A Relaxed Kačanov iteration for the p-poisson problem

In this paper we introduce and analyze an iteratively re-weighted algorithm, that allows to approximate the weak solution of the p-Poisson problem for 1<p⩽2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 < p \leqslant 2$$\end{document} by iteratively solving a sequence of linear elliptic problems. The algorithm can be interpreted as a relaxed Kačanov iteration, as so-called in the specific literature of the numerical solution of quasi-linear equations. The main contribution of the paper is proving that the algorithm converges at least with an algebraic rate.


Introduction
In this paper we approach the numerical solution of the p-Poisson problem − div(|∇u| p−2 ∇u) = f in , u = 0 on ∂ , (1.1) where ⊂ R d is open and bounded and 1 < p < ∞. The solution might be scalar or vector-valued. 1 Nonlinear problems of this type appear in many applications, e.g. non-Newtonian fluid theory [21], turbulent flow of a gas in porous media, glaciology or plastic modeling. Moreover, the p-Laplacian has a similar model character for nonlinear problems as the ordinary Laplace operator for linear problems; see [22] for an introduction.
As usual we are looking for the weak solution of (1.1). In particular, we are searching for a function u ∈ W 1, p 0 ( ) such that |∇u| p−2 ∇u · ∇ξ dx = f , ξ ∀ξ ∈ W 1, p 0 ( ), (1.2) where in the most general case f ∈ (W 1, p 0 ( )) * . It is well-known that the solution is unique and coincides with the minimizer of the energy J : W 1, p Due to the nonlinearity of the problem it is harder to obtain efficient numerical solutions of this problem with a guaranteed performance. Our goal is to construct solutions of (1.2) by means of a numerically accessible algorithm. In particular, we construct an iterative algorithm that approximates solutions of (1.2), where in each step only a linear elliptic problem has to be solved. Primarily, we focus here on the iteration on the infinite dimensional space W 1, p 0 ( ). However, the same algorithm will immediately apply also to discretized versions of the p-Poisson problem, e.g., by means of finite elements or wavelets. This approach would coincide with the one adopted, for instance, in [5] of first finding an iteration on the infinite-dimensional solution space and then discretizing in space. We will consider in subsequent work the effect of the discretization and its adaptation to error estimators.
In this paper we also restrict ourselves to the case p ∈ (1, 2], since we are in particular interested in relatively small values of p, also because the case of p > 2 is already addressed to a certain extent in [5]. We will see, e.g., in Example 20 that our algorithm actually only works properly for the range of p ∈ (1,2]. Coming from the weak formulation (1.2) one can interpret the problem as a weighted Poisson problem a p−2 ∇u · ∇ξ dx = f , ξ ∀ξ ∈ W 1,2 0 ( ) (1.4) for the given f , where a : → R and a = |∇u|. This suggests to iteratively calculate for a given function v n the new iterate v n+1 as the solution of |∇v n | p−2 ∇v n+1 · ∇ξ dx = f , ξ ∀ξ ∈ W 1,2 0 ( ).
The advantage of this step is that the calculation of v n+1 only requires solving a linear problem. This allows invoking relatively standard appraoches to discretize this step and solve it numerically with guaranteed performances. The problem with this approach, however, is that the weighted Poisson problem is only well posed if a is bounded from above and from below away from zero. However, the weight |∇v n | p−2 may be degenerating, at points where |∇u| = 0 or |∇u| = ∞.
To overcome this problem we will use a relaxation arguments. Therefore, we introduce in our algorithm two relaxation parameters − , + ∈ (0, ∞) with − + that ensure that the weight is truncated properly from below and above. In particular, we replace a by its truncation − ∨ a ∧ + := max { − , min {a, + }}.
Note that this is just the (pointwise) closest point projection of a to the truncation interval [ − , + ]. The limits − 0 and + ∞ will recover the unrelaxed or original problem. We also write := [ − , + ] and interpret both as a pair { − , + } and as the truncation interval [ − , + ]. We will write → [0, ∞] as a short version of − 0 and + ∞. We will see later, see Corollary 15, that for f in the Lorentz space L d,1 ( ) the lower parameter − is the crucial one.
According to these considerations we propose the following algorithm:
This algorithm is not completely new in the realm of quasi-linear equations. Such an iterative linearization approach is in fact called the Kačanov method in [17,18] and we refer to those papers for additional references related to the history of this method for solving numerically quasi-linear equations. It was also proposed and analyzed to solve total variation minimization problems in image processing, which can be formally related to the 1-Laplace differential operator in [6,27].
Unfortunately, the results obtained in these aforementioned papers cannot be applied straightforwardly to justify the convergence of the Kačanov iteration for equations involving the p-Laplace operator. In particular, to obtain quantitative estimates of convergence with precise rates, as we do in this paper, one needs to employ several finer tools, which have been explored in, e.g., [2,11,12,24], precisely to handle singularities in nonlinear differential operators such as the p-Laplacian. In particular, the theory of N-functions, Orlicz spaces [19], shifted N-functions [11] and Lipschitz truncations, see [13] and [4] have been used systematically in the analysis of such nonlinear operators, allowing the development of a potential theory analogous to the one known of linear equations.
Besides these tools from nonlinear potential theory, the variational formulation of the algorithm, as introduced first in [6], and further used to analyze other related iteratively re-weighted least squares algorithms [10,16], offers the right framework for the analysis also of the p-Kačanov iteration.
Taking inspiration from [6,10], in Sect. 2 we provide the variational derivation of this algorithm based on the alternating minimization of a relaxed energy with two parameters.
If we apply the algorithm with fixed relaxation parameter independent on n, i.e. 0 < − + < ∞, then our iterates v n converge to the unique minimizer u of another one-parameter relaxed energy J . We study this limit in Sect. 4 and present (linear) exponential rates of convergence.
In Sect. 3 we study how the minimizers u of the relaxed energy J converge to the minimizer u of the original problem. This convergence can also be interpreted as a limit in the sense of -convergence [3,9]. Differently, e.g., from [6], we use a novel argument based on the Lipschitz truncation technique to establish a recovery sequence for the − lim sup. In particular, thanks to the finer tools mentioned above, we can go beyond a pure compactness argument as provided by the -limit and derive precise rates of convergence depending on .
Finally, in Sect. 5 we combine the estimates of the two previous sections to deduce an overall error analysis with algebraic rates.

