A remark on the denoising of greyscale images using energy densities with varying growth rates

TL;DR

This paper proves the solvability of a class of variational problems with variable growth energy densities, extending the TV-model for greyscale image denoising to more general functional forms.

Contribution

It introduces a generalized framework for denoising models with energy densities of varying growth rates, broadening the applicability of TV-based methods.

Findings

Proves existence of solutions in Sobolev spaces for the generalized models.

Extends the TV-model to include energy densities with (1, p)-growth.

Provides a mathematical foundation for more flexible denoising functionals.

Abstract

We prove the solvability in Sobolev spaces for a class of variational problems related to the TV-model proposed by Rudin, Osher and Fatemi in [1] for the denoising of greyscale images. In contrast to their approach we discuss energy densities with variable growth rates depending on $|\nabla u|$ in a rather general form including functionals of $(1, p)$-growth.

Equations

$$I_1[w]:={\int }_{\Omega }|\nabla w|\mathrm{d}x+\frac{\lambda }{2}{\int }_{\Omega }|f-w{|}^2\mathrm{d}x\to \min$$

$$0\le f\le 1\text{ a.e. on }\Omega ,$$

$$I_p[w]:={\int }_{\Omega }|\nabla w{|}^p\mathrm{d}x+\frac{\lambda }{2}{\int }_{\Omega }|w-f{|}^2\mathrm{d}x\to \min$$

$$J_{\mu }[w]:={\int }_{\Omega }{F}_{\mu }(\nabla w)\mathrm{d}x+\frac{\lambda }{2}{\int }_{\Omega }|w-f{|}^2\mathrm{d}x\to \min$$

$$F_{\mu }(\xi ):={\Phi }_{\mu }(|\xi |),\xi \in {\mathbb{R}}^2,$$

$${\Phi }_{\mu }(t):={\int }_0^t{\int }_0^s(1+r{)}^{-\mu }\mathrm{d}r\mathrm{d}s,t\ge 0,$$

$$\{\begin{array}{rl}& {\Phi }_{\mu }(t)=\frac{1}{\mu -1}t+\frac{1}{\mu -1}\frac{1}{\mu -2}(t+1{)}^{-\mu +2}-\frac{1}{\mu -1}\frac{1}{\mu -2},\mu \ne 2,\\ & {\Phi }_2(t)=t-\ln (1+t),t\ge 0.\end{array}$$

$$c_{1}\big{(}|\xi|-1\big{)}\leq F_{\mu}(\xi)\leq c_{2}\big{(}|\xi|+1\big{)}\,,\ \xi\in\mathbb{R}^{2}\,,$$

$$c_{3}\big{(}1+|\xi|\big{)}^{-\mu}|\eta|^{2}\leq D^{2}F_{\mu}(\xi)(\eta,\eta)\leq c_{4}\big{(}1+|\xi|\big{)}^{-1}|\eta|^{2}$$

$$\underset{\mu \to \infty }{\lim }\left(\mu -1\right)F_{\mu }\left(\xi \right)=\left|\xi \right|,\xi \in {\mathbb{R}}^2,$$

$$p,\mu \in (1,\infty )$$

$$c_5{\left(1+|\xi |\right)}^{-\mu }|\eta {|}^2\le D^{2}F(\xi )(\eta ,\eta )\le c_6{\left(1+|\xi |\right)}^{p-2}|\eta {|}^2,$$

$$c_7|\xi |-{\tilde{c}}_7\le F(\xi )\le c_8\left(|\xi {|}^p+1\right).$$

$$F(\xi ):={\int }_0^{|\xi |}{\int }_0^s(\epsilon +r{)}^{\phi (r)-2}\mathrm{d}r\mathrm{d}s,\xi \in {\mathbb{R}}^2,$$

$$\phi :[0,\infty )\to [2-\mu ,p],\phi (0)=p,\underset{r\to \infty }{\lim }\phi (r)=2-\mu .$$

$$J[w]:={\int }_{\Omega }F(\nabla w)\mathrm{d}x+\frac{\lambda }{2}{\int }_{\Omega -D}|w-f{|}^2\mathrm{d}x,$$

$$0\le {\mathcal{L}}^2(D)<{\mathcal{L}}^2(\Omega ),$$

$$\mu ,p<2.$$

$$J[w]\to \min \text{mbox}in{W}^{1,1}(\Omega )$$

$$c_9|\nabla w{|}^s-{\tilde{c}}_9\le F(\nabla w)\le c_{10}\left({\left|\nabla w\right|}^q+1\right)$$

