Existence for accreting viscoelastic solids at large strains

TL;DR

This paper establishes the existence of solutions for a model describing the growth of viscoelastic solids at large strains, incorporating boundary accretion, incompatible strains, and a regularization approach.

Contribution

It introduces a novel existence proof for a coupled accretion and viscoelastic equilibrium model with large strains and boundary growth.

Findings

Existence of solutions for the accreting viscoelastic solid model.

Regularization via a fictitious surrounding material ensures well-posedness.

Model captures growth driven by deformation and incompatible strains.

Abstract

By revisiting a model proposed in [45], we address the accretive growth of a viscoelastic solid at large strains. The accreted material is assumed to accumulate at the boundary of the body in an unstressed state. The growth process is driven by the deformation state of the solid. The progressive build-up of incompatible strains in the material is modeled by considering an additional backstrain. The model is regularized by postulating the presence of a fictitious compliant material surrounding the accreting body. We show the existence of solutions to the coupled accretion and viscoelastic equilibrium problem.

Figures

Equations

$$\Omega (t):=\{x\in {\mathbb{R}}^d|\theta (x)<t\}.$$

$$\gamma (\nabla y(\theta (x),x))|\nabla \theta (x)|=1\text{for}x\in \Omega (T)\setminus \overline{{\Omega }_0},$$

$$\theta (x)=0\text{for}x\in {\Omega }_0.$$

$$\nabla y=F_{\mathrm{e}}A$$

$$A(x)=\nabla y(\theta (x),x)\text{for}x\in \Omega (T)\setminus \overline{{\Omega }_0}.$$

$$-\mathrm{div}\left(\mathrm{D}W(\nabla y{A}^{-1})A^{-\top }+\mathrm{D}{V}^J(\nabla y)+{\partial }_{\nabla \dot{y}}R(\nabla y,\nabla \dot{y})-\mathrm{div}\mathrm{D}H({\nabla }^{2}y)\right)=f$$

$$\gamma (\nabla y(\theta (x)\wedge T,x))|\nabla \theta (x)|=1\text{for}x\in U\setminus \overline{{\Omega }_0}$$

$$\theta (x)=0\text{for}x\in {\Omega }_0,$$

$$A(t,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x)=\left\{\begin{array}[]{ll}A_{0}(x)&\text{for}\ x\in\overline{\Omega_{0}},\\ \nabla y(\theta(x),x)&\text{for}\ x\in\Omega(t\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\setminus\overline{\Omega_{0}},\\ I&\text{for}\ x\in{U}\setminus\overline{\Omega(t\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})},\end{array}\right.$$

$$\displaystyle-\operatorname{div}\!\left(h(\theta(x){-}t){\rm D}W(\nabla y(t,x)A^{-1}(t,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x))A^{-\top}(t,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x)+{\rm D}V^{J}(\nabla y(t,x))\right)$$

$$-\mathrm{div}\left(h(\theta (x)-t){\partial }_{\nabla \dot{y}}R(\nabla y(t,x),\nabla \dot{y}(t,x))-\mathrm{div}\mathrm{D}H({\nabla }^{2}y(t,x))\right)$$

$$=h(\theta (x)-t)f(t,x)\text{for}(t,x)\in (0,T)\times U.$$

$$\displaystyle{\rm D}H(\nabla^{2}y){:}(\nu\otimes\nu)=0\ \ \text{on}\ \ (0,T)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times\partial U,$$

$$\displaystyle y={\rm id}\ \ \text{on}\ \ (0,T)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times\partial U,$$

$$\displaystyle y={\rm id}\ \ \text{on}\ \ (0,T)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times\omega,$$

$$y(0,\cdot )=y_0\text{on}U.$$

$$\mathrm{SO}(d)$$

$${\mathrm{GL}}_{+}(d)$$

$$\displaystyle\mathcal{A}\coloneqq\Big{\{}y\in W^{2,p}_{\omega}(U;\mathbb{R}^{d})\mid\nabla y\in{\rm GL}_{+}(d)\text{ a.e. in }U\Big{\}},$$

$$W_{\omega }^{2,p}(U;{\mathbb{R}}^d):=\{y\in W^{2,p}(U;{\mathbb{R}}^d)\mid y\equiv \mathrm{id}\text{ on }\omega \cup \partial U\}.$$

$$\Vert y{\Vert }_{W^{2,p}(U;{\mathbb{R}}^d)}\le c\left(1+\Vert {\nabla }^{2}y{\Vert }_{L^p(U;{\mathbb{R}}^{d\times d\times d})}\right)\forall y\in W_{\omega }^{2,p}(U;{\mathbb{R}}^d).$$

$$0=W(I)\leq W(F)\leq\frac{1}{c_{W}}(|F|^{p}+1)\quad\forall\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}F\in\mathbb{R}^{d\times d}.$$

$$W(Q\nabla y(t,x)\nabla y^{-1}(\theta (x),x)Q^{\top })=W(\nabla y(t,x)\nabla y^{-1}(\theta (x),x)).$$

$$V^{J}(F)\geq\frac{c_{J}}{|\det F|^{q}}-\frac{1}{c_{J}}\quad\forall F\in{\rm GL}_{+}(d).\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$$

$$h(\sigma)=\begin{cases}1\qquad&\text{ if }\sigma\leq 0,\\ \delta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\qquad&\text{ if }\sigma>0.\end{cases}$$

$$\mathcal{W}(\sigma ,y;A):={\int }_{U}h(\sigma )W(\nabla y{A}^{-1})+V^J(\nabla y)\mathrm{d}x.$$

$$\mathcal{H}(y):={\int }_{U}H({\nabla }^{2}y)\mathrm{d}x$$

$$\displaystyle c_{H}|G|^{p}-\frac{1}{c_{H}}\leq H(G)\leq\frac{1}{c_{H}}(1+|G|^{p}),\quad|DH(G)|\leq\frac{1}{c_{H}}(1+|G|^{p-1})\quad\forall G\in\mathbb{R}^{d\times d\times d}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$$

