Obstacle problems in fractal domains have been studied in [10] in the framework of reinforcement problems.
To face the numerical approach to the solutions of obstacle problems in fractal domains, it is natural to consider the solutions of obstacle problems in pre-fractal approximating domains and the corresponding FEM-solutions and to evaluate the approximation error.
We consider the pre-fractal Koch Islands ${\mathrm{\Omega}}_{\alpha}^{n}$ that are polygonal domains having as sides pre-fractal Koch curves. We start by a regular polygon and we replace each side by a pre-fractal Koch curve (see Figures 3 and 4); we refer to [12, Section 2] for the definition and details.

In [12, Section 3], we showed that, assuming some natural conditions, the solutions ${u}_{n}$ of the obstacle problem in ${\mathrm{\Omega}}_{\alpha}^{n}$ converge to the *fractal* solution of the obstacle problem in the Koch Island ${\mathrm{\Omega}}_{\alpha}$.

For any (fixed) *n*, the number of reentrant angles is fixed and hence we can prove, for the solution ${u}_{n}$ of the obstacle problem in ${\mathrm{\Omega}}_{\alpha}^{n}$, all the results of previous sections with $\chi =\frac{\omega}{\pi}$, where

$\omega =\{\begin{array}{cc}\pi +\theta (\alpha )\hfill & \text{if the sides of the polygons are obtained by outward curves,}\hfill \\ \pi +2\theta (\alpha )\hfill & \text{if the sides of the polygons are obtained by inward curves.}\hfill \end{array}$(5.1)

We recall that by $\theta (\alpha )$ we denote the opening of the rotation angle of the similarities involved in the construction of the Koch curve, that is,

$\theta (\alpha )=\mathrm{arcsin}\left(\frac{\sqrt{\alpha (4-\alpha )}}{2}\right).$

Then $\chi \in (1,\frac{3}{2})$ in the case of outward curves or $\chi \in (1,2)$ in the case of inward curves.

Figure 3 Pre-fractal Koch Islands ${\mathrm{\Omega}}_{\alpha}^{2}$ with $\alpha =2.1$, $\alpha =3$ and $\alpha =3.75$, respectively.

Figure 4 Pre-fractal Koch Islands ${\mathrm{\Omega}}_{\alpha}^{2}$ with $\alpha =2.1$, $\alpha =3$ and $\alpha =3.75$, respectively.

In this framework, the involved weighted Sobolev space is

${H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})=\{v\in {W}^{1,2}({\mathrm{\Omega}}_{\alpha}^{n}):{D}^{\beta}v\in {L}_{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})\text{for all}|\beta |=2\},\beta =({\beta}_{1},{\beta}_{2}),{\beta}_{1},{\beta}_{2},\in \mathbb{N},$

which is a Hilbert space with the norm

${\parallel v\parallel}_{{H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}={\left\{\sum _{|\beta |=2}{\parallel {D}^{\beta}v\parallel}_{{L}_{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}^{2}+{\parallel v\parallel}_{{W}^{1,2}({\mathrm{\Omega}}_{\alpha}^{n})}^{2}\right\}}^{1/2}.$

Here
${L}_{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})$ is the completion of the space $C({\overline{\mathrm{\Omega}}}_{\alpha}^{n})$ with respect to the norm

${\parallel v\parallel}_{{L}_{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}={\left\{{\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}{|v|}^{2}{\rho}^{2\mu}\mathit{d}x\right\}}^{1/2}$

and $\rho ={\rho}_{n}(x)$ denotes the distance function from the set of vertices of the *reentrant* corners of ${\mathrm{\Omega}}_{\alpha}^{n}$.
In this setting, we state the following theorems.

#### Theorem 5.1.

*We assume*

$\{\begin{array}{cccc}\hfill {\phi}_{i}& \in {W}^{1,p}({\mathrm{\Omega}}_{\alpha}^{n}),\hfill & & \hfill i=1,2,\\ \hfill {\phi}_{1}& \le {\phi}_{2}\mathit{\text{in}}{\mathrm{\Omega}}_{\alpha}^{n},\hfill & & \hfill {\phi}_{1}\le 0\le {\phi}_{2}\mathit{\text{in}}\partial {\mathrm{\Omega}}_{\alpha}^{n},\end{array}$(5.2)

