https://doi.org/10.1007/s00211-020-01107-1
Numerische Mathematik
A Relaxed Kaˇcanov iteration for the p -poisson problem
L. Diening1·M. Fornasier2·R. Tomasi3 ·M. Wank3
Received: 23 December 2018 / Revised: 18 July 2019 / Published online: 25 February 2020
© The Author(s) 2020
Abstract
In this paper we introduce and analyze an iteratively re-weighted algorithm, that allows to approximate the weak solution of thep-Poisson problem for 1 < p2 by itera- tively solving a sequence of linear elliptic problems. The algorithm can be interpreted as a relaxed Kaˇcanov 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.
Mathematics Subject Classification 35J70·65L60
1 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⊂Rdis open and bounded and 1< p <∞. The solution might be scalar or vector-valued.1
1 All our results also hold for thep-Poisson system, where the functions are vector-valued. To this end sometimesRdhas to be replaced byRN×d.
We acknowledge support for the Article Processing Charge by the Deutsche Forschungsgemeinschaft and the Open Access Publication Fund of Bielefeld University. This research was supported by the DFG project “Optimal Adaptive Numerical Methods for p-Poisson Elliptic Equations”.
B
L. Dieninglars.diening@uni-bielefeld.de M. Fornasier
massimo.fornasier@ma.tum.de
1 Faculty of Mathematics, Bielefeld University, Postfach 10 01 31, 33501 Bielefeld, Germany 2 Department of Mathematics, TUM, Boltzmannstraße 3, 85748 Garching, Munich, Germany 3 Osnabrück University, Osnabrück, Germany
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 nonlin- ear 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 functionu ∈W01,p()such that
|∇u|p−2∇u· ∇ξd x= f, ξ ∀ξ ∈W01,p(), (1.2)
where in the most general case f ∈(W01,p())∗. It is well-known that the solution is unique and coincides with the minimizer of the energyJ : W01,p()→ Rdefined by
J(v):= 1p
|∇v|pd x− f, v. (1.3)
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 W01,p(). 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 coin- cide 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 ofp, also because the case of p>2 is already addressed to a certain extent in [5]. We will see, e.g., in Example20that 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
ap−2∇u· ∇ξd x= f, ξ ∀ξ ∈W01,2() (1.4)
for the given f, wherea:→Randa= |∇u|. This suggests to iteratively calculate for a given functionvnthe new iteratevn+1as the solution of
|∇vn|p−2∇vn+1· ∇ξd x= f, ξ ∀ξ ∈W01,2().
The advantage of this step is that the calculation of vn+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 ifa is bounded from above and from below away from zero. However, the weight|∇vn|p−2 may be degenerating, at points where|∇u| =0 or|∇u| = ∞.
To overcome this problem we will use a relaxation arguments. Therefore, we intro- duce in our algorithm two relaxation parameters−, +∈(0,∞)with− +that ensure that the weight is truncated properly from below and above. In particular, we replaceaby its truncation
−∨a∧+:=max{−,min{a, +}}.
Note that this is just the (pointwise) closest point projection ofa to the truncation interval [−, +]. The limits − 0 and + ∞ will recover the unrelaxed or original problem. We also write:= [−, +]and interpretboth as a pair{−, +} and as the truncation interval[−, +]. We will write→ [0,∞]as a short version of−0 and+ ∞. We will see later, see Corollary15, that for f in the Lorentz spaceLd,1()the lower parameter−is the crucial one.
According to these considerations we propose the following algorithm:
Algorithm:The relaxedp-Kaˇcanov algorithm Data: Given f ∈(W01,p())∗,v0∈W01,2();
Result: Approximate solution of thep-Poisson problem (1.2);
Initialize:ε0= [ε0,−, ε0,+] ⊂(0,∞),n=0;
whiledesired accuracy is not achieved yetdo Definean:=εn,−∨ |∇vn| ∧εn,+
end
;
Calculatevn+1by means of
(εn,−∨ |∇vn| ∧εn,+)p−2∇vn+1· ∇ξd x= f, ξ ∀ξ ∈W01,2();
Choose new relaxation intervalεn+1⊃n; Increasenby 1;
Since 0< n,−n,+<∞the equation forvn+1in the algorithm is always well defined, since it is uniformly elliptic (with constant depending onn).