$$F(\nabla w)=F({\partial }_{1}w{\partial }_{2}w)=F_1({\partial }_{1}w)+F_2({\partial }_{2}w)$$

$$K[w]:={\int }_{\Omega }F(\nabla w)\mathrm{d}x+{\int }_{\Omega -D}\rho \left(|w-f|\right)\mathrm{d}x,$$

$$1<\mu <3/2,$$

$$1<p<\mu .$$

$$K[w]\to \min \text{ in }{W}^{1,1}(\Omega )$$

$$\left|D^{2}F(\xi )\right|{\left|\xi \right|}^2\le c_{11}\left(F\left(\xi \right)+1\right),\xi \in {\mathbb{R}}^2,$$

$$F(\xi )={\int }_0^1(1-t)D^{2}F(t\xi )(\xi ,\xi )\mathrm{d}t,\xi \in {\mathbb{R}}^2.$$

$$c_{12}{\int }_0^1(1-t)(1+t|\xi |{)}^{-\mu }\mathrm{d}t|\xi {|}^2\le F(\xi )\le c_{13}{\int }_0^1(1-t)(1+t|\xi |{)}^{p-2}\mathrm{d}t|\xi {|}^2,$$

$$c_{12}{\int }_0^1(1-t)(1+t|\xi |{)}^{-\mu }\mathrm{d}t|\xi {|}^2\ge c_{12}{\int }_0^{1/|\xi |}(1-t)(1+t|\xi |{)}^{-\mu }\mathrm{d}t|\xi {|}^2$$

$$\ge c_{12}{\int }_0^{1/|\xi |}(1-t)(1+1{)}^{-\mu }\mathrm{d}t|\xi {|}^2\ge c_{14}{\int }_{1/2|\xi |}^{1/|\xi |}(1-t)\mathrm{d}t|\xi {|}^2$$

$$\ge c_{14}{\int }_{1/2|\xi |}^{1/|\xi |}\left(1-\frac{1}{|\xi |}\right)\mathrm{d}t|\xi {|}^2\ge c_{14}{\int }_{1/2|\xi |}^{1/|\xi |}\frac{1}{2}\mathrm{d}t|\xi {|}^2=c_{15}|\xi |,$$

$$0\le u\le 1\text{mbox}a.e.on\Omega .$$

Full text

AMS-classification: 49N60, 49J45, 49Q20

Keywords: variational problems with nonstandard growth, TV-regularization, denoising and inpainting of images

1 Introduction

In 1992 Rudin, Osher and Fatemi proposed (compare [1]) to study the variational problem

[TABLE]

as a model for the restoration of a noisy greyscale image $f$. In this setting (and throughout our paper) $\Omega$ is a bounded Lipschitz region in $\mathbb{R}^{2}$, the function $f:\Omega\to\mathbb{R}$ represents the noisy data, for which we assume

[TABLE]

and $\lambda>0$ denotes a parameter being under our disposal. As a matter of fact, problem (1.1) has to be discussed in the space $BV(\Omega)$ of functions with finite total variation (see, e.g., [2] or [3] for a definition and further properties of this class) admitting a unique solution $u$ which in addition satisfies (1.2). From the analytical point of view, the functional $I_{1}$ from (1.1) does not behave very nicely: the energy density $|\nabla w|$ is neither differentiable nor strictly convex (“elliptic”) so that no additional information on the minimizer $u$ are available. One common alternative used in the variational approach towards the denoising of images is to replace (1.1) by

[TABLE]

for some power $p>1$, where the choice $p=2$ already occurs in the work of Arsenin and Tikhonov [4], we refer to the monograph [5] for more information on the subject including references. The natural space for problem (1.3) is the Sobolev class $W^{1,p}(\Omega)$ (compare [6] for details), and from nowadays standard results on nonlinear elliptic equations (see the references stated in Chapter 3.2 of [7]) going back to e.g. Uralt’seva, Uhlenbeck, Evans, Di Benedetto and many other prominent authors it follows that the unique solution of problem (1.3) is at least of class $C^{1}$ on the interior of the domain $\Omega$. However, from the point of view of applications, a high degree of regularity of the minimizer is not always favourable (“effect of oversmoothing”), which means that in certain cases one should discuss a linear growth model but with better ellipticity properties in comparison to the functional $I_{1}$. This is the subject of the papers [8, 9, 10], in which we studied the problem

[TABLE]

(including even inpainting) with density

[TABLE]

the function $\Phi_{\mu}:[0,\infty)\to[0,\infty)$ being defined through

[TABLE]

with explicit formula

[TABLE]

In the case $\mu>1$ the density $F_{\mu}$ is of linear growth in the sense that

[TABLE]

with constants $c_{1},c_{2}>0$. Formally we can also consider values $\mu<1$, but then (1.4) reduces to (1.3) for the choice $p=2-\mu$. The density $F_{\mu}$ is of class $C^{2}$ satisfying in case $\mu>1$ the condition of $\mu$-ellipticity, i.e.