$$\displaystyle c_{H}|G-G^{\prime}|^{p}\leq(DH(G)-DH(G^{\prime}))\vdots(G-G^{\prime})\quad\forall G,\,G^{\prime}\in\mathbb{R}^{d\times d\times d};\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$$

$$\mathcal{R}(\sigma ,y,\dot{y}):={\int }_{U}h(\sigma )R(\nabla y,\nabla \dot{y})\mathrm{d}x$$

$$R(F,\dot{F})\coloneqq\frac{1}{2}\dot{C}{:}\mathbb{D}(C)\dot{C}\quad\forall F,\,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\dot{F}\in\mathbb{R}^{d\times d}$$

$$c_{R}|\dot{C}|^{2}\leq\dot{C}{:}\mathbb{D}(C)\dot{C}\quad\forall C,\,\dot{C}\in\mathbb{R}^{d\times d}_{\rm sym}.\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$$

$${\partial }_{\dot{F}}R(F,\dot{F})=2F\left(\mathbb{D}(C):\dot{C}\right)=2F\mathbb{D}(F^{\top }F):({\dot{F}}^{\top }F+F^{\top }\dot{F}).$$

$${\int }_{U}W(\nabla y_0{A}_0^{-1})\text{mathbbm}{1}_{{\Omega }_0}+W(\nabla y_0)\text{mathbbm}{1}_{U\setminus {\Omega }_0}+V^J(\nabla y_0)+H({\nabla }^2{y}_0)\mathrm{d}x<\infty .$$

$$(\theta ,y)\in C^{0,1}(\overline{U})\times \left(L^{\infty }(0,T;W^{2,p}(U;{\mathbb{R}}^d))\cap H^1(0,T;H^1(U;{\mathbb{R}}^d))\right)$$

Full text

1. Introduction

Growth is a fundamental process in all biological systems, as well as in a variety of natural, technological, and social ones. Among the many different dynamics, accretive growth occurs when growth is realized via a progressive accumulation, addition, or layering of material at the boundary of the system. This paradigm is of paramount relevance in numerous situations. The formation of horns, teeth, and seashells [33, 39, 42], coral reefs [26], bacterial colonies [21], trees [12, 14], and cell motility due to actin growth [20] are biological examples of accretive growth. In geophysics, sedimentation and glacier formations are also accretive processes, as is planet formation [6]. Furthermore, accretive growth is a key aspect in many technological applications, including metal solidification [38], crystal growth [29, 44], additive manufacturing [17, 22, 30], layering, coating, and masonry, just to mention a few.

In this note, we consider the evolution in time of a viscoelastic solid under accretive growth. Correspondingly, the reference configuration of the body $\Omega(t)\subset\mathbb{R}^{d}\ (d\geq 2)$ is time-dependent and the deformation $y(t,\cdot):\Omega(t)\to\mathbb{R}^{d}$ of the solid is defined on a time-dependent domain. Although in some cases the map $t\in[0,T]\mapsto\Omega(t)$ can be rightfully assumed to be given (this is for instance the case for 3D printing) the evolution in time of the reference configuration is not a-priori known in general, but is rather influenced by the mechanical process. In this paper, we assume $t\mapsto\Omega(t)$ to be unknown and we tackle its specification by adopting a level-set approach [13, 40] and setting

[TABLE]

The map $x\mapsto\theta(x)\in[0,\infty)$ is called time-of-attachment function: the value $\theta(x)$ corresponds to the instant in time at which the point $x\in\mathbb{R}^{d}$ the accreting body reaches the point $x\in\mathbb{R}^{d}$. We assume that accretion occurs at a positive growth rate $\gamma(\cdot)>0$ and in the outward pointing normal direction to the body.

This implies that the time-of-attachment function $\theta$ solves the following external problem for the generalized eikonal equation [4, 11]

[TABLE]

Here, $\Omega_{0}\subset\mathbb{R}^{d}$ is the given initial reference configuration of the accreting solid. As growth is often driven by the mechanical state of the accreting body [18], we let the growth rate $\gamma$ by setting $\gamma=\gamma(\nabla y(\theta(x),x))$. Specifically, $\gamma\colon\mathbb{R}^{d\times d}\rightarrow[0,\infty)$ is assumed to be Lipschitz continuous and such that $c_{\gamma}\leq\gamma(\cdot)\leq C_{\gamma}$ for some $0<c_{\gamma}\leq C_{\gamma}$.

In order to track the progressive accumulation of incompatibilities due to growth [38, 39, 45], we follow the classical approach of finite plasticity [23, 24] and postulate the multiplicative decomposition

[TABLE]

where $F_{\rm e}\in{\rm GL}_{+}(d)$ corresponds to the elastic part of the deformation gradient, whereas $A\in{\rm GL}_{+}(d)$ is the backstrain originated by the build-up of incompatibilities [18] during growth. The elastic energy density $W\colon\mathbb{R}^{d\times d}\to[0,\infty)$ of the accreting medium is assumed to be depending on $F_{\rm e}=\nabla yA^{-1}$. In order to specify a constitutive relation for $A$, we follow [45], see also [11, 43], and assume that the material is added to the accreting body in an unstressed state. Specifically, we assume $W$ to be minimized at the identity matrix $I$ and $F_{\rm e}(t,x)=I$ at the accreting front, i.e., for $(t,x)=(\theta(x),x)$. This entails the constitutive relation

[TABLE]

Note that the backstrain $A$ is independent of time. In particular, as the boundary $\partial\Omega(t)$ reaches point $x$, the value of $A(x)$ is stored according to (4). This reflects the intuition that growth-driven incompatibilities are recorded in the body along the process.

