Existence of strong solutions to the Dirichlet problem for the Griffith energy

Abstract

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998). In a recent paper (Chambolle and Crismale in J Eur Math Soc (JEMS), 2018) we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti et al. (Ann Inst H Poincaré Anal Non Linéaire 36:455–474, 2019) and Chambolle et al. (J Math Pures Appl, 2019. https://doi.org/10.1016/j.matpur.2019.02.001).

Appendix

In this appendix we deal with a regularity result for solutions of elliptic equations with Dirichlet boundary conditions. By [39, Theorem 4.18, (i)], the estimate (A.1) below is formally obtained (with the notations in [39], in particular $\gamma$ represents there the trace operator) by taking $G_1=B_{3 R_0/4}$, $G_2=B_{R_0}$, ${\mathcal {P}}u=\mathrm {div\,}{{\mathbb {C}}}e(u)$, $f=0$, $\gamma u=0$. However, since the dependence of $C'_{0,m}$ from the other relevant known constant is not clearly specified in [39], and it is very important for Theorem 2.6 and its consequences, we give an outline of the proof. We refer to the notation of Sect. 2, in particular recall (2.5) and (2.6).

Theorem A.1

Let $\gamma \in [0, 1/2]$, $u\in H^1(B_1;{{{\mathbb {R}}}}^n)$ be a local minimiser of $E_{0,\gamma }(\cdot , B_1)$, and $R_0<1$ be such that $\frac{3}{4}R_0 > \gamma$. Then for every $m\in {{\mathbb {N}}}$ and $\varrho \le R_0$ there exists $C_{0,m}'$ depending on ${{\mathbb {C}}}$, m, and $R_0$, such that

$$\begin{aligned} \Vert u\Vert _{H^m(B_{3R_0/4};{{{\mathbb {R}}}}^n)}\le C'_{0,m} \Vert e(u)\Vert _{L^2(B_{R_0};{{\mathbb {M}}^{n\times n}_{sym}})}. \end{aligned}$$

(A.1)

Proof

First let us prove that for any $\varrho < R_0$ it holds

$$\begin{aligned} \Vert u\Vert _{H^2(B_\varrho )} \le C_{{{\mathbb {C}}}} (R_0- \varrho )^{-2} \Vert e(u)\Vert _{L^2(B_{R_0})}. \end{aligned}$$

(A.2)

Let us fix $\varrho < R_0$ and take a cut-off function $\psi$ between $B_{\varrho }$ and $B_{R_0}$, that is $\psi {:}\,B_1 \rightarrow [0,1]$, $\psi \in C^\infty _c(B_{R_0})$, $\psi = 1$ in $B_{\varrho }$, such that

$$\begin{aligned} \Vert \nabla \psi \Vert ^2_{L^\infty (B_1)} + \Vert \mathrm {D}^2 \psi \Vert _{L^\infty (B_1)} \le C (R_0-\varrho )^{-2}, \end{aligned}$$

(A.3)

for a universal constant C. For any $w {:}\,B_1 \rightarrow {{\mathbb {R}}}^s$, $s \ge 1$, and $x \in B_{1-h}$ we denote

$$\begin{aligned} \nabla _{l,h} w (x) := \frac{w(x+h e_l)- w(x)}{h} \end{aligned}$$

the difference quotient in the direction $e_l$, where $e_l$ is the l-th element of the canonical basis of ${{{\mathbb {R}}}}^n$. Since $\psi u$ has compact support in $B_{R_0}$ and ${{\mathbb {C}}}$ has constant coefficients (with respect to x) we have that for h small

$$\begin{aligned} \int \limits _{B_{R_0}} {{\mathbb {C}}}\, \nabla _{l,h}(e(\psi u)) {:}\,e(v) \, \mathrm {d} x= - \int \limits _{B_{R_0}} {{\mathbb {C}}}e(\psi u) {:}\,\nabla _{l,-h} (e(v)) \, \mathrm {d} x\end{aligned}$$