[TABLE]

with $c_{3},c_{4}>0$ and for all $\xi,\eta\in\mathbb{R}^{2}$. From (1.7) it follows

[TABLE]

and (1.9) together with (1.10) shows that “$(1-\mu)F_{\mu}(\nabla w)$” is a reasonable approximation of the TV-density “$|\nabla w|$” occurring in problem (1.1). Moreover, it turns out that the degree of regularity of the solution $u_{\mu}\in\mbox{BV}(\Omega)$ of problem (1.4) can be controlled in terms of the parameter $\mu$. Precisely it holds

THEOREM 1.1**.**

Let $f$ satisfy (1.2), fix $\mu>1$ and define $F_{\mu}$ according to (1.5), (1.6).

1. a)

If $\mu<2$, then the solution $u_{\mu}$ of (1.4) belongs to the Sobolev space $W^{1,1}(\Omega)$ and is of class $C^{1}$ in the interior of $\Omega$. 2. b)

In case $\mu>2$ there are simple examples of data $f$ for which $u_{\mu}\not\in W^{1,1}(\Omega)$.

For part a) we refer to [8, 9, 10, 11, 12], a discussion of b) even for the one-dimensional case $\Omega=(0,1)$ can be found in [13]. Up to now, all our energy functionals are of uniform power growth in the sense that the regularizing part involving $\nabla w$ can be estimated from above and below by the quantity $\int_{\Omega}|\nabla w|^{q}\,\mathrm{d}x$ for some power $q\in[1,\infty)$, and the purpose of the present paper is to introduce - at least to some extend - energy functionals and densities $F$, which allow some flexibility of the growth rate, which means that the growth rate of $F(\nabla w)$ can be prescribed in terms of $|\nabla w|$. To be precise, we consider a density $F:\mathbb{R}^{2}\to[0,\infty)$ of class $C^{2}$ satisfying $F(0)=0$ and $DF(0)=0$. For numbers $c_{5},c_{6}>0$ and for exponents

[TABLE]

we assume the validity of $(\eta,\xi\in\mathbb{R}^{2})$

[TABLE]

and in Lemma 2.1 we will show that (1.12) yields the growth estimate $(c_{7},\tilde{c}_{7},c_{8}>0)$

[TABLE]

The reader should note that (1.12) implies (1.9), if we allow the choice $p=1$. An example of a density $F$ with (1.12) is given by $(\varepsilon>0)$

[TABLE]

for a continuous and decreasing function

[TABLE]

A discussion of (1.14) together with further examples can be found in Section 5. Assuming (1.12) we then look at the variational problem

[TABLE]

where $D$ is a measurable subset of $\Omega$ such that

[TABLE]

i.e. we study an inpainting problem combined with simultaneous denoising, where $D$ is the inpainting region and the choice $D=\emptyset$ corresponds to the case of pure denoising. We have the following results:

THEOREM 1.2**.**

Let (1.2), (1.11) and (1.12) hold together with (1.16). Assume in addition that

[TABLE]

Then the variational problem

[TABLE]

with $J$ defined in (1.15) admits a unique solution $u$. This solution additionally satisfies $0\leq u\leq 1$ a.e. on $\Omega$ as well as $u\in W^{1,s}_{\mathrm{loc}}(\Omega)$ for any finite $s$.

REMARK 1.1**.**

Once having established the local higher integrability result $|\nabla u|\in L^{s}_{\mathrm{loc}}(\Omega),s<\infty$, we think that actually $u\in C^{1,\alpha}(\Omega)$, $0<\alpha<1$, can be deduced along similar lines as in [11], where densities $F$ satisfying (1.9) for some exponent $\mu\in(1,2)$ are considered.

