On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type

Abstract

We discuss nonlinear homogeneous eigenvalue problems and the variational characterization of their eigenvalues. We focus on the Ljusternik-Schnirelmann method, present one possible alternative to this method and compare it with the Courant-Fischer minimax principle in the linear case. At the end we present a special nonlinear eigenvalue problem possessing an eigenvalue which allows the variational characterization but is not of Ljusternik-Schnirelmann type.

The question of characterization of the set of all eigenvalues of nonlinear homogeneous eigenvalue problem is a very important problem of the calculus of variations that has been open since the 1970s, see, for example, [1]. While the well-known Courant-Fischer minimax principle provides complete characterization of the set of eigenvalues of the linear eigenvalue problem, its nonlinear counterparts, like the Ljusternik-Schnirelmann method, do not exhaust the entire set of eigenvalues in general. The purpose of this paper is to discuss this issue in the context of general homogeneous nonlinear eigenvalue problem (1). In particular, we want to explain that “variational characterization” is not a synonym for “Ljusternik-Schnirelmann characterization.” Indeed, this obfuscation is quite common in the literature (cf. [2]) and might lead to many misunderstandings.

We present a variational argument in order to find an eigenvalue that is not characterized by the Ljusternik-Schnirelmann formula. For this purpose we present the periodic problem for the one-dimensional 𝑝-Laplacian from [3] as a concrete example and heuristically enlighten technically involved estimates. We show the local character of our variational characterization contrary to the global character of the Ljusternik-Schnirelmann method. This illustrates the striking difference between the linear and nonlinear problems. Note that also this “local procedure” does not yield all the eigenvalues of the problem in question.

The novelty of this paper is thus to point out some connections might be hidden in more technically complicated papers and to remind some challenging problems of the calculus of variations. We also rigorously show that the second eigenvalue of Ljusternik-Schnirelmann type for (1) can be found as a minimax over a special class of curves rather than the family of sets of genus greater than or equal to 2. This fact makes the construction of the second eigenvalue more geometrically transparent and might be useful in numerical approach to find its concrete value.

Let 𝐽,𝑆∶𝑋→𝑋^∗ be odd and (𝑝−1)-homogeneous operator from a Banach space 𝑋 into its dual, 𝑝>1. A real number 𝜆 and an element 𝑢∈𝑋, 𝑢≠0, for which𝐽(𝑢)−𝜆𝑆(𝑢)=0 (1) are called an eigenvalue and an eigenvector of the eigenvalue problem (1). Set𝐹(𝑢)∶=⟨𝐽(𝑢),𝑢⟩,𝐺(𝑢)∶=⟨𝑆(𝑢),𝑢⟩, (2) where ⟨⋅,⋅⟩ stands for the duality pairing between 𝑋^∗ and 𝑋. Assume that 𝐽 and 𝑆 are such that 𝐹 and 𝐺 are 𝐶¹-functionals from 𝑋 into ℝ,𝐹^′(𝑢),𝑣=⟨𝐽(𝑢),𝑣⟩,𝐺^′(𝑢)=⟨𝑆(𝑢)⟩,{(𝑢)},𝑣,𝑣𝒮∶=𝑢∈𝑋∶𝐺=1 (3) is a 𝐶¹-manifold without boundary that does not contain the zero element 0∈𝑋. The Lagrange multiplier method implies that 𝜆₀ is an eigenvalue of (1) and 𝑢₀ is the corresponding eigenvector if and only if 𝑢₀ is a critical point of 𝐹|_𝒮 and 𝐹(𝑢₀)=𝜆₀ (i.e., 𝜆₀ is the corresponding critical level). There are several methods to find critical points of 𝐹 on 𝒮. However, it seems that with the exception of some special situations (e.g., 𝐽 and 𝑆 are linear operators) none of the “nonlinear” methods describes the entire set of critical levels and critical points of 𝐹|_𝒮. One of the most frequently used methods is based on the notion of the Krasnoselskii genus.

Set 𝒜∶={𝐴⊂𝑋∶𝐴isclosedandsymmetric}. The Krasnoselskii genus of a set 𝐴,  𝛾(𝐴), is defined as follows: 𝛾(𝐴)∶=inf𝑚∈ℕ∶∃ℎ∈𝐶⁰𝐴,ℝ^𝑚⧵{0},ℎisodd,∞if𝑚∈ℕ∶∃ℎ∈𝐶⁰𝐴,ℝ^𝑚⧵{0},ℎisodd=∅. (4) For any 𝑘∈ℕ consider a family of sets defined byℱ_𝑘∶=𝐴∈𝒜∶𝐴⊂𝒮,𝛾(𝐴)≥𝑘. (5) Assume that 𝛾(𝒮)=∞. Then, ℱ_𝑘≠∅ for any 𝑘∈ℕ and it is invariant under any continuous and odd map from 𝒮 into itself.