The viscoelastic evolution of the accreting solid is determined by the equilibrium system

[TABLE]

to be solved in the noncylindrical domain $\cup_{t\in[0,T]}\{t\}\times\Omega(t)$. Here, $V^{J}\colon\mathbb{R}^{d\times d}\rightarrow[0,\infty]$ is an additional elastic energy term specifically penalizing self-interpenetration of matter, i.e., $V^{J}(F)\rightarrow\infty$ as $\det F\rightarrow 0^{+}$ and $V^{J}(F)<\infty$ if and only if $\det F>0$. Moreover, R:{\rm GL}_{+}(d)\times\mathbb{R}^{d\times d}\rightarrow[0,\infty)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0} is the instantaneous dissipation potential, which is assumed to be quadratic in $\nabla\dot{y}^{\top}\nabla y+\nabla y^{\top}\nabla\dot{y}$ where the dot denotes the partial time derivative. The term $H\colon\mathbb{R}^{d\times d\times d}\rightarrow[0,\infty)$ qualifies the accreting solid as a second-grade, nonsimple material and $f=f(t,x)$ is and external-force density.

The model (1)–(5) is of free-boundary type, as equations are posed on the unknown sets $\Omega(t)$ from (1). This creates significant difficulties for the analysis, forcing us to reduce the model to a fixed, ambient setting, see Figure 1. In particular, we ask that $\Omega(t)\subset U$ for all $t\in[0,T]$, for a fixed, open, and bounded container $U\subset\mathbb{R}^{d}$. Problem (2)–(3) can be reduced to the fixed-boundary setting by considering

[TABLE]

instead of (2)–(6). This modification is actually immaterial as we will check that the restriction to $\Omega(T)=\{x\in U\ |\ \theta(x)<T\}$ of a solution $\theta$ to (3)–(7) solves (2)–(3), as well. Note nonetheless that (6) requires to introduce the minimum $\theta(x)\wedge T$, as $\theta(x)>T$ at some points of $U$.

In order to reduce the mechanical problem (4)–(5) to the fixed container, we assume that the complement $U\setminus\overline{\Omega(t)}$ to the accreting body is filled by a second, fictitious medium of viscoelastic type. By letting such fictitious medium be very compliant, one expects that the behavior of the regularized model to approximate the original, free-boundary one. Correspondingly, the constitutive equation (4) may be extended to $U$ by posing

[TABLE]

where we have additionally specified the initial backstrain $A_{0}$ on $\Omega_{0}$ and defined $A=I$ in the region {U}\setminus\overline{\Omega(t\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})} which is not accessed by the accreting medium at time $t$, by simplicity. Compared with (4), the backstrain $A(t,x)$ is now depending on $t$, as well. This is an artifact of the extension of the model to the container $U$.

Eventually, the viscoelastic equilibrium (5) also need to be extended to the whole container. At all $(t,x)\in[0,T]\times U$, we may distinguish the accreting medium and the fictitious one by the sign of $\theta(x)-t$. More precisely, if $\theta(x)-t\leq 0$ we have that $x\in\overline{\Omega(t)}$ (accreting medium), whereas $\theta(x)-t>0$ implies that $x\in U\setminus\overline{\Omega(t)}$ (fictitious material). The sharp transition in the material parameters between the two media will be described by a function $(x,t)\mapsto h(\theta(x)-t)$ with $h(\theta(x)-t)=1$ for $\theta(x)-t\leq 0$ and h(\theta(x)-t)=\delta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0} for $\theta(x)-t>0$ with \delta\in(0,1)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0} very small. By using $h$, the elastic energy density and the instantaneous dissipation density of combined medium are considered to be $h(\theta(x)-t)W(\nabla yA^{-1})$ and $h(\theta(x)-t)R(\nabla y,\nabla\dot{y})$, respectively, and the viscoelastic equilibrium system takes the form

[TABLE]

Notice that the second-order-potential density $H$ and the term $V^{J}$ are assumed to be the same for both the accreting and the fictitious media. On the other hand, the force density $h(\theta(x){-}t)f(t{,}x)$ distinguishes between the accreting and the fictitious medium, as it would be the case for gravity for different densities.

The aim of this paper is to show that model (6)–(9) admits solutions. To this end, system (6)–(9) is complemented by the boundary and initial conditions

[TABLE]

Specifically, at the boundary $\partial U$ of the container $U$ we prescribe the homogeneous Neumann and Dirichlet conditions (10)–(11). These choices are motivated by simplicity, reflecting the mere instrumental role of the container $U$ in the model. In fact, other options would be viable, as well. On the other hand, the anchoring condition (12) fixes the position of the body at a portion $\omega\subset\subset\Omega_{0}$ of the starting configuration $\Omega_{0}$ see [11]. Independently of (10)–(11), this anchoring condition will allow to use a Poincaré-type inequality, cf. (14), which turns out to be crucial for the analysis.

Our main result consists in proving existence of solutions $(\theta,y)$ to the fully coupled system (6)–(13), see Definition 3.1 and Theorem 3.1. More precisely, we find $\theta\in C(\overline{U})$ solving the external problem for the generalized eikonal equation (6)–(7) in the viscosity sense [8], and $y\in L^{\infty}(0,T;W^{2,p}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$ satisfying the viscoelastic equilibrium system (9)–(13) in the weak sense, with the constitutive relation (8) pointwise fulfilled.

Before proceeding, let us mention that the applied literature on the mechanics of growth is well developed. Comprehensive accounts can be found in the monographs [18, 42], as well as in [15, 32, 34, 35, 41], among many others. In comparison, rigorous mathematical results mechanical growth models are just a few. For instance, an elastic growing body and the coupled dynamics of the morphogen is studied in [5]. In the setting of bulk growth, where growth is realized by addition of material in the bulk of the solid, existence results are available in both one [3] and three [10, 16] space dimensions.

In the specific setting of accretive growth, which is the focus of this paper, a first existence result in the context of linearized elasticity has been obtained in [11]. There, the constitutive relation (2) cannot be directly considered due to the limited regularity of the solution [9], and an additional regularization via a mollification is introduced, which can be interpreted as a diffused-interface, phase-field approximation. By neglecting the backstress $A$, the viscoelastic growth model (2)–(9) has been considered in [7], where an existence result is obtained, both in the mollified setting (as above) and the limiting sharp-interface case.