REMARK 1.2**.**

Energy densities $F$, for which

[TABLE]

holds or for which an appropriate variant of (1.12) is true, have been extensively discussed for instance in the papers [14, 15, 16, 17, 18, 19, 20, 21, 22] dealing even with the higher-dimensional case including vector-valued functions. Roughly speaking it is shown in the above mentioned papers and the references quoted therein, that (1.19) provides some additional regularity of (local) minimizers, provided $s>1$ and $q$ is not too far away from $s$, we refer to [23] for a survey. Recalling that (1.12) implies (1.13), Theorem 1.2 covers the case “$s=1$”, and (1.17) expresses the fact that the upper bound $p$ satisfies “$p<2s$”. Note that the latter requirement turns out to be a sufficient condition for the regularity of minimizers in the setting of [21].

REMARK 1.3**.**

Variational problems of mixed linear/superlinear growth are the subject of Section 6 in [23]. Here the density $F$ is of splitting form in the sense that

[TABLE]

with $F_{1}$ growing linearly in $|\partial_{1}w|$, whereas $F_{2}(\partial_{2}w)$ behaves as $|\partial_{2}w|^{p}$ with power $p>1$. From the point of view of image restoration condition (1.20) seems to be unnatural, however, if $F_{1}$ satisfies (1.9) with $\mu\in(1,2)$ and if $p<2$, then regularity results are available, thus our hypothesis (1.17) naturally occurs in the splitting case (1.20).

Next let $\rho:[0,\infty)\to[0,\infty)$ denote a function of class $C^{1}$ being strictly increasing and strictly convex, e.g. $\rho(t)=\sqrt{1+t^{2}}-1$, and let

[TABLE]

which means that we consider more general data terms.

THEOREM 1.3**.**

With $\rho$ from above let $f$, $F$ and $D$ satisfy (1.2), (1.12) and (1.16), respectively, and assume in addition that $\displaystyle\limsup_{t\to\infty}\frac{\rho(t)}{t^{m}}<\infty$ for some $m\geq 1$. Moreover, let

[TABLE]

Then the variational problem

[TABLE]

with $K$ from (1.21) has a unique solution $u$. It holds $0\leq u\leq 1$ a.e. on $\Omega$, moreover, $|\nabla u|$ is in $L^{s}_{\mathrm{loc}}(\Omega)$ for any finite $s$. If the density $F$ is balanced in the sense that

[TABLE]

holds for some constant, then (1.23) can be replaced by the requirement $p\in(1,2)$ (compare (1.17)).

REMARK 1.4**.**

We conjecture that in the balanced case (1.25) the results of Theorem 1.2 and 1.3 extend to any exponent $p\geq 2$, we refer to Remark 3.1.

Our paper is organized as follows: in Section 2 we collect some preliminary material and discuss regularized problems approximating (1.18) and (1.24). Section 3 is devoted to the proof of Theorem 1.2, and Theorem 1.3 is established in Section 4. Finally, in Section 5 we present some examples of densities $F$ satisfying (1.12) including the model from (1.14).

2 Some preliminary results and discussion of regularized problems

We start with a growth estimate for densities $F$ satisfying (1.12).

Lemma 2.1**.**

Suppose that we have the ellipticity condition (1.12) for $F:\mathbb{R}^{2}\to[0,\infty)$ with exponents $p,\mu$ according to (1.11). Then $F$ is of $(1,p)$-growth in the sense of inequality (1.13).

Proof. We just consider the case $p\geq 2$. For $p<2$ the following arguments can be easily adjusted. We recall that $F$ should satisfy $F(0)=0$, $DF(0)=0$, thus we obtain from Taylor’s theorem (applied to $t\mapsto F(t\xi)$)

[TABLE]

Applying (1.12) to the r.h.s. of (2.1) we find

[TABLE]

and from $(1+t|\xi|)^{p-2}|\xi|^{2}\leq(1+|\xi|)^{p}$ (in case $p\geq 2$) we immediately deduce the second inequality in (1.13). If $|\xi|\leq 2$, then the first inequality in (1.13) is obvious by an appropriate choice of $c_{7},\tilde{c}_{7}>0$. In case $|\xi|\geq 2$ we observe for the l.h.s. of (2.2)

[TABLE]

thus the first inequality of (1.13) extends to the case $|\xi|\geq 2$ after adjusting $c_{7},\tilde{c}_{7}$. ∎

REMARK 2.1**.**

The requirement $DF(0)=0$ is essential for deducing the lower bound on $F$ stated in (1.13) from the condition of $\mu$-ellipticity, i.e. from the first inequality in (1.12).

