+}|\nabla (v^n-w^n)|^2+ C\int _{B_1^+}(g-q)^2\\&\qquad +C\int _{B_1^+\cap \{x_n=0, x_1<0\}}(g-q)^2+\frac{1}{C}\int _{B_1^+\cap \{x_n=0, x_1<0\}}(v^n-w^n)^2\\&\quad \le \frac{1}{C}\frac{1}{C}\int _{B_1^+}|\nabla (v^n-w^n)|^2+C\epsilon ^2, \end{aligned}$$

where we used the trace Theorem in the last inequality. Rearranging terms implies that

$$\begin{aligned} \int _{B_1^+}|\nabla (v^n-w^n)|^2\le C\epsilon ^2. \end{aligned}$$

$\square$

We will show that $v^n$ can be expressed as a series of homogeneous functions.

Since q is a polynomial, say of degree N, it follows that $h=\frac{\partial ^{N+1} v^n}{\partial x_i^{N+1}}$ is harmonic for each $i=2,3,\dots ,n-1$. Thus, if any solution h to (154) can be represented a series of i / 2 homogeneous harmonic functions so can $\frac{\partial ^{N+1} v^n}{\partial x_i^{N+1}}$. Notice that taking the derivative means reducing the homogeneity by one. And since we may freely differentiate in the $i=2,3,\dots ,n-1$ directions we directly see that $v^n$ is can be written as a series of i / 2 homogeneous functions modulo a harmonic function depending only on $x_1$ and $x_n$.

Step 2

Any solution to (154) may be written as the sum of i / 2 homogeneous harmonic functions.

Proof of Step 2

It is well known (basically the spectral theorem on the sphere, see [17]) that the eigenfunctions of the Laplace equation on the upper half sphere with homogeneous Dirichlet/Neumann boundary conditions form a complete orthogonal basis for $L^2$ on the sphere. Moreover any eigenfunction may be extended homogeneously into the ball. We may thus write any solution of (154) as a infinite series of harmonic homogeneous functions. Therefore, we only need to prove that the homogeneous functions have homogeneity i / 2 for $i\in \mathbb {N}$.

To that end we differentiate an eigenfunction, say l(x) of homogeneity $\kappa$, in the $j=2,3,\dots ,n-1$ directions $k>\kappa$ times. Since the homogeneity decreases by one of each derivative we get $D^\alpha l(x)$ is homogeneous of order $\kappa -k<0$ for any multiindex $\alpha$ of length $|\alpha |=k$ and $\alpha _1=\alpha _n=0$. But a simple difference quotient shows that $D^\alpha l(x)$ is in $W^{1,2}$ and since the Wiener criteria is satisfied on $x_1=x_n=0$ and every other point with Dirichlet data is smooth it follows that $D^\alpha l(x)$ is bounded. Boundedness and $\kappa -k<0$ homogeneity implies that $D^{\alpha }l(x)=0$. By integrating $D^\alpha l$, using that the multiindex is arbitrary again, we may conclude that any eigenfunction l has homogeneity equal to the homogeneity of a harmonic solution depending only on $x_1$ and $x_n$ plus an integer. $\square$

Step 3

Any homogeneous solution to (154) that depend only on $x_1$ and $x_n$ is homogeneous of order i / 2 for $i\in \mathbb {N}$. We may thus conclude that $w^n$ can be arbitrarily well approximated by i / 2 homogeneous functions.

Proof of Step 3

It is easy to verify that the functions $r^{\frac{2i+1}{2}}\cos (\frac{2i+1}{2}\phi )$ in polar coordinates forms a basis. That $w^n$ may be approximated by i / 2 homogeneous functions follows from Steps 1 and 2. $\square$

Step 4

The functions $w^1,w^2,w^3,\dots ,w^{n-1}$ may be written as a sum of i / 2 homogeneous solutions.

Proof of Step 4

Fix a $j=1,2,\dots ,n-1$ and consider the system

$$\begin{aligned} \begin{array}{ll} \Delta v^n=-\frac{2+\lambda }{2}\frac{\partial q(x)}{\partial x_n} &{} \quad \hbox {in } B_1^+ \\ \Delta v^j=-\frac{2+\lambda }{2}\frac{\partial q(x)}{\partial x_j} &{} \quad \hbox {in } B_1^+ \\ v^n(x_1,x_2,0)=0 &{} \quad \hbox {on } \{x_1>0\} \cap \Pi \\ \frac{\partial v^j}{\partial x_n}+\frac{\partial v^n}{\partial x_j}=0 &{} \quad \hbox {on } \Pi \\ \frac{\partial w^n}{\partial x_n}=-\frac{\lambda }{4}q(x) &{} \quad \hbox {on } \{x_1<0\}\cap \Pi \\ v^j(x)=w^j(x) &{} \quad \hbox {on } \partial B_1\cap \{x_n>0\} \end{array} \end{aligned}$$

where q(x) is as in Step 1. We also let $v^n$ be given by a finite sum of i / 2 homogeneous solutions such that $\Vert v^n-w^n\Vert _{W^{1,2}}<\epsilon$—such an approximation exists due to Steps 1–3.

The argument to show that $v^j$, for $j=1,2,\dots ,n-1$, can be written as a series of i / 2-homogeneous solutions is now very similar to the case for $v^n$. In particular, for $\alpha$ a multiindex such that $\alpha _1=\alpha _n=0$ we see that $D^\alpha v^j$ is harmonic and $D^\alpha \frac{\partial v^j}{\partial x_n}=0$ on $\Pi$, since large enough derivative of $v^n$ vanishes on $\Pi$. In particular $D^\alpha v^j$ can be written as a series of harmonic polynomials. Integrating $|\alpha |$ times will yield the result. $\square$

Remark

We do not use that we know that $\mathbf{w }$ can be written as a series of i / 2 homogeneous solutions of the form $q(x) p_{k+\frac{1}{2}}$ where q(x) is a polynomial. But the proof actually shows that for any $\epsilon >0$, if we denote the projection of $\mathbf{w }$ into the linear span of all functions $q(x)p_{k+\frac{1}{2}}(x)$ that are homogeneous of order less than M by $\mathbf{w }_M$, then $\Vert \mathbf{w }-\mathbf{w }_M\Vert _{W^{1,2}(B_1^+)}< \epsilon$ if M is large enough.