The paper is structured as follows. In Section 2, we specify notation and the assumptions on the ingredients of the model. The definition of solution to the problem is detailed in Section 3, where the main existence result, Theorem 3.1, is also stated. Section 4 is then devoted to the proof of Theorem 3.1. The proof strategy is iterative. At first, we show that, for given $y$, we can find a viscosity solution $\theta$ to (6)–(7), see Proposition 4.1. Then, in Proposition 4.2 we check the existence of $y$ solving (8)–(13) for given $\theta$. Eventually, we combine these results and iteratively define a sequence $(y_{k},\theta_{k})_{k\in\mathbb{N}}$ converging, up to subsequences, to a solution to the fully coupled problem.

2. Notation and setting

We devote this section to specifying notation and assumptions.

2.1. Notation

In what follows, we denote by $\mathbb{R}^{d\times d}$ the Euclidean space of $d{\times}d$ real matrices, $d\geq 2$, by $\mathbb{R}^{d\times d}_{\rm sym}$ the subspace of symmetric matrices, and by $I$ the identity matrix. Given $A\in\mathbb{R}^{d\times d}$, we indicate by $A^{\top}$ its transpose and by $|A|^{2}:=A{:}A$ its Frobenius norm, where the contraction product between two matrices $A,\,B\in\mathbb{R}^{d\times d}$ is defined as $A{:}B:=A_{ij}B_{ij}$ (we use the summation convention over repeated indices). Analogously, let $\mathbb{R}^{d\times d\times d}$ be the set of real $3$-tensors, and define their contraction product as $A\smash{\vdots}B:=A_{ijk}B_{ijk}$ for $A,B\in\mathbb{R}^{d\times d\times d}$. Moreover, given a real 4-tensor $\mathbb{C}\in\mathbb{R}^{d\times d\times d\times d}$ and the matrix $A\in\mathbb{R}^{d\times d}$ we indicate by $\mathbb{C}{:}A\in\mathbb{R}^{d\times d}$ and $A{:}\mathbb{C}\in\mathbb{R}^{d\times d}$ the matrices given in components by $(\mathbb{C}{:}A)_{ij}=\mathbb{C}_{ijk\ell}A_{k\ell}$ and $(A{:}\mathbb{C})_{ij}=A_{k\ell}\mathbb{C}_{k\ell ij}$, respectively. We shall use the following matrix sets

[TABLE]

The scalar product of two vectors $a,\,b\in\mathbb{R}^{d}$ is classically indicated by $a{\cdot}b$. The symbol $B_{R}\subset\mathbb{R}^{d}$ denotes the open ball of radius $R>0$ and center $0\in\mathbb{R}^{d}$, $|E|$ indicates the Lebesgue measure of the Lebesgue-measurable set $E\subset\mathbb{R}^{d}$, and $\mathbbm{1}_{E}$ is the corresponding characteristic function, namely, $\mathbbm{1}_{E}(x)=1$ for $x\in E$ and $\mathbbm{1}_{E}(x)=0$ otherwise. For $E\subset\mathbb{R}^{d}$ nonempty and $x\in\mathbb{R}^{d}$ we define ${\rm dist}(x,E):=\inf_{e\in E}|x{-}e|$. We denote by $\mathcal{H}^{d-1}$ the $(d{-}1)$-dimensional Hausdorff measure and by $\mathcal{L}^{d+1}$ the Lebesgue measure in $\mathbb{R}^{d+1}$.

In the following, we indicate by $c$ a generic positive constant, possibly depending on data but independent of the time discretization step $\tau$, to be used in the proof of Proposition 4.2. Specifically, $c$ may depend on $\delta>0$, defined in (15) below. Note that the value of $c$ may change from line to line.

2.2. Setting

We start by posing the following.

1. (H1)

Let $T>0$ be a fixed final time, $U\subset\mathbb{R}^{d}$ be nonempty, open, connected, bounded, and Lipschitz, $\Omega_{0}\subset\subset U$ and $\omega\subset\subset\Omega_{0}$ be nonempty and open, and $p>d$.

We define $Q\coloneqq(0,T)\times U$.

2.2.1. Admissible deformations

The set of admissible deformations is given as

[TABLE]

where

[TABLE]

Deformations $y$ are locally invertible and orientation preserving. Moreover, they satisfy the anchoring condition $y\equiv{\rm id}$ in $\omega$ for almost every $t\in(0,T)$. In particular, this condition entails the validity of the following Poincaré-type inequality

[TABLE]

2.2.2. Energy

The elastic energy density $W\colon\mathbb{R}^{d\times d}\rightarrow[0,\infty)$ of the accreting medium is asked to satisfy

1. (H2)

$W\in C^{1}(\mathbb{R}^{d\times d})$; 2. (H3)

there exists $c_{W}>0$ such that

[TABLE]

Although it is not strictly needed for the analysis, we additionally assume

1. (H3)

frame indifference:   $W(QF)=W(F)$ for all $F\in\mathbb{R}^{d\times d}$ and $Q\in{\rm SO}(d)$; 2. (H4)

isotropy:   $W(FQ)=W(F)$ for all $F\in\mathbb{R}^{d\times d}$ and $Q\in{\rm SO}(d)$.