Lemma 2.2**.**

Under the conditions on the data stated in Theorem 1.2 and 1.3, respectively, but for arbitrary choices of $p,\mu\in(1,\infty)$, the variational problems (1.18) and (1.24) admit at most one solution $u\in W^{1,1,}(\Omega)$. We have

[TABLE]

Proof. From “strict convexity” (for the density $F$ this property follows from the first inequality in (1.12)) we get

[TABLE]

for minimizers $u,v\in W^{1,1}(\Omega)$. But then $u=v$ is a consequence of (1.16). Replacing $u$ by $\min(u,1)$ and $\max(u,0)$ we see by an elementary calculation (compare, e.g., [9]) that (2.3) holds for the minimizer $u$, since otherwise we could decrease the energy. ∎

During the proofs of Theorem 1.2 and 1.3 we will essentially benefit from

Lemma 2.3**.**

Suppose that we are in the situation of Theorem 1.2 or 1.3, where here we allow in both cases exponents $p\in(1,2)$ and $\mu\in(1,\infty)$. For $\delta>0$ let $u_{\delta}\in W^{1,2}(\Omega)$ denote the solution of either

[TABLE]

or

[TABLE]

with $J$ and $K$ from (1.15) and (1.21), respectively. It holds:

1. i)

$0\leq u_{\delta}\leq 1$*  a.e.  on $\Omega$.* 2. ii)

The functions $u_{\delta}$ are of class $W^{2,2}_{\mathrm{loc}}(\Omega)\cap W^{1,\infty}_{\mathrm{loc}}(\Omega)$. 3. iii)

We have the uniform bound $\displaystyle\sup_{\delta>0}\|u_{\delta}\|_{W^{1,1}(\Omega)}<\infty$. 4. iv)

Suppose that we can find an exponent $q>1$ such that for each subdomain $\Omega^{*}\Subset\Omega$

[TABLE]

Then $u_{\delta}\to:u$ in $L^{1}(\Omega)\cap W^{1,q}_{\mathrm{loc}}(\Omega)$ as $\delta\to 0$ for a function $u\in W^{1,1}(\Omega)$, and $u$ solves the variational problem (1.18), respectively (1.24).

Proof. i) follows as inequality (2.3) in Lemma 2.2, ii) is immediate from elliptic regularity theory, and iii) is a consequence of the first inequality in (1.13). Let us discuss iv): from i), iii) and assumption (2.4) we deduce the existence of $u\in\mbox{BV}(\Omega)\cap W^{1,q}_{\mathrm{loc}}(\Omega)\subset W^{1,1}(\Omega)$ such that

[TABLE]

(at least for a subsequence) as $\delta\to 0$. From De Giorgi’s theorem on lower semicontinuity (see, e.g., [24] Theorem 2.3, p.18) we see that (2.5) and (2.6) yield

[TABLE]

if we are in the situation of Theorem 1.2, whereas

[TABLE]

in the setting of Theorem 1.3. Since for $v\in W^{1,2}(\Omega)$ it holds

[TABLE]

we obtain from (2.7) (recall the definition of $J_{\delta}$ in $(1.18)_{\delta}$)

[TABLE]

and by approximation $(W^{1,2}(\Omega)\ni v_{k}\to v$ in $W^{1,1}(\Omega))$, inequality (2.9) extends to $v\in W^{1,1}(\Omega)$. If the $u_{\delta}$ are the solutions of problem $(1.24)_{\delta}$, then by the same arguments it follows

[TABLE]

Consider $v\in W^{1,1}(\Omega)$. In case $K[v]=+\infty$, i.e.

[TABLE]

there is nothing to prove. In the other case, due to the growth of $\rho$ at infinity and by (1.2), we see that $v$ is in the space $L^{m}(\Omega-D)$ and according to [25], Lemma 2.1, we find a sequence $v_{k}\in C^{\infty}(\overline{\Omega})$ such that

[TABLE]

as $k\to\infty$, hence $K[v_{k}]\to K[v]$, and since $K[u]\leq K[v_{k}]$ by (2.10), we finally have shown that $u$ solves (1.24). ∎

3 Proof of Theorem 1.2

In this section we assume that all the hypotheses of Theorem 1.2 are valid and define $u_{\delta}$ as in Lemma 2.3 as the unique solution of problem $(1.18)_{\delta}$. Let $F_{\delta}(\xi):=\frac{\delta}{2}|\xi|^{2}+F(\xi),\ \xi\in\mathbb{R}^{2}$. For $\eta\in C^{1}_{0}(\Omega)$ with $0\leq\eta\leq 1$ we have (by passing to the differentiated version of the Euler equation associated to $(1.18)_{\delta}$ and by quoting Lemma 2.3 ii))