This algorithm is not completely new in the realm of quasi-linear equations. Such an iterative linearization approach is in fact called the Kaˇcanov 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 Kaˇcanov iteration for equa- tions 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 sev- eral 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-Kaˇcanov iteration.
Taking inspiration from [6,10], in Sect.2we 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 parameterindependent onn, i.e.
0 < − + < ∞, then our iteratesvn converge to the unique minimizer u of another one-parameter relaxed energyJ. We study this limit in Sect.4and present (linear) exponential rates of convergence.
In Sect.3we study how the minimizersuof the relaxed energyJ converge to the minimizeruof 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.5we combine the estimates of the two previous sections to deduce an overall error analysis with algebraic rates.
2 Variational formulation of the algorithm
In this section we show that the algorithm can be deduced from an alternating min- imization 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 useW1,p()andW01,p()for the Sobolev space without and with zero boundary values. We usecfor a generic positive constant whose value may change from line to line. We use f gfor f c g. We also write f gfor f gandg f.
The most important feature of the algorithm is that it only needs to solve linear sub- problems, 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 parametera : → [0,∞)such that the new functional is quadratic with respect tov. In particular, we define
J(v,a):=
1
2ap−2|∇v|2+
1 p−21
apd x− f, v.
This energy is well-defined for allv ∈ W01,p()and measurablea : → [0,∞) but might take the value∞.
This relaxed energy is convex with respect to(v,a). This follows from the fact that β(t,a):= 12ap−2t2is convex on[0,∞)2, since
(∇2β)(t,a)=
ap−2 (p−2)ap−3t (p−2)ap−3t 12(p−2)(p−3)ap−4t2
is nonnegative definite asap−20 and det((∇2β)(t,a))=a2p−6t2(2−p)(p−1) 0. Notice that in the latter lower bound we specifically used 1<p2.
Remark 1 If p >2, then the relaxed energyJ(v,a)is neither bounded from below nor convex with respect toa. Therefore, the algorithm derived below using the min- imization with respect toa does not lead to a feasible problem for p >2. See also Remark21.
Note thatJ(v,a)(for fixeda) is quadratic with respect tovand a minimization with respect tovleads formally to the elliptic equation
−div(ap−2∇v)= f, see (1.4) for its weak form.
Unfortunately, the ellipticity of this system degenerates fora(x)→0 anda(x)→
∞. To overcome this problem we restrict the minimization with respect to a (for fixedv) to functions with values within a relaxation interval[−, +] ⊂(0,∞), i.e.
− a(x) +. This minimization with respect to a (for fixed v) has a simple solution, namely
arg min
a:−a+
J(v,a)=−∨ |∇v| ∧+, (2.1)
where∨denotes the maximum and∧the minimum, since
∂∂a
1
2ap−2|∇v|2+(1p−12)ap
=2−2pap−3(a2− |∇v|2).
This allows us to define for fixed= [ε−, ε+] ⊂ [0,∞]another relaxed energy J(v):=J(v, ε−∨ |∇v| ∧ε+)= min
a:−a+J(v,a). (2.2)
This immediately implies that the relaxed energyJ(v)is monotonically decreasing with respect to , i.e., an increasing interval in terms of inclusion decreases the energyJ(v).
This new relaxed energyJ somehow “hides” the constrained minimization with respect toa. We can writeJ : W01,p()→R∪ {∞}explicitly as
J(v)=
κ(|∇v|)d x− f, v
withκ : R0→Rgiven by
κ(t):=
⎧⎪
⎨
⎪⎩
1
2ε−p−2t2+(1p−12)ε−p fort ε− 1
ptp forε−tε+
1
2ε+p−2t2+(1p−12)ε+p fort ε+.
Note that 1ptp κ(t)for allt 0 and 1ptp = lim→[0,∞]κ(t)for allt 0.
Since κ(t) +p−2t2 for large t, we see that J(v) < ∞ if and only if v ∈ W01,2(). Moreover, lim→[0,∞]J(v) = J(v)for allv ∈ W01,2()andJ(v) lim inf→[0,∞]J(v)for allv∈W01,p().