We remark that both (H3) and (H4) are required for the model to be frame indifferent. Indeed, taking (8) into account, for all rotation $Q\in{\rm SO}(d)$ we have

[TABLE]

The density $V^{J}\colon{\rm GL}_{+}(d)\rightarrow[0,\infty)$ is asked to be such that

1. (H5)

$V^{J}\in C^{1}({\rm GL}_{+}(d))$; 2. (H6)

there exist $q>pd/(p-d)$ and $c_{J}>0$ such that

[TABLE]

Finally, for some fixed $\delta>0$ we recall that $h\colon\mathbb{R}\rightarrow[0,1]$ is given by

[TABLE]

In order to specify the stored elastic energy of the combined accreting-fictitious medium, we define the functional \mathcal{W}\colon C(\overline{U})\times\mathcal{A}\times L^{\infty}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{U};{\rm GL}_{+}(d))\rightarrow[0,\infty) as

[TABLE]

Here, $A\in L^{\infty}(U;{\rm GL}_{+}(d))$ is a placeholder for the backstrain tensor given by (8) and $\sigma\in C(\overline{U})$ is a placeholder for $x\mapsto\theta(x){-}t$, whose sublevel set $\{x\in U\mid\theta(x)-t<0\}$ identifies the location of the accreting medium at time $t$. In particular, in the accreting medium the latter energy density reads W+V^{J}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}, whereas in the fictitious medium it is {\delta}W+V^{J}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}. Choosing $\delta$ small hence corresponds to assuming that the fictitious material is highly elastically compliant. Recall nonetheless that we assume that both accreting and fictitious materials have the same V^{J}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}, modeling a comparable response to extreme compression.

We additionally consider a second-order potential $\mathcal{H}\colon W^{2,p}_{\omega}(U;\mathbb{R}^{d})\rightarrow[0,\infty)$ given by

[TABLE]

where $H:\mathbb{R}^{d\times d\times d}\rightarrow[0,\infty)$ is such that

1. (H7)

$H\in C^{1}(\mathbb{R}^{d\times d\times d})$ is convex; 2. (H8)

there exists a positive constant $c_{H}>0$ such that

[TABLE] 3. (H9)

$H(QG)=H(G)$ for all $G\in\mathbb{R}^{d\times d\times d},Q\in{\rm SO}(d)$.

Frame indifference (H9) of $H$ is assumed to guarantee physical consistency, albeit being not necessary for the analysis. By including $\mathcal{H}$ in the total energy of the combined medium, we are indeed modeling a second-grade, nonsimple material [28, Sec. 2.5]. The inclusion of this second-order potential is primarily motivated by the need to ensure sufficient compactness in the problem and the corresponding length-scale constant $c_{H}$ is ideally assumed to be very small. Moving from these considerations, the second-order energy density $H$ is taken to be identical for both the accreting and the fictitious materials, for simplicity.

2.2.3. Viscous dissipation

The dissipation potential is defined via $\mathcal{R}\colon C(\overline{U})\times W^{2,p}_{\omega}(U;\mathbb{R}^{d})\times H^{1}(U;\mathbb{R}^{d})\rightarrow[0,\infty)$ given by

[TABLE]

where $R:\mathbb{R}^{d\times d}\times\mathbb{R}^{d\times d}\rightarrow[0,\infty)$ is specified as

[TABLE]

with $C\coloneqq F^{\top}F$ and $\dot{C}\coloneqq\dot{F}^{\top}F+F^{\top}\dot{F}$. We assume

1. (H10)

$\mathbb{D}\in C(\mathbb{R}^{d\times d}_{\rm sym};\mathbb{R}^{d\times d\times d\times d})$ is such that $\mathbb{D}_{ijk\ell}=\mathbb{D}_{jik\ell}=\mathbb{D}_{k\ell ij}$ for every $i,j,k,\ell=1,\dots,d$; 2. (H11)

there exists a positive constant $c_{R}>0$ such that

[TABLE]

The very structure of $R$ guarantees that it is frame indifferent [1]. Notice that by the definition of $R$, we have that $\partial_{\dot{F}}R$ is linear in $\dot{F}$. More precisely, we have that

[TABLE]

2.2.4. Loading and initial data

We denote by $f\colon[0,T]\times U\rightarrow\mathbb{R}^{d}$ a given body-force density, and we require

1. (H12)

$f\in W^{1,\infty}\left(0,T;L^{2}(U;\mathbb{R}^{d})\right)\cap L^{\infty}(Q;\mathbb{R}^{d})$.

We moreover assume that the initial backstrain $A_{0}$ and the initial deformation $y_{0}$ satisfy

1. (H13)

$A_{0}\in C(\overline{\Omega_{0}};{\rm GL}_{+}(d))$, $y_{0}\in\mathcal{A}$, and

[TABLE]

Here and in the following, $W(\nabla y_{0}A_{0}^{-1})\mathbbm{1}_{\Omega_{0}}$ indicates the trivial extension of $W(\nabla y_{0}A_{0}^{-1})$ to $U$.

2.2.5. Growth

Concerning the growth problem, we assume the following

1. (H14)

$\gamma\in C^{0,1}({\rm GL}_{+}(d))$ is such that $c_{\gamma}\leq\gamma(\cdot)\leq C_{\gamma}$ for some $0<c_{\gamma}\leq C_{\gamma}$; 2. (H15)

$\Omega_{0}+B_{C_{\gamma}T}\subset\subset U$.

We remark assumption (H15) guarantees that the accreting material does not reach the boundary of the cointainer $U$ by the final time $T$, see (38) below.

3. Notion of solution and main results

3.1. Notion of solution

We now specify our notion of solution to (6)–(13).

Definition 3.1** (Solution).**

We say that a pair

[TABLE]

is a solution to the initial-boundary-value problem (6)–(13) if

1. (1)

$\theta$* is a viscosity solution to*

[TABLE]

namely, for any $\varphi\in C^{1}(\mathbb{R}^{d})$, we have that $\gamma(\nabla y(\theta(x)\wedge T,x))|\nabla\varphi(x)|\leq(\geq)\,1$ at any local minimum (maximum, respectively) point $x\in U\setminus\overline{\Omega_{0}}$ of $\varphi-\theta$. 2. (2)

$y(t,\cdot)\in\mathcal{A}$* for almost every $t\in(0,T)$; $y(0,\cdot)=y^{0}(\cdot)$ in $U$, and*

[TABLE]

with backstrain tensor $A$ defined as

[TABLE]

where $\Omega(t):=\{x\in U\ |\ \theta(x)<t\}$ for $t\in(0,T]$.

3.2. Main result

Our main result is the following.