[TABLE]

where here and it what follows the sum in taken w.r.t. $\alpha=1,2$. It holds

[TABLE]

[TABLE]

where we have used (1.2) as well as Lemma 2.3 i), $c_{k}$ denoting a positive constant independent of $\delta$. Recalling in addition Lemma 2.3 iii) we get from (3.1)

[TABLE]

Applying the Cauchy-Schwarz inequality to the bilinear form $D^{2}F_{\delta}(\nabla u_{\delta})$ and using Young’s inequality, the estimate (3) yields

[TABLE]

hence using (1.12) for $D^{2}F$ (dropping the $\delta$-term on the l.h.s.)

[TABLE]

We remark the validity of $\displaystyle\sup_{\delta>0}\,\delta\int_{\Omega}\left|\nabla u_{\delta}\right|^{2}\,\mathrm{d}x<\infty$ and assume w.l.g. $\|\nabla\eta\|_{L^{\infty}(\Omega)}\geq 1$. Then we obtain

[TABLE]

On the r.h.s. of (3) we use Young’s inequality twice recalling (1.17) and (1.11):

[TABLE]

Inserting these estimates into (3), choosing $\eta$ such that $\eta\equiv 1$ on $B_{r_{1}}(x_{0}),\eta\equiv 0$ outside $B_{r_{2}}(x_{0}),B_{r_{1}}(x_{0})\subset B_{r_{2}}(x_{0})\Subset\Omega$, we obtain after appropriate choice of $\varepsilon$ and $\tau$

[TABLE]

where for the moment we just neglect $\int_{\Omega}\eta^{2}(1+|\nabla u_{\delta}|)^{-\mu}|\nabla^{2}u_{\delta}|^{2}\,\mathrm{d}x$ and $\alpha$ denotes a suitable positive number. Applying Lemma 3.1, p.161, from [24] to estimate (3.4) we find that (2.4) from Lemma 2.3 holds with the choice $q=2$, and we can quote iv) of Lemma 2.3 yielding a unique $W^{1,1}(\Omega)$- solution $u$ of (1.18).

Going back to (3), recalling the estimates stated after (3) and applying our bound (2.4) valid for $q=2$, it follows

[TABLE]

for any $\Omega^{\ast}\Subset\Omega$, thus ($\varphi_{\delta}:=(1+|\nabla u_{\delta}|)^{1-\mu/2}$)

[TABLE]

which by Sobolev’s theorem implies

[TABLE]

for any $s<\infty$. This proves the last claim of Theorem 1.2. ∎

REMARK 3.1**.**

Suppose that $F$ satisfies the condition (1.25). In this case we estimate

[TABLE]

and observe $\displaystyle\sup_{\delta>0}\int_{\Omega}F_{\delta}(\nabla u_{\delta})\,\mathrm{d}x<\infty$. Thus we can replace $\int_{\operatorname{spt}\eta}\left(1+|\nabla u_{\delta}|\right)^{p}\,\mathrm{d}x$ in (3) through a constant ending up with

[TABLE]

hence we obtain (3.5) just assuming $\mu\in(1,2)$. Thus the bound (1.17) imposed on $p$ at this stage does not enter, however during our proof we work with the quadratic regularization $(1.18)_{\delta}$, which requires $p\leq 2$. In other words: under the assumption (1.25) the claims of Theorem 1.2 extend to exponents $p>2$ (keeping the bound $1<\mu<2$) and a proof can be carried out by working with the regularization

[TABLE]

for some exponent $\overline{p}>p$. We leave the details to the reader.

4 Proof of Theorem 1.3

Let the assumptions of Theorem 1.3 hold. In place of equation (3.1) we have

[TABLE]

where $u_{\delta}$ is the solution of problem $(1.24)_{\delta}$ (see Lemma 2.3). From (1.2) and Lemma 2.3 i) it follows

[TABLE]

and clearly (recall Lemma 2.3 iii))

[TABLE]

where we use the symbol $c_{k}(\eta)$ to denote constants proportional to $\|\nabla\eta\|^{\alpha}_{L^{\infty}(\Omega)}$ for some positive exponent $\alpha$. Applying Young’s inequality to the integral on the r.h.s. of (4.2) and discussing the l.h.s. of (4) as done after (3) we find