for any $v\in H^1(B_{R_0})$. Then for h small and $v \in H^1_0(B_{R_0}; {{{\mathbb {R}}}}^n)$, $v=0$ in $B_{R_0}{\setminus }H_\gamma$

$$\begin{aligned}&\bigg | \int \limits _{B_{R_0}} {{\mathbb {C}}}\, \nabla _{l,h}(e(\psi u)) {:}\,e(v) \, \mathrm {d} x\bigg | = \bigg | \int \limits _{B_{R_0}} {{\mathbb {C}}}e(\psi u) {:}\,\nabla _{l,-h} (e(v)) \, \mathrm {d} x\bigg | \nonumber \\&\quad \le \bigg | \int \limits _{B_{R_0}} {{\mathbb {C}}}(\nabla \psi \odot u) {:}\,e(\nabla _{l,-h} v) \, \mathrm {d} x\bigg | = \bigg | \int \limits _{B_{R_0}} {{\mathbb {C}}}\, \nabla _{l,h}(\nabla \psi \odot u) {:}\,e(v) \, \mathrm {d} x\bigg | \nonumber \\&\quad \le C_{{{\mathbb {C}}}} \Big ( \Vert \mathrm {D}^2 \psi \Vert _{L^{\infty }(B_1)} \Vert u\Vert _{L^2(B_{R_0})} + \Vert \nabla \psi \Vert _{L^\infty (B_1)} \Vert e(u)\Vert _{L^2(B_{R_0})} \Big ) \Vert e(v)\Vert _{L^2(B_{R_0})} \nonumber \\&\quad \le C_{{{\mathbb {C}}}} (R_0- \varrho )^{-2} \Vert e(u)\Vert _{L^2(B_{R_0})} \Vert e(v)\Vert _{L^2(B_{R_0})}, \end{aligned}$$

(A.4)

where in the first inequality we have used that $e(\psi u) = \psi \, e(u) + \nabla \psi \odot u$ and the Euler equation for minimisers of (2.6)

$$\begin{aligned} \int \limits _{B_{R_0}} {{\mathbb {C}}}e(u) {:}\,e(v) \, \mathrm {d} x= 0 \quad \text {for any }v \in H^1_0(B_{R_0}; {{{\mathbb {R}}}}^n),\, v=0\text { in }B_{R_0}{\setminus }H_\gamma , \end{aligned}$$

and the last inequality follows from (A.3) plus Poincaré’s and Korn’s inequality for $H^1_0$ functions [cf. also below in (A.5)]. We now take, for $l=1, \dots , n-1$, $v:= \nabla _{l,h}(\psi u)$ as test function; indeed, by the form of $H_\gamma$ we have that

$$\begin{aligned} \nabla _{l,h}(\psi u) \in H^1_0(B_{R_0}; {{{\mathbb {R}}}}^n),\quad \nabla _{l,h}(\psi u) =0\text { in }B_{R_0}{\setminus }H_\gamma . \end{aligned}$$

With this choice, we get

$$\begin{aligned} \bigg | \int \limits _{B_{R_0}} {{\mathbb {C}}}\, \nabla _{l,h}(e(\psi u)) {:}\,e(v) \, \mathrm {d} x\bigg | \ge C_{{{\mathbb {C}}}} \Vert e(\nabla _{l,h}(\psi u))\Vert ^2_{L^2(B_{R_0})} \ge C_{{{\mathbb {C}}}} \Vert \nabla _{l,h} u \Vert ^2_{H^1(B_\varrho )}, \end{aligned}$$