Theorem 3.1** (Existence).**

Under assumptions (H1)–(H15), there exists a solution $(\theta,y)$ to problem (6)–(13).

The proof of Theorem 3.1 is given in Section 4. As already mentioned in the Introduction, Proposition 4.1 allows us to find a solution $\theta$ to (16)–(17) for all given $y\in L^{\infty}(0,T;W^{2,p}_{\omega}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$. Then, in Proposition 4.2 we check that, given $\theta\in C(\overline{U})$, by defining $\Omega(t)$ as in (1) we can find a solution $y\in L^{\infty}(0,T;W^{2,p}_{\omega}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$ to (2)–(19). This allows us to implement an iterative procedure. The proof of Theorem 3.1 follows by checking that such iterations converge, up to subsequence, to a solution.

4. Proof of Theorem 3.1

We begin by recalling [31, Thm. 3.15], which ensures the well-posedness of the external problem for the generalized eikonal equation in the whole $\mathbb{R}^{d}$.

Proposition 4.1** **(Well-posedness of the growth

subproblem).

Assume to be given $\widehat{\gamma}\in C(\mathbb{R}^{d})$ with $c_{\gamma}\leq\widehat{\gamma}(\cdot)\leq C_{\gamma}$ for some $0<c_{\gamma}\leq C_{\gamma}$ and $\Omega_{0}\subset\mathbb{R}^{d}$ nonempty, open, and bounded. Then, there exists a unique nonnegative continuous $\theta$ viscosity solution to

[TABLE]

namely, for any $\varphi\in C^{1}(\mathbb{R}^{d})$, we have that $\widehat{\gamma}(x)|\nabla\varphi(x)|\leq(\geq)\,1$ at any local minimum (maximum, respectively) point $x\in\mathbb{R}^{d}\setminus\overline{\Omega_{0}}$ of $\varphi-\theta$. Such $\theta$ is given by the representation formula

[TABLE]

In particular, $\theta\in C^{0,1}(\mathbb{R}^{d})$ with

[TABLE]

The representation formula (22) and the bounds (H14) ensure that

[TABLE]

Before moving to the proof of Theorem 3.1, let us show that, for all given $\theta\in C(\overline{U})$, there exists a deformation $y$ satisfying (2)–(19).

Proposition 4.2** (Existence for the equilibrium subproblem).**

Let (H1)–(H15) hold, $\theta\in C(\overline{U})$, and $\Omega(t)$ be defined as in (1) for all $t\in[0,T]$. Then, there exists $y\in L^{\infty}(0,T;W^{2,p}_{\omega}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$ such that $y(t,\cdot)\in\mathcal{A}$ for almost every $t\in[0,T]$, and $y(0,\cdot)=y_{0}(\cdot)$ in $U$, satisfying (2)–(19).

Proof.

We follow the blueprint of [2] or [27] and argue by time-discretization. Let $\tau\coloneqq T/{N_{\tau}}>0$ with $N_{\tau}\in\mathbb{N}$ given and consider the corresponding uniform partition of the time interval $[0,T]$ given by $t_{i}\coloneqq i\tau$, for $i=0,\dots,N_{\tau}$. Moreover, set $A^{0}_{\tau}\coloneqq A_{0}\mathbbm{1}_{\Omega_{0}}+I\mathbbm{1}_{U\setminus\Omega_{0}}$. For $i=1,\dots,N_{\tau}$, assume to know $y^{j}_{\tau}\in\mathcal{A}$ for $j=0,1,\dots,i-1$ and define $A^{i}_{\tau}:U\to\mathbb{R}^{d\times d}$ as

[TABLE]

Notice that (H13), the definition of $\mathcal{A}$, and the fact that $p>d$, imply that $A^{i}_{\tau}\in L^{\infty}(U;{\rm GL}_{+}(d))$. We find $y^{i}_{\tau}\in\mathcal{A}$ by solving

[TABLE]

Under the growth conditions (H6), (H8), and (H11), the regularity and convexity assumptions (H2), (H5), (H7), (H10), and (H12), and by using the Poincaré inequality (14), the existence of $y^{i}_{\tau}\in\mathcal{A}$ for $i=1,\dots,N_{\tau}$ easily follows by the Direct Method of the calculus of variations. Moreover, every minimizer $y_{\tau}^{i}$ satisfies the time-discrete Euler–Lagrange equation

[TABLE]

for every z^{i}\in C^{\infty}(\overline{U}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0};\mathbb{R}^{d}) with $z^{i}\equiv 0$ on $\omega\cup\partial U$, and for every $i=1,\dots,N_{\tau}$.

From the minimality of $y^{i}_{\tau}$ we get that

[TABLE]

Summing over $i=1,\dots,n\leq N_{\tau}$, we obtain

[TABLE]

The growth conditions (H6), (H8), and (H11), and the definition (15) of $h$ ensure that

[TABLE]

In order to obtain an a-priori estimate, we now control the various terms in the above right-hand side. The initial-value term is directly bounded by (H12)–(H13).

The second term in the right-hand side of (27) can be handled as follows

[TABLE]

since $A^{i-1}_{\tau}(x)=I$ if $\theta(x)>t_{i-1}$. By the growth condition (H3), we then have

[TABLE]

where we also the Poincaré inequality (14) and the continuous embedding of $L^{\infty}(U)$ into $W^{1,p}(U)$ for $p>d$.