[TABLE]

We specify $\eta$ as in Section 3 and let

[TABLE]

Then (4) shows (with suitable $\alpha_{1}>0$)

[TABLE]

Next we observe (quoting Sobolev’s inequality)

[TABLE]

where we have used that $\sup_{\delta>0}\int_{\Omega}\Psi_{\delta}\,\mathrm{d}x<\infty$ on account of Lemma 2.3 iii). We discuss the remaining integral on the r.h.s. of (4.5) observing that $\Psi_{\delta}=\varphi^{\mu/(2-\mu)}_{\delta}$ and using Hölder’s inequality:

[TABLE]

We have

[TABLE]

with exponent $2\mu-2\in(0,1)$, which follows from (1.22). Quoting Lemma 2.3 iii) one more time, another application of Hölder’s inequality gives (for some $\alpha_{2}>0$)

[TABLE]

We insert (4.6) into (4.5) giving the bound

[TABLE]

With (4.7) we return to (4.4) and assume that the radius $r_{2}$ is sufficiently small, thus

[TABLE]

Up to now we have not used our hypothesis (1.23), which enters next:

[TABLE]

where in the last estimate we applied Hölder’s inequality and use the smallness of $r_{2}$ to get the factor $1/2$. As outlined after (3.4) we deduce (2.4) with value $q:=\mu$. Moreover, using this information in (4.4), we see

[TABLE]

for any subdomain $\Omega^{*}\Subset\Omega$, thus (3.5) holds, and we get all the results of Theorem 1.3 as described in Section 3, where for the balancing case we refer to Remark 3.1. ∎

5 Examples

In this section we focus on energy densities depending on the modulus of $\nabla u$, a situation for which the following observations are helpful.

Proposition 5.1**.**

Let $g:[0,\infty)\to[0,\infty)$ denote a $C^{2}$-function for which $g(0)=g^{\prime}(0)=0,\ g^{\prime\prime}\geq 0$. Then

[TABLE]

is a convex function of class $C^{2}$ for which $G(0)=0$, $DG(0)=0$ and

[TABLE]

Proof. We just note that (5.2) follows from the formula

[TABLE]

∎

If $G:\mathbb{R}^{2}\to[0,\infty)$ is a non-negative function of class $C^{2}$, we recall the balancing condition (see (1.25)):

[TABLE]

Proposition 5.2**.**

Let $g$ satisfy the assumptions of Proposition 5.1. Assume further that

[TABLE]

Then $G$ from (5.1) satisfies (5.3).

Proof. This is an immediate consequence of (5.2) and (5.4). ∎

Proposition 5.3**.**

If $g$ is a function as in Proposition 5.1 such that

[TABLE]

and if $g$ satisfies the $(\Delta 2)$-condition, i.e.

[TABLE]

then inequality (5.4) holds. Thus $G(\xi):=g(|\xi|)$, $\xi\in\mathbb{R}^{2}$, is balanced in the sense of (5.3).

Proof. We have (recalling $g(0)=0$ as well as $g^{\prime}(t)\geq 0$)

[TABLE]

where the last inequality follows from the fact that $g^{\prime}$ is increasing on account of our hypothesis $g^{\prime\prime}\geq 0$. Thus we get

[TABLE]

(see (5.6)), and together with (5.5) we arrive at (5.4). ∎

All our energy densities “$G(\nabla u)$” discussed below are of the principal form (compare (1.6) and (1.14))

[TABLE]

for a continuous function $\omega:[0,\infty)\to[0,\infty)$ such that $\omega(s)>0$ for $s>0$. Note that $g$ is strictly increasing and strictly convex implying the strict convexity of $G$ on $\mathbb{R}^{2}$.

Example 5.1**.**

Consider a continuous function $\eta:[0,\infty)\to[0,1]$ and define $G$ according to (5.7) with the choice

[TABLE]

for exponents $p,\mu\in(1,\infty)$.

Proposition 5.4**.**

The density $G$ satisfies (1.12).