Variational formulation of the algorithm
In this section we show that the algorithm can be deduced from an alternating minimization of a relaxed energy. Recall that 1 < p 2 throughout this article. Since the case p = 2 is just the standard Laplace problem, it suffices in the following to consider the case 1 < p < 2 only.
Let us introduce some standard notation. We use W 1, p ( ) and W 1, p 0 ( ) for the Sobolev space without and with zero boundary values. We use c for a generic positive constant whose value may change from line to line. We use f g for f c g. We also write f g for f g and g f . The most important feature of the algorithm is that it only needs to solve linear subproblems, which carry their own energy depending on the weight. Therefore, very much inspired by the work [6,10] and with appropriate adjustments, we extend the energy by an additional parameter a : → [0, ∞) such that the new functional is quadratic with respect to v. In particular, we define This energy is well-defined for all v ∈ W 1, p 0 ( ) and measurable a : This relaxed energy is convex with respect to (v, a). This follows from the fact that is nonnegative definite as a p−2 0 and det((∇ 2 β)(t, a)) = a 2 p−6 t 2 (2− p)( p −1) 0. Notice that in the latter lower bound we specifically used 1 < p 2.

Remark 1
If p > 2, then the relaxed energy J (v, a) is neither bounded from below nor convex with respect to a. Therefore, the algorithm derived below using the minimization with respect to a does not lead to a feasible problem for p > 2. See also Remark 21.
Note that J (v, a) (for fixed a) is quadratic with respect to v and a minimization with respect to v leads formally to the elliptic equation where ∨ denotes the maximum and ∧ the minimum, since ∂ ∂a This allows us to define for fixed = [ε − , ε + ] ⊂ [0, ∞] another relaxed energy This immediately implies that the relaxed energy J (v) is monotonically decreasing with respect to , i.e., an increasing interval in terms of inclusion decreases the energy J (v). This new relaxed energy J somehow "hides" the constrained minimization with respect to a. We can write J : W Note that 1 p t p κ (t) for all t 0 and 1 Since κ (t) Based on the above observations it is natural to iteratively minimize J (v, a) alternating between v and a. Certainly, we have also to increase the relaxation interval . Thus our algorithm reads as follows: Algorithm: The relaxed p-Kačanov algorithm (variational formulation) Data: Given f ∈ (W 1, p 0 ( )) * , v 0 ∈ W 1,2 0 ( ); Result: Approximate solution of the p-Poisson problem (1.2); Initialize: ε 0 = [ε 0,− , ε 0,+ ] ⊂ (0, ∞), n = 0; while desired accuracy is not achieved yet do Calculate a n by means of a n := arg min Calculate v n+1 by means of v n+1 := arg min Choose a new relaxation interval ε n+1 ⊃ n ; Increase n by 1; end This is just the algorithm given in the introduction written in different form.