Based on the above observations it is natural to iteratively minimizeJ(v,a)alter- nating betweenvanda. Certainly, we have also to increase the relaxation interval. Thus our algorithm reads as follows:
Algorithm:The relaxedp-Kaˇcanov algorithm (variational formulation) Data: Given f ∈(W01,p())∗,v0∈W01,2();
Result: Approximate solution of thep-Poisson problem (1.2);
Initialize:ε0= [ε0,−, ε0,+] ⊂(0,∞),n=0;
whiledesired accuracy is not achieved yetdo Calculateanby means of
an := arg min
a:−a+
J(vn,a);
Calculatevn+1by means of
vn+1:= arg min
v∈W01,2()
J(v,an);
Choose a new relaxation intervalεn+1⊃n; Increasenby 1;
end
This is just the algorithm given in the introduction written in different form.
3 Convergence in the relaxation parameter
In this section we show that the minimizersu of the relaxed energyJ converge to the minimizeruofJ for → [0,∞]and derive an upper bound for the relaxation error.
SinceJ(v)J(v)andJ isW01,p()coercive, it follows thatJis also coercive inW01,p(). However, J(v) <∞requiresv ∈ W01,2(), as we have seen above.
Certainly, there is a gap between the spaceW01,p()andW01,2(). 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 functionφ : R0 → Ris called an N-function if and only if there is a right- continuous, positive on the positive real line, and non-decreasing functionφ:R0→ Rwithφ(0) = 0 and limt→∞φ(t) = ∞ such that φ(t) = t
0φ(τ)dτ. An N- function is said to satisfy the2-condition if and only if there is a constantc > 1 such thatφ(2t) cφ(t). For an N-function satisfying the 2-condition we define the Orlicz space to consist of those functionsv∈ L1loc()with
φ(|v|)d x<∞. It becomes a Banach space with the normfφ:=inf{γ >0 :
φ(|v|/γ )d x 1}.
The Orlicz-Sobolev spaceW1,φ()then consists of thosev∈ Lφsuch that the weak derivative∇vis also inLφ, equipped with the normvφ+∇vφ. The spaceW01,φ() denotes the subspace of those functions fromW1,φ()with zero boundary trace, which coincides with the closure ofC0∞()inW1,φ(). For example choosingφ(t):= 1ptp we haveLφ()=Lp()andW01,φ()=W01,p().
The functionκcannot be an N-function, sinceκ(0)=0, . However, if we define φ(t):=κ(t)−κ(0), (3.1)
thenφis actually an N-function. It can be verified thatφsatisfies the2-condition with a constant independent of.
Sinceφ(t) +p−2t2for larget andis bounded, we have Lφ() L2(). However, the constant of the embedding Lφ() → L2()depends on, so this equivalence is not of much use. Instead we use the chain of embeddings
L2() →Lφ() →Lp(), (3.2)
with constants independent of. This follows from the fact that the Simonenko indices ofφare within[p,2]. We refer the reader to, e.g., [28, Chapter 2] for the details.
Sinceφis strictly convex andκ(t)=φ(t)+κ(0), the energyJadmits a unique minimizeru∈W01,φ()whose Euler-Lagrange equation is
(ε−∨ |∇u| ∧ε+)p−2∇u· ∇ξd x = f, ξ ∀ξ ∈W01,φ(). (3.3)
At this we used that
φ(t)
t =(ε−∨t∧+)p−2. (3.4)
Remark 2 Let us consider the special case ε+ = ∞. Then, the derivative of the truncated function reads φ(t) = (ε− ∨t)p−2t. A different modification, namely (ε−+t)p−2t, would lead to the so-called shifted N-function of 1ptp, 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 1ptpon the truncation interval(the derivatives agree there). See the Appendix for more information on uniformly convex Orlicz functions and their shifted verions.
Lemma 3 The functionsφandφare uniformly convex Orlicz functions in the sense of Sect.Bfrom the Appendix. The convexity constants are independent of.
Proof The uniform convexity ofφfollows from φφ((tt))t =(p−1)and Lemma (B.2).
Now, fort ∈(−, +)we have φφ(t()t)t =(p−1), while fort ∈(0,∞)\ [−, +]we haveφφ(t)t
(t) =1. Hence, the claim forφfollows again by Lemma (B.2).