As for the third term in the right-hand side of (27), we notice that, for $x\in\overline{\Omega_{0}}$, $A^{i}_{\tau}(x)=A^{i-1}_{\tau}(x)=A_{0}$. For $x\in U\setminus\overline{\Omega_{0}}$ with $\theta(x)\leq t_{i-2}$ for $i>2$, there exists $k\in\{1,\dots,i-2\}$ such that $\theta(x)\in(t_{k-1},t_{k}]$, and thus $A^{i}_{\tau}(x)=A^{i-1}_{\tau}(x)=\nabla y^{k}_{\tau}$. Similarly, for $x\in U\setminus\overline{\Omega_{0}}$ such that $\theta(x)>t_{i-1}$ for $i\geq 2$ we have $A^{i}_{\tau}(x)=A^{i-1}_{\tau}(x)=I$. Hence, the integrand is nonzero only for $x\in U$ such that $t_{i-2}<\theta(x)\leq t_{i-1}$ for $i\geq 2$. For such $x$ we have $A^{i}_{\tau}(x)=\nabla y^{i-1}_{\tau}$ and $A^{i-1}_{\tau}(x)=I$, so that

[TABLE]

Eventually, the regularity (H12) ensures that

[TABLE]

where we also used the embedding $W^{2,p}(U;\mathbb{R}^{d})\subset L^{\infty}(U;\mathbb{R}^{d})$, the Poincaré inequality (14), and the fact that $p>2$.

By collecting (28)–(29) in (27) we hence have that

[TABLE]

The Discrete Gronwall Lemma [28, (C.2.6), p. 534] and the Poincaré inequality (14) allow us to conclude that

[TABLE]

Let us now introduce the following notation for the time interpolants of a vector $(u_{0},...,u_{N_{\tau}})$ over the interval $[0,T]$: We define the backward-constant interpolant $\overline{u}_{\tau}$, the forward-constant interpolant $\underline{u}_{\tau}$, and the piecewise-affine interpolant $\widehat{u}_{\tau}$ on the partition $(t_{i})_{i=0}^{N_{\tau}}$ as

[TABLE]

Making use of this notation, we can rewrite (30) as

[TABLE]

By the Sobolev embedding of $W^{2,p}(U;\mathbb{R}^{d})$ into $C^{1,1-d/p}(\overline{U};\mathbb{R}^{d})$ and by the classical result [19, Thm. 3.1], the bound (31) implies

[TABLE]

Moreover, by the Poincaré inequality (14), the generalization of Korn’s first inequality by [36] and [37, Thm. 2.2], and the uniform positivity of the determinant (32), it follows that

[TABLE]

Thus, the classical Poincaré inequality applied to $\dot{y}$ proves that

[TABLE]

Hence, the estimates above yield

[TABLE]

for $\alpha\in(0,1-d/p)$, as $\tau\to 0$, up to not relabeled subsequences. In particular, these convergences imply $\det\nabla\overline{y}_{\tau}\to\det\nabla y$ uniformly and, together with the lower bound (32), that $\nabla y\in{\rm GL}_{+}(d)$ everywhere, i.e., $y(t,\cdot)\in\mathcal{A}$ for every $t\in(0,T)$.

Summing up the time-discrete Euler–Lagrange equations (4) for $i=1,\dots,N_{\tau}$ and rewriting in terms of the time interpolants, we get

[TABLE]

We now pass to the limit in (4) as $\tau\rightarrow 0$. Let $z\in C^{\infty}(\overline{Q};\mathbb{R}^{d})$ with $z\equiv 0$ on (0,T)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times(\omega\cup\partial U) be given and let $(z^{i}_{\tau})_{i=1}^{N_{\tau}}\subset W^{2,p}(U;\mathbb{R}^{d})$ be such that $z^{i}_{\tau}\equiv 0$ on $\omega\cup\partial U$ for every $i=1,...,N_{\tau}$, and $\overline{z}_{\tau}\to z$ strongly in $L^{\infty}(0,T;W^{2,p}(U;\mathbb{R}^{d}))$. First, notice that, by the coarea formula and the Lipschitz continuity (23) of $\theta$, we have

[TABLE]

Thus, $|\partial\Omega(t)|=0$ for almost every $t\in(0,T)$. We hence have

[TABLE]

which implies that $h(\theta(x){-}\overline{t}_{\tau}(t))\to h(\theta(x){-}t)$ for almost every $(t,x)\in Q$. By (H12), it thus follows

[TABLE]

Similarly, for the dissipation, we find

[TABLE]

by the convergences (34)–(36), and (H10). By the continuity (H5) and the lower bound on the determinant (32), we also have

[TABLE]

Moreover, convergence (36) guarantees that for almost every $(t,x)\in Q$, $\overline{A}_{\tau}$ converges to $A$ given by (19). Indeed, let $(t,x)\in Q$, $(t_{i_{\tau}})_{\tau}$ be such that $t\in(t_{i_{\tau}-1},t_{i_{\tau}}]$ for every $\tau>0$, and $t_{i_{\tau}}\rightarrow t$, as $\tau\rightarrow 0$. Thus, $\overline{A}_{\tau}(t,x)=A^{i_{\tau}}_{\tau}(x)$. If $x\in\overline{\Omega_{0}}$, then A^{i_{\tau}}_{\tau}(x)=A_{0}(x)=A(t,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x), whereas if x\in U\setminus\overline{\Omega(t)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}, then $\theta(x)\geq t>t_{i_{\tau}-i}$ and thus, by definition (25), A^{i_{\tau}}_{\tau}(x)=I=A(t,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x). On the other hand, if x\in\Omega(t)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\setminus\overline{\Omega_{0}}, then there exists $s\in(0,t)$ such that $\theta(x)=s$ and there exist $k_{\tau}\in\mathbb{N}$, $k_{\tau}\geq 1$, for every $\tau>0$ such that $s\in(t_{k_{\tau}-1},t_{k_{\tau}}]$. Since $s<t$, we can assume $t_{k_{\tau}}\leq t_{i_{\tau}-1}$, so that A^{i_{\tau}}_{\tau}(x)=\nabla y^{k_{\tau}}_{\tau}(x)\rightarrow\nabla y(s,x)=\nabla y(\theta(x),x)=A(t,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x), by convergence (36). Hence, by the continuity (H2) and the bound (H3) on $W$, convergences (34)–(36), and dominated convergence, we have

[TABLE]

