Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

Abstract

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in a large natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in $\mathbb {R}^2$ or $\mathbb {R}^3$ we also validate the approximation of the solution of the Westervelt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.

Appendices

Proof of Theorem 8

As in Ref. [17] let us define for every $m\in \mathbb {N}^*$, $t\ge 1$ the function

$$\begin{aligned} G_{t,m}(\xi ):=\left\{ \begin{array}{ll} 0 &{} \hbox {if }\xi \le 0,\\ \xi ^t &{} \hbox {if }\xi \in ]0,m[,\\ m^{t-1}u &{} \hbox {if }\xi \ge m, \end{array}\right. \end{aligned}$$

(77)

which by its definition is piece wise smooth and has a bounded derivative. This implies that $G_{t,m}(u)\in V(\varOmega )$ for $u\in V(\varOmega )$ by Theorem 7.8 of Ref. [29]. For some fixed $m\ge 1$ and $q\ge 2$ we introduce the following notations:

$$\begin{aligned} v:=G_{q-1,m}(u), \quad w:=G_{\frac{q}{2},m}(u). \end{aligned}$$

Using again Theorem 7.8 in Ref. [29] we obtain that

$$\begin{aligned} \partial _{x_i}w\partial _{x_j}w=\left\{ \begin{array}{ll} \frac{q^2}{4(q-1)}\partial _{x_i}u\partial _{x_j}v, &{}\hbox { if } u(x)\le m\\ \partial _{x_i}u\partial _{x_j}v, &{} \hbox { if }u(x)\ge m. \end{array} \right. \end{aligned}$$

Consequently we find

$$\begin{aligned} \Vert \nabla w\Vert _{L^2(\varOmega )}^2\le&\frac{q^2}{4(q-1)} (\nabla u,\nabla v)_{L^2(\varOmega )}\\ \le&q [(\nabla u,\nabla v)_{L^2(\varOmega )}+a\int _{\varGamma _R}\mathrm {Tr}_{\varGamma _R}u\;\mathrm {Tr}_{\varGamma _R}v dm_d]\\ \le&q (f,v)_{L^2(\varOmega )}\\ \le&q \Vert f\Vert _{L^2(\varOmega )} \Vert v\Vert _{L^2(\varOmega )}. \end{aligned}$$

Using estimate (16) we obtain

$$\begin{aligned} \Vert w\Vert _{L^6(\varOmega )}^2\le C \Vert \nabla w\Vert _{L^2(\varOmega )}^2 \le C q \Vert f\Vert _{L^2(\varOmega )} \Vert v\Vert _{L^2(\varOmega )}, \end{aligned}$$

where $C>0$ depends only on $\varOmega$ in the same way as in Proposition 2. Then we use the fact that $0\le v\le w^{\frac{2(q-1)}{q}}$ to deduce

$$\begin{aligned} \Vert w^{\frac{2}{q}}\Vert _{L^ {3q}(\varOmega )}^q\le C q \Vert f\Vert _{L^2(\varOmega )} \Vert w^{\frac{2}{q}}\Vert _{L^{2(q-1)}(\varOmega )}^{q-1}. \end{aligned}$$

(78)

Let us denote by $u^+$ and $u^-$ the positive and negative parts of u, $u^{\pm }:=\max (0,\pm u)$. The sequence of functions $w^{\frac{2}{q}}= [G_{\frac{q}{2},m}(u)]^{\frac{2}{q}}$ is increasing as m increases and converges to $u^+$ as m goes to infinity. Thus, if we take $\overline{u}=\frac{u^+}{M}$ with $M=C \Vert f\Vert _{L^2(\varOmega )}$, from (78) with the help of the monotone convergence theorem we have

$$\begin{aligned} \Vert \overline{u}\Vert _{L^ {3q}(\varOmega )}^q\le q \Vert \overline{u}\Vert _{L^{2(q-1)}(\varOmega )}^{q-1}. \end{aligned}$$

(79)