*and*

$\{\begin{array}{cc}& k\ne 0\hfill \\ & f,{A}_{p}({\phi}_{i})\in {L}^{\mathrm{\infty}}({\mathrm{\Omega}}_{\alpha}^{n}),i=1,2,\hfill \\ & {A}_{p}({\phi}_{2})\wedge f\ge 0.\hfill \end{array}$(5.3)

*Then the solution ${u}_{n}$ of obstacle problem (2.1) in ${\mathrm{\Omega}}_{\alpha}^{n}$ belongs to the weighted Sobolev space*

${H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n}),\mu >1-\gamma ,$(5.4)

*where*

$\gamma =\gamma (p,\chi )=1+\frac{p{(1-\chi )}^{2}+(1-\chi )\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}}{2\chi (2-\chi )(p-1)}$(5.5)

*with $\chi \mathrm{=}\frac{\omega}{\pi}$ and ω in (5.1).*

*Moreover,*

${\parallel {u}_{n}\parallel}_{{H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}\le C\left\{1+{\parallel f\parallel}_{{L}^{\mathrm{\infty}}({\mathrm{\Omega}}_{\alpha}^{n})}+{\parallel {A}_{p}({\phi}_{1})\parallel}_{{L}^{\mathrm{\infty}}({\mathrm{\Omega}}_{\alpha}^{n})}+{\parallel {A}_{p}({\phi}_{2})\parallel}_{{L}^{\mathrm{\infty}}({\mathrm{\Omega}}_{\alpha}^{n})}\right\}.$(5.6)

If $k=0$, then an analog of Theorem 4.3 holds.

#### Theorem 5.2.

*We assume (5.2) and*

$\{\begin{array}{cc}& k=0\hfill \\ & f,{A}_{p}({\phi}_{i})\in {L}^{\mathrm{\infty}}({\mathrm{\Omega}}_{\alpha}^{n}),i=1,2,\hfill \\ & {A}_{p}({\phi}_{2})\wedge f\ge {c}^{*}>0.\hfill \end{array}$(5.7)

*If the solution ${u}_{n}$ of obstacle problem (2.1) in ${\mathrm{\Omega}}_{\alpha}^{n}$ belongs to the space ${H}^{\mathrm{2}\mathrm{,}\mu}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\alpha}^{n}\mathrm{)}$, then for any $q\mathrm{\ge}\mathrm{1}$ and $p\mathrm{>}\mathrm{2}$ we obtain*

${|\nabla {u}_{n}|}^{-\frac{(p-t)q}{t-q}}\in {L}^{1}({\mathrm{\Omega}}_{\alpha}^{n}),$(5.8)

*with*

$t\ge \frac{q(p+(p-2)2)}{q+(p-2)2}.$

We introduce the triangulation of the domain ${\mathrm{\Omega}}_{\alpha}^{n}$ in order to define the approximate solutions ${u}_{h}$ according to the Galerkin method.
Let ${T}_{h}$ be a partitioning of the domain ${\mathrm{\Omega}}_{\alpha}^{n}$ into disjoint, open regular triangles τ, each side being bounded by *h* so that ${\overline{\mathrm{\Omega}}}_{\alpha}^{n}={\bigcup}_{\tau \in {T}_{h}}\overline{\tau}$.
Associated with ${T}_{h}$, we consider the finite-dimensional spaces

${S}_{h}=\{v\in C({\overline{\mathrm{\Omega}}}_{\alpha}^{n}):{v|}_{\tau}\text{is affine for all}\tau \in {T}_{h}\}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{S}_{h,0}=\{v\in {S}_{h}:v=0\text{on}\partial {\mathrm{\Omega}}_{\alpha}^{n}\}.$

By ${\pi}_{h}$ we denote the interpolation operator ${\pi}_{h}:C({\overline{\mathrm{\Omega}}}_{\alpha}^{n})\to {S}_{h}$ such that ${\pi}_{h}v({P}_{i})=v({P}_{i})$ for any vertex ${P}_{i}$ of the partitioning ${T}_{h}$.

#### Definition 5.3.

The family of triangulations ${T}_{h}$
is *adapted* to the ${H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})$-regularity if the following conditions hold:

•

The vertices of the polygonal curves $\partial {\mathrm{\Omega}}_{\alpha}^{n}$ are nodes of
the triangulations.

•

The meshes are conformal and regular.