Convergence in the relaxation parameter
In this section we show that the minimizers u of the relaxed energy J converge to the minimizer u of J for → [0, ∞] and derive an upper bound for the relaxation error. Since 0 ( ), as we have seen above. Certainly, there is a gap between the space W 1, p 0 ( ) and W 1,2 0 ( ). To close this gap we need a finer analysis of the energies, which requires the use of Orlicz spaces. We state in the following some standard results for these spaces, see for Example [19].
A function φ : R 0 → R is called an N-function if and only if there is a rightcontinuous, positive on the positive real line, and non-decreasing function φ : dτ . An Nfunction is said to satisfy the 2 -condition if and only if there is a constant c > 1 such that φ(2t) c φ(t). For an N-function satisfying the 2 -condition we define the Orlicz space to consist of those functions The function κ cannot be an N-function, since κ (0) = 0, . However, if we define then φ is actually an N-function. It can be verified that φ satisfies the 2 -condition with a constant independent of . Since φ (t) p−2 + t 2 for large t and is bounded, we have L φ ( ) L 2 ( ). However, the constant of the embedding L φ ( ) → L 2 ( ) depends on , so this equivalence is not of much use. Instead we use the chain of embeddings with constants independent of . This follows from the fact that the Simonenko indices of φ are within [ p, 2]. We refer the reader to, e.g., [28,Chapter 2] for the details.
Since φ is strictly convex and κ (t) = φ (t)+κ (0), the energy J admits a unique minimizer u ∈ W 1,φ 0 ( ) whose Euler-Lagrange equation is At this we used that Remark 2 Let us consider the special case ε + = ∞. Then, the derivative of the truncated function reads would lead to the so-called shifted N-function of 1 p t p , as introduced in more generality in [11], which has similar properties as our truncation functions. However, the version from this paper is more suitable for our energy relaxation, since it is closer to the original function 1 p t p on the truncation interval (the derivatives agree there). See the Appendix for more information on uniformly convex Orlicz functions and their shifted verions.
, which contains both u and u, it is natural to consider all energies J and J as functionals on W Let us recall that the goal of this section is to show that u converges to u in W 1, p 0 ( ). Since W 1, p 0 ( ) is uniformly convex, strong convergence is a consequence of weak convergence and norm convergence, or equivalently, in this case, energy convergence J (u ) → J (u). It is possible to show the weak convergence as well as that of the energy by means of -convergence. Indeed, we will see in Remark 11 that J → J in the sense of -convergence. However, we will derive in the following much stronger results that provide us with a precise rate of convergence for the energies. This energy convergence implies strong convergence of the sequence, see the proof of Corollary 10.
Let us turn to the convergence of the energies Since J is monotonically decreasing with respect to , it follows from the minimizing properties of u and u that Therefore, it suffices to prove the stronger claim In fact, we will later need this stronger estimate in the other sections.
It follows from the minimizing property of u that So it would be natural to estimate J (u) − J (u) in terms of and u. However, the solution u is unfortunately a priori only a W 1, p 0 -function, so J (u) might be infinity. Hence, we cannot assure that this difference is small. This is only possible if we assume higher regularity of u. In order to treat arbitrary right-hand sides f ∈ (W 1, p 0 ( )) * at this point, we have to use a much more subtle argument. For this we need a result from [13, Subsection 3.5] and [4, Theorem 2.7], which allows to change u on a small set such that it becomes a Lipschitz function. This technique is known as the Lipschitz truncation technique. Its origin goes back to [1]. As a tool we need the Hardy-Littlewood operator, e.g. [25], Then, there exists an approximation T λ v ∈ W 1,∞ 0 ( ) of v with the following properties: All our convergence results concerning the relaxation parameter ε are based on the following result, which shows how the energy relaxation depends on the truncation interval ε.