We take $q_0=2$ and $q_{n+1}=1+ \eta q_n$ with $\eta =\frac{3}{2}$ for all $n\in \mathbb {N}$, what allows us thanks to estimate (79) to find

$$\begin{aligned} \Vert \overline{u}\Vert _{L^ {3q_{n+1}}(\varOmega )}^{q_{n+1}}\le q_{n+1} \Vert \overline{u}\Vert _{L^{3q_n}(\varOmega )}^{\eta q_n}. \end{aligned}$$

From the last estimate we obtain by induction that

$$\begin{aligned} \Vert \overline{u}\Vert _{L^ {3q_{n+1}}(\varOmega )}\le \left( \prod _{k=1}^{n+1} q_k^{\frac{\eta ^{n+1-k}}{q_{n+1}}} \right) \Vert \overline{u}\Vert _{L^6(\varOmega )}^{2\frac{\eta ^{n+1}}{q_{n+1}}}. \end{aligned}$$

As $\eta =\frac{3}{2} >1$ we see that $\eta \le \frac{q_{n+1}}{q_n} \le 2\eta$, which by induction implies that $q_{n+1}=4\eta ^{n+1}-2$. Consequently,

$$\begin{aligned} \Vert \overline{u}\Vert _{L^ {3q_{n+1}}(\varOmega )}\le 2^{\sum _{k=1}^{n+1}\eta ^{-k}} (2\eta )^{\frac{1}{2}\sum _{k=1}^{n+1}k\eta ^{-k}}\Vert \overline{u}\Vert _{L^6(\varOmega )}^{2\frac{\eta ^{n+1}}{4\eta ^{n+1}-2}}. \end{aligned}$$

Since $\eta >1$ we can pass to the limit for $n\rightarrow +\infty$:

$$\begin{aligned} \Vert \overline{u}\Vert _{L^{\infty }(\varOmega )}\le K \Vert \overline{u}\Vert _{L^6}^{\frac{1}{2}}, \end{aligned}$$

where

$$\begin{aligned} K= 2^{\sum _{k=1}^{+\infty }\eta ^{-k}} (2\eta )^{\frac{1}{2}\sum _{k=1}^{+\infty }k\eta ^{-k}}<+\infty . \end{aligned}$$

Taking into account that

$$\begin{aligned} \Vert \overline{u}\Vert _{L^{\infty }(\varOmega )}\le K \vert \varOmega \vert ^{\frac{1}{12}}\Vert \overline{u}\Vert _{L^{\infty }(\varOmega )}^{\frac{1}{2}}, \end{aligned}$$

we conclude in

$$\begin{aligned} \Vert \overline{u}\Vert _{L^{\infty }(\varOmega )}\le K^2 \vert \varOmega \vert ^{\frac{1}{6}}. \end{aligned}$$

Finally, by definition of $\overline{u}$ we obtain

$$\begin{aligned} \Vert u^+\Vert _{L^{\infty }(\varOmega )}\le C \Vert f\Vert _{L^2(\varOmega )}, \end{aligned}$$

where $C>0$ depends only on $\varOmega$ in the same way as in Proposition 2. As $u^-=(-u)^+$, and by linearity $-u$ is the solution of the Poisson problem (3) with f replaced by $-f$, then we also have

$$\begin{aligned} \Vert u^-\Vert _{L^{\infty }(\varOmega )}\le C \Vert f\Vert _{L^2(\varOmega )}, \end{aligned}$$

which finishes the proof.

Scale irregular Koch curves and the Strong Open Set Condition

Koch mixtures [13] can give a typical example of a self-similar fractal boundary in $\mathbb {R}^2$.