•

There exists ${\sigma}^{*}>0$ such that, as $h\to 0$,

${h}_{\tau}\le {\sigma}^{*}{h}^{\frac{1}{1-\mu}}$$\mathrm{\hspace{1em}}\text{for all}\tau \in {T}_{h}\text{such that one of the vertices of}\tau \text{belongs to}{\mathcal{\mathcal{R}}}^{n},$${h}_{\tau}\le {\sigma}^{*}h\cdot \underset{\tau}{inf}{\rho}^{\mu}$$\mathrm{\hspace{1em}}\text{for all}\tau \in {T}_{h}\text{with no vertix in}{\mathcal{\mathcal{R}}}^{n}.$

Here $h=sup\{{h}_{\tau}=\mathrm{diam}(\tau ):\tau \in {T}_{h}\}$ is the size of the
triangulation and $\rho ={\rho}_{n}(x)$ denotes the distance of the point
*x* from the set ${\mathcal{\mathcal{R}}}^{n}$ of the vertices of the *reentrant* corners of ${\mathrm{\Omega}}_{\alpha}^{n}$.

The construction of triangulations ${T}_{h}$
*adapted* to the ${H}^{2,\mu}$-regularity was introduced by Grisvard in [20]. This tool has been fruitfully used for the FEM approximation of linear problems in pre-fractal domains by [38, 39, 23, 1, 13, 14].

Consider the two obstacle problem in the finite-dimensional space ${S}_{h,0}$:

$\text{find}u\in {\mathcal{\mathcal{K}}}_{h}\text{such that}\mathit{\hspace{1em}}{a}_{p}(u,v-u)-{\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}f(v-u)\mathit{d}x\mathit{d}y\u2a7e0\mathit{\hspace{1em}}\text{for all}v\in {\mathcal{\mathcal{K}}}_{h},$(5.9)

where

${a}_{p}(u,v)={\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}{({k}^{2}+{|\nabla u|}^{2})}^{\frac{p-2}{2}}\nabla u\nabla vdxdy\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mathcal{\mathcal{K}}}_{h}=\{v\in {S}_{h,0}:{\phi}_{1,h}\u2a7dv\u2a7d{\phi}_{2,h}\text{in}{\mathrm{\Omega}}_{\alpha}^{n}\},$

with ${\phi}_{1,h}={\pi}_{h}{\phi}_{1}$ and ${\phi}_{2,h}={\pi}_{h}{\phi}_{2}$.

#### Proposition 5.4.

*Let us assume hypothesis (5.2). Then, for any $f\mathrm{\in}{L}^{{p}^{\mathrm{\prime}}}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\alpha}^{n}\mathrm{)}$, there exists a unique function ${u}_{h}$ that solves problem (5.9).
Moreover,*

${\parallel {u}_{h}\parallel}_{{W}^{1,p}({\mathrm{\Omega}}_{\alpha}^{n})}\le C\left\{|k|+{\parallel f\parallel}_{{L}^{{p}^{\prime}}({\mathrm{\Omega}}_{\alpha}^{n})}^{{p}^{\prime}/p}+{\parallel {\phi}_{1}\parallel}_{{W}^{1,p}({\mathrm{\Omega}}_{\alpha}^{n})}+{\parallel {\phi}_{2}\parallel}_{{W}^{1,p}({\mathrm{\Omega}}_{\alpha}^{n})}\right\}.$

As previously, the solution ${u}_{h}$ to problem (5.9) realizes the minimum on the convex ${\mathcal{\mathcal{K}}}_{h}$ of the functional ${J}_{p}(\cdot )$, i.e.,

${J}_{p}(u)=\underset{v\in {\mathcal{\mathcal{K}}}_{h}}{\mathrm{min}}{J}_{p}(v),\text{where}\mathit{\hspace{1em}}{J}_{p}(v)=\frac{1}{p}{\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}{({k}^{2}+{|\nabla v|}^{2})}^{\frac{p}{2}}\mathit{d}x\mathit{d}y-{\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}fv\mathit{d}x\mathit{d}y.$

#### Theorem 5.5.

*Let us denote by ${u}_{n}$ and ${u}_{h}$ the solutions of problems (2.1) in ${\mathrm{\Omega}}_{\alpha}^{n}$ and (5.9), respectively. Let us assume hypotheses (5.2), (5.3) and*

${\phi}_{i}\in {H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n}),i=1,2.$(5.10)