SinceW01,p()is the largest space, see (3.2), which contains bothuandu, it is natural to consider all energiesJ andJ as functionals onW01,p()with possible value∞.
Let us recall that the goal of this section is to show thatuconverges touinW01,p(). Since W01,p()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 Remark11thatJ→Jin 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 Corollary10.
Let us turn to the convergence of the energiesJ(u)→ J(u)for → [0,∞]. SinceJis monotonically decreasing with respect to, it follows from the minimizing properties ofuanduthat
0J(u)−J(u)J(u)−J(u). (3.5) Therefore, it suffices to prove the stronger claim
J(u)−J(u)→0 as→ [0,∞]. (3.6) In fact, we will later need this stronger estimate in the other sections.
It follows from the minimizing property ofu that J(u)−J(u)J(u)−J(u).
So it would be natural to estimate J(u)−J(u)in terms of andu. However, the solution u is unfortunately a priori only a W01,p-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 ofu. In order to treat arbitrary right-hand sides f ∈ (W01,p())∗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 changeu on a small set such that it becomes a Lipschitz function. This technique is known as theLipschitz truncation technique. Its origin goes back to [1]. As a tool we need the Hardy-Littlewood operator, e.g. [25],
(Mg)(x):=sup
r>0
−
Br(x)|g|d x:=sup
r>0
1
|Br(x)|
Br(x)|g|d x where|Br(x)|denote the Lebesgue measure ofBr(x).
Theorem 4 [Lipschitz trunction [4,13]]Letv ∈W01,p()and for allλ >0define Oλ:=Oλ(v):= {x∈ : M(∇v)(x) > λ},
Then, there exists an approximation Tλv∈ W01,∞()ofvwith the following proper- ties:
(a) {v=Tλv} ⊂Oλ. (b) TλvLp()vLp(). (c) ∇TλvLp()∇vLp().
(d) |∇Tλv|λχOλ+ |∇v|χ\Oλ λalmost everywhere.
(e) ∇(v−Tλv)Lp()∇vLp(Oλ). (f) vλ→vin W01,p()asλ→ ∞.
All our convergence results concerning the relaxation parameterεare based on the following result, which shows how the energy relaxation depends on the truncation intervalε.
Theorem 5 The estimate
J(uε)−J(u)ε−p+
Oλ(u)
|∇u|pd x (3.7)
holds for allλε+/c1, where c1is the (hidden) constant from Theorem4(d).
Proof Letλε+/c1and letTλube the Lipschitz truncation ofu. Then
|∇Tλu|c1λε+.
Using the minimizing property ofuεand the equation foruwe get J(uε)−J(u)J(Tλu)−J(u)=
κ(|∇Tλu|)−1p|∇u|p
d x− f,Tλu−u
=
κ(|∇Tλu|)−1p|∇u|p d x−
|∇u|p−2∇u· ∇(Tλu−u)d x.
Using|∇Tλu|ε+,κ(t)= 1ptpfort∈ [ε−, ε+],κ(t) 1pε−p fort ∈ [0, ε−], and Tλu =uoutside ofOλwe get
κ(|∇Tλu|)− 1p|∇u|p
⎧⎪
⎨
⎪⎩
1
pε−p on{|∇Tλu|ε−},
0 on
\Oλ
∩ {|∇Tλu|> ε−},
1
p|∇Tλu|p onOλ∩ {|∇Tλu|> ε−}.
This, the previous estimate and Theorem4(e) imply J(uε)−J(u)|| 1pε−p +
Oλ(u) 1
p|∇Tλu|pd x+
Oλ(u)
|∇u|p−1|∇(Tλu−u)|d x
ε−p+
Oλ(u)
|∇Tλu|pd x+
Oλ(u)
|∇u|pd x+
Oλ(u)
|∇(Tλu−u)|pd x
ε−p+
Oλ(u)
|∇u|pd x.
This proves the claim.
Corollary 6 J(uε)→J(u)andJ(uε)→J(u)asε→ [0,∞].