(A.5)

since

$$\begin{aligned} \Vert \nabla _{l,h} u \Vert ^2_{H^1(B_\varrho )}=\Vert \nabla _{l,h}(\psi u) \Vert ^2_{H^1(B_{\varrho })} \le C \Vert e(\nabla _{l,h}(\psi u))\Vert ^2_{L^2(B_{\varrho })} \end{aligned}$$

for a universal constant C: this holds by the combination of Korn’s inequality in $H^1_0(B_{\varrho })$

$$\begin{aligned} \Vert \nabla \big (\nabla _{l,h}(\psi u) \big ) \Vert ^2_{L^2(B_\varrho )} \le 2 \Vert e(\nabla _{l,h}(\psi u))\Vert ^2_{L^2(B_\varrho )}, \end{aligned}$$

and Poincaré’s inequality in $H^1_0(B_{\varrho })$

$$\begin{aligned} \Vert \nabla _{l,h}(\psi u) \Vert ^2_{H^1(B_\varrho )} \le \big (1 + 9 R_0^2 / 16 \big ) \Vert \nabla \big (\nabla _{l,h}(\psi u) \big ) \Vert ^2_{L^2(B_\varrho )}, \end{aligned}$$

being $\varrho \le \frac{3}{4}R_0$.

As usual, to prove regularity of solutions to elliptic equations, the derivative $\partial _{n\,n} u$ is estimated by looking at the equation in weak form $\mathrm {div\,} {{\mathbb {C}}}e(u)=0$, that gives

$$\begin{aligned} \Vert \partial _{n\,n} u\Vert _{L^2(B_\varrho )} \le C_{{{\mathbb {C}}}} \left( \Vert u\Vert _{H^1(B_\varrho )} + \sum _{l=1}^{n-1} \Vert \partial _l u\Vert _{H^1(B_\varrho )} \right) . \end{aligned}$$

Combining the estimate above with (A.4) and (A.5) (and standard properties of difference quotients), we obtain (A.2). Arguing in a similar way it is possible to show that if $\mathrm {div}\, {{\mathbb {C}}}e(w)=f$ in $B_{R_0} \cap H_\gamma$ and $w= 0$ in $B_{R_0} {\setminus }H_\gamma$, then

$$\begin{aligned} \Vert w\Vert _{H^2(B_\varrho )} \le C_{{{\mathbb {C}}}} (R_0- \varrho )^{-2} \Vert e(w)\Vert _{L^2(B_{R_0})} + C_{{{\mathbb {C}}}} \Vert f\Vert _{L^2(B_{R_0})}. \end{aligned}$$

(A.6)

Now it is proven by induction that

$$\begin{aligned} \Vert u\Vert _{H^{m+1}(B_\varrho )} \le C_{{{\mathbb {C}}}} (R_0- \varrho )^{-2m} \Vert e(u)\Vert _{L^2(B_{R_0})}, \end{aligned}$$

(A.7)

the case $m=1$ being (A.2). For $l=1, \dots , n-1$ we have that $\mathrm {div}\, {{\mathbb {C}}}e(\partial _l u)$ is expressed in terms of derivatives of u of order at most $m+1$ (cf. [39, Lemma 4.13]) and $\partial _l u= 0$ in $B_{R_0} {\setminus }H_\gamma$, so that (A.6) and the induction assumption for m give

$$\begin{aligned} \Vert \partial _l u\Vert _{H^{m+1}(B_\varrho )} \le C_{{{\mathbb {C}}}} (R_0- \varrho )^{-2(m+1)} \Vert e(u)\Vert _{L^2(B_{R_0})}. \end{aligned}$$

(A.8)

It lasts to estimate $\partial ^{m+2}_n u$, that is the derivative of u taken $m+2$ times with respect to $e_n$. In order to do so, it is enough to apply $\partial ^m_n$ to the explicit expression of $\partial _{n\,n}$ in terms of the other second order derivatives obtained from $\mathrm {div\,} {{\mathbb {C}}}e(u)=0$: then $\partial ^{m+2}_n$ is a linear combination (trough combination of coefficients of ${{\mathbb {C}}}$) of the other derivatives of order $m+2$, already estimated in (A.8). We conclude (A.1) by taking $\varrho = \frac{3}{4} R_0$ in (A.7). $\square$

Remark A.2

From Theorem A.1, employing the Sobolev embedding $H^m \hookrightarrow C^1$ for any $m> 2 + n/2$ and recalling (1.1), we obtain Theorem 2.6.