Theorem 5 The estimate
holds for all λ ε + /c 1 , where c 1 is the (hidden) constant from Theorem 4 (d).
Proof Let λ ε + /c 1 and let T λ u be the Lipschitz truncation of u. Then Using the minimizing property of u ε and the equation for u we get This, the previous estimate and Theorem 4 (e) imply This proves the claim.
Proof Due to (3.5) it suffices to prove J (u ε ) → J (u). Consider the right-hand side of (3.7) with λ := ε + /c 1 . The first term goes to zero as ε − → 0. Now consider the second term.
Before we continue we need the following natural quantities, see [11].
The following two lemmas are modifications of similar results of [11, Lemma 3] and [12,Lemma 16]. In fact, they follow from the properties of uniformly convex Orlicz functions; see Sect. B from the Appendix for more details.
where the constants can be chosen independently of ε.
Proof This follows directly from the uniform convexity of φ , see Lemma 3, Lemma 41 and Lemma 40.

Lemma 9
The following estimates hold for arbitrary v ∈ W 1,φ 0 ( ) and u ε being the minimizer of J : In particular, for the case where ε = [0, ∞] the statement actually implies also Proof This is just Lemma 42 applied to φ .
We are now prepared to show the convergence of minimizers u of J to u.
It follows from the shift-change-lema, see Corollary 44 or [12,Corollary 26], that for all δ > 0 there exists c δ > 0 such that So the properties of the Lipschitz truncation, see Theorem 4 (f), imply that the righthand side goes to zero as → [0, ∞]. Hence, T ε + /c 1 v is a recovery sequence of v. Moreover, J J , so the standard theory of -convergence [3,9] proves u u in W To our knowledge this is the first time that the Lipschitz truncation is used to construct a recovery sequence for the −lim sup in a -convergence argument related to energies on W 1, p 0 ( ).
Up to now, we discussed the convergence of u → u without any additional assumptions on the data f ∈ (W 1, p 0 ( )) * and the domain . If f is more regular and ∂ is suitably smooth, then we obtain specific rates for the convergence. The rates of convergence will follow from the regularity of ∇u in terms of the weak-L q spaces L q,∞ ( ), which consists of all functions v such that
To exemplify the consequences of Lemma 12 we combine it with the regularity results of [7,14]:

Convergence of the Kačanov-iteration
In this section we study the convergence of the Kačanov-iteration for fixed relaxation parameter = [ε − , ε + ]. In particular, for v 0 ∈ W 1,2 0 ( ) arbitrary we calculate recursively v n+1 by We will show that v n converges to the minimizer u of the relaxed energy J . In particular, we show exponential decay of the energy error J (v n ) − J (u ). The proof is based on the following estimate, proved below.

Theorem 18
There is a constant c K > 1 such that This theorem says that in each iteration we reduce the energy by a certain part of the remaining energy error. This implies As a direct consequence we will obtain the following exponential convergence result.

Corollary 19
There is a constant c K > 1 such that Let us get to the proof of Theorem 18.
Proof (Proof of Theorem 18) Using Lemma 9, the equation (3.3) for u ε , the equation (4.1) for v n+1 , and Young's inequality (see Remark 32) we get, for arbitrary γ > 0, Let us define For the first term I we calculate with the equation (4.1) for v n+1 To establish the inequality above, we used the fact that For the second term I I we use ε s , for any s, t 0, Lemma 8 and Lemma 9 to get Putting all estimates together we get

Example 20 (Peak function) Let
Then the minimizer of J is given by u(x) = 1 − |x|, which look like a peak. Since |∇u| ≡ 1, the factor |∇u| p−2 in the p-Laplace operator does not appear for the minimizer. So in this case u also minimizes every J as long as ε − 1 and ε + 1. This follows from Let us see how our algorithm performs with the starting value v 0 := 0. It is easy to see that v n = α n u with α 0 := 0 and α n+1 := (ε − ∨ α n ∧ ε + ) 2− p . (4.3) Since p ∈ (1, 2) one can show α n = ε (2− p) n − by induction and Note that Moreover, This estimate with s := (2 − p) n ∈ (0, 1] and t := ε s − ∈ (0, 1] gives 2n for large n, in view of (4.4). This shows that it is impossible to get an energy reduction as in (4.2) with δ independent of . Indeed, Corollary 19 would imply which contradicts the above asymptotic estimate (4.5). Nevertheless, our asymptotic shows that in this particular case and v n still converges to u.