Proof Due to (3.5) it suffices to proveJ(uε)→J(u). Consider the right-hand side of (3.7) withλ:=ε+/c1. The first term goes to zero asε− →0. Now consider the second term. SinceOλ(u)⊂ {M(∇u) > λ}and∇u ∈Lp()we get by the weakLp- estimate of the maximal operator|Oλ(u)| λ−p∇upp. Therefore|Oλ(u)| → 0 asε+→ ∞, which implies
Oλ(u)|∇u|pd x→0 asε+→ ∞. Before we continue we need the following natural quantities, see [11].
Definition 7 ForP ∈Rdwe define
A(P):=
φ
(|P|)
|P| P ifP=0
0 ifP=0 and V(P):=
φ
(|P|)
|P| P ifP=0
0 ifP=0.
Moreover, by A:= A[0,∞]andV :=V[0,∞]we denote the unrelaxed versions.
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.Bfrom the Appendix for more details.
Lemma 8 For P,Q∈Rd
(A(P)−A(Q))·(P−Q) φ(|P| ∨ |Q|)
|P| ∨ |Q| |P−Q|2|V(P)−V(Q)|2. where the constants can be chosen independently ofε.
Proof This follows directly from the uniform convexity ofφ, see Lemma3, Lemma41
and Lemma40.
Lemma 9 The following estimates hold for arbitraryv∈W01,φ()and uεbeing the minimizer ofJ:
J(v)−J(uε)
(A(∇v)−A(∇uε))· ∇(v−uε)d x
|V(∇v)−V(∇uε)|2d x J(v)−J(uε).
In particular, for the case whereε= [0,∞]the statement actually implies also J(v)−J(u)
|V(∇v)−V(∇u)|2d x.
Proof This is just Lemma42applied toφ.
We are now prepared to show the convergence of minimizersuofJtou.
Corollary 10 u →u in W01,p()asε→ [0,∞].
Proof Due to Corollary6we haveJ(u)−J(u)→0 as→ [0,∞]. Now Lemma9 for the case where= [0,∞]andv =uε impliesV(∇u)→ V(∇u)inL2(). It follows from the shift-change-lema, see Corollary44or [12, Corollary 26], that for allδ >0 there existscδ>0 such that
|∇u− ∇u|pcδ|V(∇u)−V(∇u)|2+δ|∇u|p.
This andV(∇u)→V(∇u)inL2()implies∇u → ∇uinLp().
Remark 11 [-convergence] It is also possible to deduceJ(u)→J(u)andu→u in W01,p()by means of -convergence: As the underlying topological space we choose W01,p()equipped with the weak topology. Then the Lipschitz truncation
provides a recovery sequence for v ∈ W01,p()implying - lim→[0,∞]J = J. Indeed, it follows as in the proof of Theorem5that for allv∈W01,p()
J(Tε+/c1v)−J(v)ε−p +
Oε+/c1(v)|∇v|pd x+f,Tε+/c1v−v. So the properties of the Lipschitz truncation, see Theorem4(f), imply that the right- hand side goes to zero as→ [0,∞]. Hence,Tε+/c1vis a recovery sequence ofv.
Moreover,J J, so the standard theory of-convergence [3,9] provesuu inW01,p()andJ(u)→J(u)for→ [0,∞]. The uniform convexity ofW01,p() impliesu →uinW01,p().
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 onW01,p().
Up to now, we discussed the convergence ofu →uwithout any additional assump- tions on the data f ∈ (W01,p())∗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-Lq spaces Lq,∞(), which consists of all functionsvsuch that
vLq,∞():=sup
t>0
tχ{|v|>t}Lq()<∞.
Lemma 12 Let∇u ∈Lq,∞()for some q>p. Then,
J(uε)−J(u)ε−p+ε+−(q−p)∇uqLq,∞().
Proof First note thatM :Lq,∞()→Lq,∞()is bounded. This follows for example by extrapolation theory, see [8, Theorem 1.1]. In particular,
λ|Oλ(u)|q1 λ χ{M(∇u)>λ}Lq()M(∇u)Lq,∞()∇uLq,∞(). Moreover, letLq,1()denote the usual Lorentz space, which consists of functionsv such that
vLq,1():=q ∞ 0
|{|v|>t}|1q dt=q ∞ 0
tχ{|v|>t}Lq()
dt t <∞.