The convergence of the second-gradient term follows by a standard argument [27], which we reproduce here for the sake of completeness. Let $(w_{\tau}^{i})_{i=1}^{N_{\tau}}\subset\mathcal{A}$ approximate the limiting function $y$, namely be such that $\overline{w}_{\tau}\to y$ strongly in $L^{\infty}(0,T;W^{2,p}_{\omega}(U;\mathbb{R}^{d}))$ as $\tau\rightarrow 0$. Define $\overline{z}_{\tau}\coloneqq\overline{w}_{\tau}-\overline{y}_{\tau}$. By convergences (34)–(35), it follows that $\overline{z}_{\tau}\rightarrow 0$ strongly in $L^{\infty}(0,T;H^{1}(U;\mathbb{R}^{d}))$ and $\overline{z}_{\tau}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}0$ weakly-$*$ in $L^{\infty}(0,T;W^{2,p}(U;\mathbb{R}^{d}))$. Moreover, by the strong convergence of $\nabla^{2}\overline{w}_{\tau}$ to $\nabla^{2}y$ in $L^{p}(Q;\mathbb{R}^{d\times d\times d})$ and by the boundedness of ${\rm D}H(\nabla^{2}\overline{y}_{\tau})$ in $L^{p^{\prime}}(Q;\mathbb{R}^{d\times d\times d})$ thanks to (H9), it follows

[TABLE]

Hence, the Euler–Lagrange equation (4) with test function $\overline{z}_{\tau}$ and convergences (34)–(36) entail that

[TABLE]

By the coercivity (H8), this implies that $\nabla^{2}\overline{y}_{\tau}\to\nabla^{2}y$ strongly in $L^{p}(Q;\mathbb{R}^{d\times d\times d})$. The bound on ${\rm D}H$ in (H8) ensures that, possibly passing to not relabeled subsequences, ${\rm D}H(\nabla^{2}\overline{y}_{\tau})\rightharpoonup G$ weakly in $L^{p^{\prime}}(Q;\mathbb{R}^{d\times d\times d})$ and we can identify $G={\rm D}H(\nabla^{2}y)$ as $H$ is convex. Hence, ${\rm D}H(\nabla^{2}\overline{y}_{\tau})\rightharpoonup{\rm D}H(\nabla^{2}{y})$ weakly in $L^{p^{\prime}}(Q;\mathbb{R}^{d\times d\times d})$ and thus (2) follows by passing to the limit in (4) taking $\tau\to\leavevmode\nobreak\ 0$. ∎

Having checked Propositions 4.1 and 4.2, we proceed with the proof of Theorem 3.1 by an iterative construction. We first remark that, since $y_{0}\in\mathcal{A}$ by (H13), $\nabla y_{0}$ is Hölder continuous and, thus, so is the mapping $x\in U\mapsto\gamma(\nabla y_{0}(x))$. Denoting by $\widehat{\gamma}$ be any continuous extension of such mapping to $\mathbb{R}^{d}$ with $c_{\gamma}\leq\widehat{\gamma}(\cdot)\leq C_{\gamma}$, by Proposition 4.1 there exists a unique nonnegative viscosity solution $\theta_{0}\in C(\overline{U})$ to problem

[TABLE]

satisfying (23) in $\overline{U}$. Given $\theta=\theta_{0}$, on the other hand, Proposition 4.2 provides the existence of $y^{1}\in L^{\infty}(0,T;W^{2,p}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$ satisfying (2).

For $k\geq 1$, given $y^{k}\in L^{\infty}(0,T;W^{2,p}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$, the map $x\in U\mapsto\gamma(y^{k}(\nabla y^{k}(\theta^{k}(x){\wedge}T,x)))$ is Hölder continuous. As above, we extend it continuously to $\mathbb{R}^{d}$ as $\widehat{\gamma}$ with $c_{\gamma}\leq\widehat{\gamma}(\cdot)\leq C_{\gamma}$. Let $\theta$ be the unique nonnegative viscosity solution to (20)–(21) and let $\theta^{k}\in C(\overline{U})$ be its restriction to $U$. This solves

[TABLE]

in the viscosity sense and fulfills (23) in $\overline{U}$. Owing to the bounds (24) and (H15) we also have that

[TABLE]

so that the accreting material does not reach the boundary of $U$ over the time interval $[0,T]$.

For such $\theta^{k}$, Proposition 4.2 applied for $\theta=\theta^{k}$ entails the existence of a deformation $y^{k+1}\in L^{\infty}(0,T;W^{2,p}(U;\mathbb{R}^{d}))\cap H^{1}(0,T;H^{1}(U;\mathbb{R}^{d}))$ satisfying (2).

The sequence $(\theta^{k},y^{k})_{k\in\mathbb{N}}$ generated by this iterative process is in general not unique, but still uniformly bounded in

[TABLE]

thanks to the bounds (23), (31), and (33). Thus, up to subsequences, by the Banach–Alaoglu and the Ascoli–Arzelà Theorems, there exists a pair $(y,\theta)$ such that, for some $\alpha\in(0,1-d/p)$,

[TABLE]

and $\theta$ fulfills (23) in $\overline{U}$. By the Lipschitz continuity of $\gamma$ and by convergences (41)–(39), we readily have that $x\mapsto\gamma(\nabla y^{k}(\theta^{k}(x){\wedge}T,x))$ converges to $x\mapsto\gamma(\nabla y(\theta(x){\wedge}T,x))$ uniformly in $\overline{U}$. Since the eikonal equation is stable with respect to the uniform convergence of the data [25, Prop. 1.2], $\theta$ satisfies (16) with coefficient $x\mapsto\gamma(\nabla y(\theta(x){\wedge}T,x))$. Moreover, since bounds (31) and (33) are independent of $\theta$, the same arguments of the proof of Proposition 4.2 allow passing to the limit in the Euler–Lagrange equation (2), thus concluding the proof of Theorem 3.1.

5. Acknowledgements

This research was funded in whole or in part by the Austrian Science Fund (FWF) projects 10.55776/F65, 10.55776/I5149, 10.55776/P32788, and 10.55776/I4354, as well as by the OeAD-WTZ project CZ 09/2023. For open-access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.