Algebraic rate
As we learned in the last section the Kačanov iteration converges for fixed , but the rate depends badly on the choice of the relaxation interval ε = [ε − , ε + ]. Furthermore, we have algebraic convergence of the error J (u ε ) − J (u) induced by the relaxation. We will combine these results to deduce an algebraic rate of the full error J n (v n ) − J (u) in terms of n for a specific predefined choice of n . To achieve our goal we will use that |∇u| ∈ L q,∞ ( ) for some q > p, which is justified by Theorems 13 and 14.
Let us consider a sequence of solutions created by our relaxed p-Kačanov algorithm. In particular, n = [ n,− , n,+ ] is now an increasing sequence of intervals. Then exactly as in Theorem 18 we get the following estimate.

Theorem 22
There is a constant c K > 1 such that holds for δ n := 1 c K ( n,− n,+ ) 2− p . Since n ⊂ n+1 , we have J n+1 J n . This and Theorem 22 imply Now, Lemma 12 and |∇u| ∈ L q,∞ ( ) ensure the existence of c R > 0 such that This and the previous estimate therefore imply ).  1 − δ n ). On the other hand this last term is small if n,− → 0 and n,+ → ∞, so it should not bother too much. Nevertheless, the reduction factor (1 − δ n ) tends to 1 if n,− → 0 and n,+ → ∞. The idea however is the following: if δ n goes to zero slowly, then the product n i=1 (1 − δ i ) still tends to zero algebraically.

Lemma 23
There exists K = K (α, β, p, q) (which appears in the definition of G n ) and some c 3 = c 3 (α, β, p, q) 1, such that for all n ∈ N Proof Define Hence it follows by Lemma 27 in the Appendix that there exists c 2 = c 2 (α, β, p, q) 1 with In particular, ρ n satisfies a decay estimate! On the other hand it follows from Theorem 22 that We deduce from (5.1), the definition of n and ρ n that This and the previous estimate prove We finally fix K 1 : We choose K 1 so large such that which is always possible. Combining this with our previous estimates we deduce This proves the theorem with c 3 = 2 max {c K , c 2 }.
We are now able to present our convergence result.

Theorem 24
Let ∇u ∈ L q,∞ ( ) for some q > p (as given for example in Theorem 13 or Theorem 14). Then, the sequence (v n ) n∈N produced by the algorithm above described satisfies where c 3 is the constant of Lemma 23. In particular, the energy error decreases at least algebraically.

Proof
The estimate J n (v n ) − J (u) G n − G ∞ is obvious, so it remains to prove the decay of G n − G ∞ . If follows from Lemma 23 that, for n ∈ N, This proves the lemma.

Remark 25
We have seen that the choice n = [(n + 1) −α , (n + 1) β ] ensures that the error decreases at least with an algebraic rate. However, the decay of the relaxed energy error G n − G ∞ can never be faster than algebraical with this choice of n . Hence, this choice is also very restrictive. From the numerical experiments we performed, we have seen that it is possible to decrease n,− and increase n,+ much faster and still obtain convergence. Moreover, the observed convergence is much faster than algebraic and more of exponential type. We will present the details of such numerical experiments in a subsequent work. Let us summarize: the algorithm of this section ensures an algebraic convergence rate, but in practice we expect a better behavior for other, perhaps adaptive, choices of n , still to be fully investigated.