*Let ${T}_{h}$ be a triangulation of ${\mathrm{\Omega}}_{\alpha}^{n}$ adapted to the ${H}^{\mathrm{2}\mathrm{,}\mu}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\alpha}^{n}\mathrm{)}$-regularity of the solution ${u}_{n}$. Then*

${\parallel {u}_{n}-{u}_{h}\parallel}_{{W}^{1,t}({\mathrm{\Omega}}_{\alpha}^{n})}\le C{h}^{\frac{r}{t}}{\parallel {u}_{n}\parallel}_{{H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}$(5.11)

*for any*

$r\in [1,\frac{2\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}}{\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}+(\chi -1)(p-2)}),t\in [2,p].$

#### Proof.

For any $\sigma \in [0,p]$ we put

${|v|}_{(p,\sigma )}={\left({\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}{(|k|+|\nabla {u}_{n}|+|\nabla v|)}^{p-\sigma}{|\nabla v|}^{\sigma}\mathit{d}x\mathit{d}y\right)}^{\frac{1}{p}}.$(5.12)

Repeating the proof of [12, Lemma 5.2] (given for $k=0$), we prove for any ${v}_{h}\in {\mathcal{\mathcal{K}}}_{h}$ and

$v\in {\mathcal{\mathcal{K}}}_{n}:=\{v\in {W}_{0}^{1,p}({\mathrm{\Omega}}_{\alpha}^{n}):{\phi}_{1}\le v\le {\phi}_{2}\text{in}{\mathrm{\Omega}}_{\alpha}^{n}\}$

that

${|{u}_{n}-{u}_{h}|}_{(p,t)}^{p}\le C\left\{{|{u}_{n}-{v}_{h}|}_{(p,r)}^{p}+{\parallel f+{A}_{p}({u}_{n})\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}({\parallel {u}_{n}-{v}_{h}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}+{\parallel v-{u}_{h}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})})\right\},$(5.13)

where $r\in [1,2]$, $t\in [2,p]$ and the constant *C* does not depend on *h*.
Now we evaluate the terms on the right-hand side in estimate (5.13) by choosing the test functions ${v}_{h}\in {\mathcal{\mathcal{K}}}_{h}$ and $v\in {\mathcal{\mathcal{K}}}_{n}$ in an appropriate way.
According to Theorem 5.1, the function ${u}_{n}$ belongs to the weighted Sobolev space ${H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})$ for any $\mu >1-\gamma $ (see (5.4) and (5.5)).

We choose ${v}_{h}={\pi}_{h}{u}_{n}$, and by using approximation estimates of Grisvard (see [20, Section 8.4.1]), we derive

${\parallel {u}_{n}-{\pi}_{h}{u}_{n}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}\le C{h}^{2}{\parallel {u}_{n}\parallel}_{{H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}.$(5.14)

Then we choose $v={\phi}_{2}\wedge ({u}_{h}\vee {\phi}_{1})$ and, as in [12, Lemma 4.4], we have

${\parallel v-{u}_{h}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}^{2}\le {\parallel {\pi}_{h}{\phi}_{2}-{\phi}_{2}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}^{2}+{\parallel {\pi}_{h}{\phi}_{1}-{\phi}_{1}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}^{2}.$

Again using Grisvard estimates and assumption (5.10), we derive

${\parallel v-{u}_{h}\parallel}_{{L}^{2}({\mathrm{\Omega}}_{\alpha}^{n})}\le C{h}^{2}.$(5.15)

We compare the seminorm ${|{u}_{n}-{u}_{h}|}_{{W}^{1,t}({\mathrm{\Omega}}_{\alpha}^{n})}$ with ${|{u}_{n}-{u}_{h}|}_{(p,t)}^{p}$ (defined in (5.12)) and we obtain

${|{u}_{n}-{u}_{h}|}_{{W}^{1,t}({\mathrm{\Omega}}_{\alpha}^{n})}^{t}\le \frac{C}{{|k|}^{p-t}}{|{u}_{n}-{u}_{h}|}_{(p,t)}^{p}.$(5.16)

We now evaluate the term ${|{u}_{n}-{v}_{h}|}_{(p,r)}^{p}$, where ${v}_{h}={\pi}_{h}{u}_{n}$. By the embedding of weighted Sobolev spaces in the fractional Sobolev spaces (see, for instance, [33]), ${u}_{n}$ belongs to the space ${W}^{{\sigma}_{2},2}({\mathrm{\Omega}}_{\alpha}^{n})$ for any ${\sigma}_{2}<1+\gamma $. Taking into account the Sobolev embedding (see, for instance, [7]), we have