Proof. It holds on account of $\eta(t)\in[0,1]$ and $(1+t)^{-\mu}\leq(1+t)^{p-2}$ for any $t\geq 0$

[TABLE]

In case $t\geq 1$ we observe $\{\ldots\}\leq(2t)^{p-1}$, thus $\frac{1}{t}g^{\prime}(t)\leq c_{53}\ t^{p-2}\leq c_{54}(1+t)^{p-2}$, whereas in case $t\leq 1$ we use

[TABLE]

for some $T\in(0,1)$, hence

[TABLE]

and the second inequality in (5.2) gives the upper bound

[TABLE]

With analogous calculations we obtain a lower bound:

[TABLE]

Case 1: $t\geq t(\mu)$ $(\geq 1$ sufficiently large). Then we have after appropriate choice of $t(\mu)$

[TABLE]

Case 2: $t\leq t(\mu)$. Here we observe

[TABLE]

for a suitable $\widetilde{T}\in(0,t(\mu))$, hence

[TABLE]

Recalling (5.9) and (5.2), the above estimates imply (1.12) for our density $G$. ∎

REMARK 5.1**.**

An equivalent form of (5.8) is given by

[TABLE]

for a continuous function $\Theta:[0,\infty)\to[0,\infty)$ such that $\displaystyle 1\leq\Theta(t)\leq(1+t)^{\mu+p-2},\ t\geq 0$.

In fact, if $\eta$ is given, let

[TABLE]

and if we start from $\widetilde{(5.8)}$ we obtain (5.8) by defining

[TABLE]

The density $G$ defined in (5.7) with $\omega$ as in (5.8) in general does not satisfy the balancing condition (5.3): consider $\eta:[0,\infty)\to[0,1]$ as indicated in the picture below:

Here $\varepsilon_{k}$ denotes a suitable sequence going to zero, we let $\eta\equiv 1$ on each interval $[k+\varepsilon_{k},k+1-\varepsilon_{k+1}]$ with linear interpolation on $[k-\varepsilon_{k},k+\varepsilon_{k}]$ such that $\eta(k)=0$ for each $k$. Then it holds for $t\in\mathbb{N}$

[TABLE]

and at the same time (after appropriate choice of $\varepsilon_{k}$)

[TABLE]

for $s\geq 0$ sufficiently large, hence (5.4) is violated. We discuss (5.10): it holds

[TABLE]

For simplicity let us assume $p\leq 2$. Then $\{\ldots\}\leq 1$, hence(compare (1.5) - (1.8))

[TABLE]

and we obtain (5.10) from the requirement that $\sum^{\infty}_{k=1}\varepsilon_{k}<\infty$. In the case $p>2$ a slight modification is necessary still leading to (5.10). In addition, we can choose different functions $\eta$ to get $\lim_{t\to\infty}g(t)/t^{q}>0$ for a given number $q\in(1,p)$.

REMARK 5.2**.**

As a matter of fact our previous considerations extend to densities

[TABLE]

with positive numbers $\varepsilon_{1},\varepsilon_{2}$ and with weight-function $\eta$ as in (5.8).

Example 5.2**.**

We let

[TABLE]

with $\varepsilon>0$ and for a continuous and decreasing function

[TABLE]

with exponents $p,\mu>1$. Note that (5.11), (5.12) can be seen as an approximation of the density $G(\nabla u)=|\nabla u|^{p(|\nabla u|)}$ where $p(|\nabla u|)$ decreases from $p$ to $1$ as $|\nabla u|$ ranges from [math] to $\infty$, introduced by Blomgren, Chan and Mulet [26] for $p=2$.

Proposition 5.5**.**

The density $G$ from (5.11) with $\rho$ defined in (5.12) satisfies the ellipticity condition (1.12), moreover, the balancing inequality (5.3) holds.

Proof. W.l.o.g. we let $\varepsilon=1$ and observe for any $t\geq 0$

[TABLE]

moreover, it holds

[TABLE]

we refer to the proof of Proposition 5.4. Thus (1.12) follows from Proposition 5.1. Next we discuss (5.4) for $g$ by referring to Proposition 5.3: we have

[TABLE]

In case $p\leq 2$ we get by the properties of $\rho$

[TABLE]

(note: $\rho(2r)\leq\rho(r)$), hence

[TABLE]

If the value of $p>1$ is arbitrary, we write

[TABLE]

and use the fact that

[TABLE]

thus $(1+2r)^{\rho(2r)-2}\leq c_{65}(1+r)^{\rho(2r)-2}$ , and by recalling $\rho(2r)\leq\rho(r)$ we obtain as before inequality (5.6) with a suitable constant. It remains to check (5.5): we have

[TABLE]

since $\rho$ decreases. Writing

[TABLE]

and observing that

[TABLE]

we see that $g^{\prime}(t)\geq c_{66}\ \frac{t}{2}(1+t)^{\rho(t)-2}$ and (5.5) is established. ∎