Numerical experiments
We have performed numerous experiments on the basis of the adaptive finite element method with piecewise linear elements. We developed preliminary versions of error estimators that capture the effect of the truncation, the adaptivity of the mesh and the fixpoint iteration. Let v n denote the iterated solution generated by the algorithm, then we used the following ad hoc estimators: -We use to measure the effect of the upper truncation bound n,+ and to measure the effect of the lower truncation bound n,− . -We use the optimal estimators of [2,12] with the Orlicz function φ n to estimate the error due to mesh refinement, i.e. on elements T we use the estimators -To measure the error due to the fixpoint iteration (for n 1) which is in fact an upper bound for J (v n ) − J (u ε ) and J (v n−1 ) − J (u ε ).
We used these estimators to implement a fully adaptive version of our relaxed p-Kačanov iteration. Algorithm: Adaptive relaxed p-Kačanov Algorithm Define a n := ε n,− ∨ |∇v n | ∧ ε n,+ ; Calculate v n+1 by means of (ε n,− ∨ |∇v n | ∧ ε n,+ ) p−2 ∇v n+1 · ∇ξ dx = f , ξ ∀ξ ∈ W 1,2 0 ( ); Increase total costs by current degrees of freedom; Calculate and compare the error estimators η 2 |x| . This example is chosen such that V (∇u) ∈ W 1 2 L 2,∞ ( ) 3 Note that | f (x)| behaves like |x| p−2 . Thus, f ∈ L 2 for all p > 1 but f ∈ L p only for p > √ 2. This makes potential troubles with the used error estimator, but since the error estimator is also truncated with the effect is manageable. There is a nice gap between our theory and the numerical experiments that we performed. In fact, our experiments shows a significantly faster convergence rate. This shows that we are on a good track and have developed a good algorithm. The straight black line costs −1 is the optimal convergence rate (1/costs), where the costs are the accumulative sum of the degrees of freedom for each step that requires solving a linear system. (Here we have assumed a linear cost for solving the linear system, which might be possible with a multi grid method.) It is important that we use the accumulated cost instead of the degrees of freedom, since only this truly measures the effort, in particular if the number of fixpoint iterations increases.
Let us explain the numerical results in more detail.  It decreases almost optimally with respect to the accumulated cost, but the slope seems slightly worse. It is however still much faster, than our worst-case theory predicts. Overall, we see that our algorithm converges with a rate, which is optimal in many cases.
Acknowledgements Open Access funding provided by Projekt DEAL. Many thanks to Johannes Storn who did the final numerical experiments of this article. Finally, we thank the anonymous reviewer for the careful reading of the manuscript.
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A Auxiliaries
The following embedding is probably well known. However, since we could not find a reference for it and we need it for the proof of Theorem 14, we include a short proof of it. 5 In the following N 1 2 ,2 denotes the usual Nikolskiȋ space, see e.g. [20]. holds. The embeddings (see [15] respectively [23]) This proves the claim.
Moreover, in the proof Lemma 23 we used the following algebraic estimate:

B On uniformly convex Orlicz functions
In this appendix we introduce the concept of uniformly convex Orlicz functions and their shifted versions. The results presented below are modifications of [11,Lemma 3], [12,Lemma 16], and [12,Corollary 26]. However, since we use here a slightly different notion of shifted functions and less regularity for our Orlicz functions, we decided to include a proof and keep our paper self-contained. Throughout this section we assume that our Orlicz function satisfies the following assumptions.

Definition 28
Let φ be an N-function. 6 We say that φ is uniformly convex if there exist c 4 , c 5 > 0 with Proof Using s = λt with λ > 1 in (B.1) we obtain Now, we can choose λ 0 > 1 such that μ : . From this it follows by iteration (also using the monotonicity of φ ) that φ (2t) c φ (t), wherec only depends on λ 0 and μ and therefore only on c 4 . Thus, it follows that Hence, φ(2t) (1 + 2c)φ(t), which proves the claim for φ. The claim for φ * follows by duality with Lemma 29.
For each a 0 we define the shifted N-function φ a by its derivative φ a (t) := φ (t∨a) t∨a t (B.4) and φ a (t) = t 0 φ a (τ ) dτ . In the notation of Section 3 this is just φ with = (a, ∞), see also Remark 2.

Remark 34
The shifted N-functions have already been originally introduced in [11] with the modified definition φ a (t) = φ (t+a) t+a t. This original version shares almost all of the properties with the version of this paper. However, our exact formula (φ a ) * = (φ * ) φ (a) of Lemma 33 is replaced in [11] by equivalence. This is one of the advantages of our new definition. Proof For s > t we calculate  This proves the claim.

Lemma 42
The following estimates hold for arbitrary v ∈ W 1,φ 0 ( ) and u being the minimizer of J (w) := φ(|∇w|) − f w dx: Proof It follows by convexity and J (u) = 0 that This proves the claim.
The following lemma is a sharper version of Lemma 25 and Lemma 27 of [12]. Proof We begin with the proof of (B.5). The equivalence in (B.5) follows from Lemma 40. Thus the claim is symmetric in a and b and we can assume that a b.