$|\nabla {u}_{n}|\in {L}^{{r}^{*}}({\mathrm{\Omega}}_{\alpha}^{n})\mathit{\hspace{1em}}\text{with}{r}^{*}=\frac{2}{2-{\sigma}_{2}}.$(5.17)

By the Hölder inequality, we obtain

${|{u}_{n}-{v}_{h}|}_{(p,r)}^{p}\le C(r){|{u}_{n}-{\pi}_{h}{u}_{n}|}_{{W}^{1,2}({\mathrm{\Omega}}_{\alpha}^{n})}^{r},$(5.18)

where we have used estimate (5.6) with $r=\frac{2({r}^{*}-p)}{{r}^{*}-2}$. Hence, as ${\sigma}_{2}<1+\gamma $, ${r}^{*}$ is given in (5.17) and γ in (5.5), we have to choose $r<p+\frac{2-p}{\gamma}$ and we obtain by calculations that

$r<\frac{2\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}}{\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}+(\chi -1)(p-2)}.$

Now we use [20, Theorem 8.4.1.6] and we obtain

${|{u}_{n}-{\pi}_{h}{u}_{n}|}_{{W}^{1,2}({\mathrm{\Omega}}_{\alpha}^{n})}\le Ch.$(5.19)

By taking into account estimates (5.13)–(5.16), (5.18) and (5.19), we conclude the proof using once again the Poincaré inequality.
∎

We note that in Theorem 5.5 we assume $k\ne 0$; if $k=0$ the following result holds.

#### Theorem 5.6.

*Let us denote by ${u}_{n}$ and ${u}_{h}$ the solutions of problems (2.1) in ${\mathrm{\Omega}}_{\alpha}^{n}$ and (5.9), respectively. Let us assume hypotheses (5.2), (5.7), (5.10) and that the solution ${u}_{n}$ belongs to the space ${H}^{\mathrm{2}\mathrm{,}\mu}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\alpha}^{n}\mathrm{)}$.
Let ${T}_{h}$ be a triangulation of ${\mathrm{\Omega}}_{\alpha}^{n}$ adapted to the ${H}^{\mathrm{2}\mathrm{,}\mu}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\alpha}^{n}\mathrm{)}$-regularity of the solution ${u}_{n}$. Then*

${\parallel {u}_{n}-{u}_{h}\parallel}_{{W}^{1,q}({\mathrm{\Omega}}_{\alpha}^{n})}\le C{h}^{\frac{r}{t}}{\parallel {u}_{n}\parallel}_{{H}^{2,\mu}({\mathrm{\Omega}}_{\alpha}^{n})}$

*for any*

$r\in [1,\frac{2\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}}{\sqrt{{p}^{2}-\chi (2-\chi ){(p-2)}^{2}}+(\chi -1)(p-2)})$

$t\in [2,p]$
*, $q\mathrm{\in}\mathrm{[}\mathrm{1}\mathrm{,}t\mathrm{]}$, and for $q\mathrm{<}p$ we require $t\mathrm{\ge}\frac{q\mathit{}\mathrm{(}p\mathrm{+}\mathrm{(}p\mathrm{-}\mathrm{2}\mathrm{)}\mathit{}\mathrm{2}\mathrm{)}}{q\mathrm{+}\mathrm{(}p\mathrm{-}\mathrm{2}\mathrm{)}\mathit{}\mathrm{2}}$.*

#### Proof.

We proceed as in the proof of Theorem 5.5: we replace estimate (5.16) by

${|{u}_{n}-{u}_{h}|}_{{W}^{1,q}({\mathrm{\Omega}}_{\alpha}^{n})}^{t}\le {\parallel {|\nabla {u}_{n}|}^{-\frac{(p-t)q}{t-q}}\parallel}_{{L}^{1}({\mathrm{\Omega}}_{\alpha}^{n})}^{(t-q)/q}\cdot {\int}_{{\mathrm{\Omega}}_{\alpha}^{n}}{|\nabla ({u}_{n}-{u}_{h})|}^{t}{|\nabla {u}_{n}|}^{p-t}\mathit{d}x\mathit{d}y\le C{|{u}_{n}-{u}_{h}|}_{(p,t)}^{p}.$

Here we have used the Hölder inequality and estimate (5.8). ∎

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