We recall briefly some notations introduced in Section 2 page 1223 of Ref. [13] for scale irregular Koch curves built on two families of contractive similitudes. Let $\mathcal {B}=\lbrace 1,2\rbrace$: for $a\in \mathcal {B}$ let $2<l_a<4$, and for each $a\in \mathcal {B}$ let

$$\begin{aligned} \varPsi ^{(a)}=\lbrace \psi _1^{(a)},\ldots , \psi _4^{(a)}\rbrace \end{aligned}$$

be the family of contractive similitudes $\psi _i^{(a)}:\mathbb {C}\rightarrow \mathbb {C}$, $i=1, \ldots , 4$, with contraction factor $l_a^{-1}$ defined in Ref. [14].

Let $\varXi =\mathcal {B}^{\mathbb {N}}$; we call $\xi \in \varXi$ an environnent. We define the usual left shift S on $\varXi$. For $\mathcal {O}\subset \mathbb {R}^2$, we set

$$\begin{aligned} \varPhi ^{(a)}(\mathcal {O})=\bigcup _{i=1}^4 \psi _i^{(a)}(\mathcal {O}) \end{aligned}$$

and

$$\begin{aligned} \varPhi ^{(\xi )}_m(\mathcal {O})=\varPhi ^{(\xi _1)}\circ \cdots \circ \varPhi ^{(\xi _m)}(\mathcal {O}). \end{aligned}$$

Let K be the line segment of unit length with $A=(0,0)$ and $B=(1,0)$ as end points. We set, for each m in $\mathbb {N}$,

$$\begin{aligned} K^{(\xi ),m}=\varPhi ^{(\xi )}_m(K). \end{aligned}$$

$K^{(\xi ),m}$ is the so-called m-th prefractal curve. The fractal $K^{(\xi )}$ associated with the environment sequence $\xi$ is defined by

$$\begin{aligned} K^{(\xi )}=\overline{\bigcup _{m=1}^{+\infty }\varPhi ^{(\xi )}_m(\varGamma )}, \end{aligned}$$

where $\varGamma =\lbrace A,B\rbrace$. For $\xi \in \varXi$, we set $i\vert m=(i_1,\ldots ,i_m)$ and $\psi _{i\vert m}=\psi _{i_1}^{(\xi _1)}\circ \cdots \circ \psi _{i_m}^{(\xi _m)}$. We define the volume measure $\mu ^{(\xi )}$ as the unique Radon measure on $K^{(\xi )}$ such that

$$\begin{aligned} \mu ^{(\xi )}(\psi _{i\vert m}(K^{(S^m\xi )}))=\frac{1}{4^m} \end{aligned}$$

(see Section 2 in Ref. [7]) as, for each $a\in \mathcal {B}$, the family $\varPhi ^{(a)}$ has 4 contractive similitudes.

The fractal set $K^{(\xi )}$ and the volume measure $\mu ^{(\xi )}$ depend on the oscillations in the environment sequence $\xi$. We denote by $h_a^{(\xi )}(m)$ the frequency of the occurrence of a in the finite sequence $\xi \vert m$, $m\ge 1$:

$$\begin{aligned} h_a^{(\xi )}(m)=\frac{1}{m}\sum _{i=1}^m 1_{\lbrace \xi _i=a\rbrace }, \hbox { }a=1,2. \end{aligned}$$

Let $p_a$ be a probability distribution on $\mathcal {B}$, and suppose that $\xi$ satisfies

$$\begin{aligned} h_a^{(\xi )}(m)\underset{m\rightarrow +\infty }{\longrightarrow } p_a , \end{aligned}$$

(where $0\le p_a\le 1,$ $p_1+p_2=1$) and

$$\begin{aligned} \vert h_a^{(\xi )}(m)-p_a\vert \le \frac{C_0}{m}, \hbox { }a=1,2,\hbox { }(n\ge 1), \end{aligned}$$

with some constant $C_0\ge 1$, that is, we consider the case of the fastest convergence of the occurrence factors.