Since(Ls,1)∗=Ls,∞for 1<s<∞and1s +s1 =1 we obtain
Oλ(u)
|∇u|pd x|∇u|p
L
q
p,∞()χOλ(u)
L
q−pq ,1
()
∇upLq,∞()|Oλ(u)|q−qp ∇uqLq,∞()λ−(q−p),
where |Oλ|denotes the Lebesgue measure ofOλ. Applying Theorem5 withλ :=
ε+/c1yields the statement.
To exemplify the consequences of Lemma12we combine it with the regularity results of [7,14]:
Theorem 13 ([7], Theorems 1.3 and 1.4)Let⊂Rdbe convex or let its boundary
∂ ∈ W2Ld−1,1(for example∂ ∈ C2suffices2) and additionally f ∈ Ld,1().
Then∇u ∈L∞().
Theorem 14 ([14], (4.3)) Let be a polyhedral domain where the inner angle is strictly less than2π and f ∈ Lp()and 1p+ p1 =1. Then∇u∈Ldpd−1,∞().
Proof Actually, it is proven in [14] (4.3) that|∇u|2p ∈ N12,2()(Nikolski˘ı space).
Now, one can use the embedding N12,2() → Ld−12d ,∞()of Lemma 26 in the
Appendix.
Corollary 15 Under the assumptions of Theorem13we have J(uε)−J(u)ε−p.
Proof Since∇u ∈L∞(), we haveM(∇u)∈L∞()and so forλ:=ε+/c1andε+ large enough,Oλ(u)= ∅. Hence, Theorem5implies the estimate.
Corollary 16 Under the assumptions of Theorem14we have J(uε)−J(u)ε−p+ε+−d−p1
Proof Since∇u∈Ldpd−1,∞(), an application of Lemma12finishes the pf.
Remark 17 The choice f =0 and henceu =0 givesJ(u)=κ(0)||ε−p. This shows that the estimate in Corollary15is sharp.
4 Convergence of the Kaˇcanov-iteration
In this section we study the convergence of the Kaˇcanov-iteration for fixed relax- ation parameter= [ε−, ε+]. In particular, forv0 ∈W01,2()arbitrary we calculate recursivelyvn+1by
(ε−∨ |∇vn| ∧ε+)p−2∇vn+1· ∇ξd x= f, ξ ∀ξ ∈W01,2(). (4.1)
2 The condition∂ ∈W2Ld−1,1means that the weak Hessians of the local maps characterizing the boundary are in the Lorentz spaceLd−1,1.
We will show thatvn converges to the minimizer u of the relaxed energy J. In particular, we show exponential decay of the energy errorJ(vn)−J(u). The proof is based on the following estimate, proved below.
Theorem 18 There is a constant cK >1such that
J(vn)−J(vn+1)δ (J(vn)−J(uε))
holds forδ :=c1K(εε−+)2−p.
This theorem says that in each iteration we reduce the energy by a certain part of the remaining energy error. This implies
J(vn+1)−J(uε)=
J(vn)−J(uε)
−
J(vn)−J(vn+1) (1−δ)
J(vn)−J(uε)
. (4.2)
As a direct consequence we will obtain the following exponential convergence result.
Corollary 19 There is a constant cK >1such that
J(vn)−J(uε)(1−δ)n(J(v0)−J(uε)).
holds forδ :=c1K(εε−+)2−p.
Let us get to the proof of Theorem18.
Proof (Proof of Theorem18)Using Lemma9, the equation (3.3) foruε, the equa- tion (4.1) for vn+1, and Young’s inequality (see Remark 32) we get, for arbitrary γ >0,
J(vn)−J(uε)
(A(∇vn)−A(∇uε))· ∇(vn−uε)d x
=
φ(|∇vn|)
|∇vn| ∇(vn−vn+1)· ∇(vn−uε)d x γ1 12
φ(|∇vn|)
|∇vn| |∇(vn−vn+1)|2d x
=:I
+γ 12
φ(|∇vn|)
|∇vn| |∇(vn−uε)|2d x
=:I I
.
Let us define
J(v,a):=J (v, ε−∨a∧ε+) .