Under these conditions, the measure $\mu ^{(\xi )}$ has the property that there exist two positive constants $C_1$, $C_2$, such that (see Refs. [59, 60]),

$$\begin{aligned} C_1 r^{d^{(\xi )}}\le \mu ^{(\xi )}(K^{(\xi )}\cap B_r(x))\le C_2 r^{d^{(\xi )}} \;\;\;\hbox {for all } x\in K^{(\xi )},0<r\le 1, \end{aligned}$$

where $B_r(x)\subset \mathbb {R}^2$ denotes the Euclidean ball of radius r and centered at x with

$$\begin{aligned} d^{(\xi )}=\frac{\ln 4}{p_1 \ln p_1+p_2 \ln p_2} . \end{aligned}$$

According to Definition 3, it means that $K^{(\xi )}$ is a $d^{(\xi )}$-set and the measure $\mu ^{(\xi )}$ is a $d^{(\xi )}-$ dimensional measure equivalent to the $d^{(\xi )}$-dimensional Hausdorff measure $m_{d^{(\xi )}}$.

Fig. 2

An illustration for the Open Set Condition in the case of the square Koch curve, also called the Minkowski fractal. The thick dotted line outlines the set $\mathcal {O}$, which is called the 0-cell. The thin dotted lines outlines the open sets in $\varPhi _1(\mathcal {O})$, which are called 1-cells. The bottom picture illustrates the stronger form of the Open Set Condition used in Conjecture 1: the thin solid lines outline the open sets $\mathcal {O}'$ and $\varPhi _1(\mathcal {O}')$

We now discuss the more general set-up in Sect. 6.1. The standard Open Set Condition [26, Section 9.2] is satisfied for an iterated function system if there is a non-empty bounded open set O such that $\varPhi _m(O)\subset O$ with the union in the left-hand side disjoint. Note that the open set O may not be unique. Conjecture 1 assumes The Fractal Self-Similar Face Condition (Assumption 1) and a strong version of the Open Set Condition, see Fig. 2, that we introduce as follows.

Assumption 3

(A Strong Open Set Condition) We assume the Open Set Condition for the sequence $\varPhi _m$ is satisfied with two different convex open polygons $\mathcal {O}\subsetneqq \mathcal {O}'$, not depending on m, such that

$$\begin{aligned} \partial {\mathcal {O}}\cap K_0=\partial {\mathcal {O}}'\cap K_0=\partial {\mathcal {O}}\cap \partial {\mathcal {O}}'=\partial _{(n-2)}K_0. \end{aligned}$$

Conjecture 1

: If Assumptions 1 and 3 are satisfied, then $\varOmega _m$ and $\varOmega$ are uniformly exterior and interior $(\epsilon ,\infty )$-domains, that is, $\epsilon$ does not depend on m.

One possible approach to this conjecture, following Definition 2 of an $(\varepsilon ,\delta )$-domain from [38] and [36, Remark 1], condition (ii) is equivalent to saying that the $\frac{1}{\varepsilon }$-cigar

$$\begin{aligned} C(\gamma ,\varepsilon ):=\bigcup _{z\in \gamma } B(z,\varepsilon \lambda (z)),\quad \text {where}\quad \lambda (z)=|x-z|\frac{|y-z|}{|x-y|},\quad z\in \gamma , \end{aligned}$$

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is contained in $\varOmega$. Another possible approach to this conjecture is by the recent result [5, Theorem 2.15] (see also [5, Appendix A]), it is enough to prove the interior and exterior NTA conditions with uniform constants (we do not provide here the definition of an NTA domain, see [37] or [5]).

By [5, Defintion 2.12], we need to verify the Corkscrew condition [5, Defintion 2.10] and the Harnack chain condition [5, Defintion 2.12] with constants not depending on m. The Corkscrew condition, both exterior and interior, is immediately implied by the self-similarity and the Strong Open Set Condition. The essential arguments in the proof of the Harnack chain condition are similar to those in Ref. [2], where the reader can find background and detailed explanations of the techniques.

In the two dimensional case there are more straightforward arguments to show that polygonal approximations to a self-similar curve bound uniformly $(\varepsilon ,\infty )$-domains. Such arguments can be based on the Ahlfors three point condition, see [38, page 73].