Let us assume that 𝐹 satisfies the Palais-Smale condition on 𝒮. The Ljusternik-Schnirelmann method yields a sequence {𝜆_𝑘}^∞_𝑘=1⊂ℝ given by𝜆_𝑘∶=inf_{𝐴∈ℱ𝑘}sup_𝑢∈𝐴𝐹(𝑢). (6) Every 𝜆_𝑘 is a critical level of 𝐹 subject to 𝒮, and it is achieved at some 𝑢_𝑘∈𝒮, that is,𝐹|_𝒮^′𝑢_𝑘=0 (7) (cf. [4, 5]). It follows that 𝜆_𝑘 is an eigenvalue of (1) and 𝑢_𝑘 is the corresponding eigenvector. In general, the sequence {𝜆_𝑘}^∞_𝑘=1 given by (6) does not exhaust the set of all critical levels of 𝐹|_𝒮, and thus it might not be the set of all eigenvalues of (1). An eigenvalue of (1) that allows the characterization (6) is called an eigenvalue of Ljusternik-Schnirelmann type.

The model example of the abstract setting presented above is the eigenvalue problem for the Dirichlet 𝑝-Laplacian. Indeed, set 𝑋=𝑊₀^1,𝑝(Ω), 𝑝>1, and𝐹(𝑢)=_Ω|∇𝑢|^𝑝d𝑥,𝐺(𝑢)=_Ω|𝑢|^𝑝d𝑥, (8) where Ω⊂ℝ^𝑁 is a bounded domain. Then,𝒮=𝑢∈𝑋∶‖𝑢‖_𝐿𝑝(Ω)=1, (9) and a critical point 𝑢₀∈𝒮,𝐹(𝑢₀)=𝜆₀ satisfies𝐹^′𝑢₀,𝑣=𝜆₀𝐺^′𝑢₀,𝑣 (10) for any 𝑣∈𝑋. But this is a weak formulation of the problem−Δ_𝑝𝑢₀−𝜆₀𝑢₀^𝑝−2𝑢₀𝑢=0inΩ,₀=0on𝜕Ω, (11) and, therefore, 𝜆₀ is an eigenvalue and 𝑢₀ is the corresponding eigenvector (eigenfunction) of the Dirichlet 𝑝-Laplacian (11).

If 𝑝≠2 and Ω⊂ℝ^𝑁, 𝑁≥2, it is a long-standing open problem whether or not the sequence{𝜆_𝑘}^∞_𝑘=1 given by (6) exhausts the entire set of eigenvalues of (11). On the other hand, if Ω=(0,1), 𝑝>1, or Ω⊂ℝ^𝑁, 𝑁≥1, and 𝑝=2 (i.e., the problem (11) is linear), the sequence {𝜆_𝑛}^∞_𝑛=1 given by (6) is the set of all eigenvalues of (11). To prove this fact in the former case the ODE techniques (uniqueness of the solution of an associated initial value problem) are employed while in the latter case the linearity of the problem plays the key role.

There are several other variational characterizations similar to (6). Namely, the family of sets ℱ_𝑘 can be replaced by an other family ℱ_𝑘, which is invariant with respect to any continuous odd semiflow Φ∶𝒮×[0,∞)→𝒮 such that Φ(⋅,0)=id_𝒮, Φ(⋅,𝑡)∶𝒮→𝒮 is a homeomorphism for any fixed 𝑡≥0 and 𝐹(Φ(𝑢,⋅)) is nonincreasing for any fixed 𝑢∈𝒮. Then,𝜆_𝑘∶=inf_𝐴∈ℱ_𝑘sup_𝑢∈𝐴𝐹(𝑢),𝑘∈ℕ, (12) is a critical level of 𝐹|_𝒮.

We give a concrete example of ℱ_𝑘 below (cf. [6]). Let 𝑆^𝑘−1 be the unit sphere in ℝ^𝑘.

Setℱ_𝑘∶=𝐴⊂𝒮∶∃ℎ∶𝑆^𝑘−1⟶𝒮,𝐴=ℎ𝑆^𝑘−1,ℎisoddandcontinuous. (13) It is clear that ℱ_𝑘⊊ℱ_𝑘, which implies𝜆_𝑘≤𝜆_𝑘,𝑘∈ℕ. (14)

It is also obvious that 𝜆₁=𝜆₁. We claim that 𝜆₂=𝜆₂ as well. Indeed, for an 𝐴∈ℱ₂ there exists (arbitrarily small) symmetric neighborhood 𝒩(𝐴)⊂𝒮 such that 𝛾(𝒩(𝐴))≥2 (see, [7, Proposition 2.3]). Without loss of generality we may assume that 𝒩(𝐴) is a pathwise connected set in 𝒮 (otherwise 𝛾(𝒩(𝐴))=1, a contradiction). Choosing 𝑢∈𝒩(𝐴), we have −𝑢∈𝒩(𝐴)by symmetry, and there is a continuous path 𝜑∈𝒩(𝐴) from 𝑢 to −𝑢. By symmetry of 𝒩(𝐴) we also have −𝜑⊂𝒩(𝐴). But 𝜑∪(−𝜑)=ℎ(𝑆¹) for some continuous and odd map ℎ of 𝑆¹ into 𝒮. Since the neighborhood 𝒩(𝐴) can be chosen small enough, the value ofmax_{𝑢∈𝜑∪(−𝜑)}𝐹(𝑢)−sup_𝑢∈𝐴𝐹(𝑢) (15) can be made smaller than any positive number. This implies 𝜆₂≤𝜆₂, which proves the claim.

The author is not aware of any assumptions laid on 𝐽 and 𝑆 that would guarantee that also 𝜆_𝑘=𝜆_𝑘 for 𝑘≥3 (cf. [6]).

It is instructive to compare formula (12) with the classical Courant-Fisher minimax principle:𝑐_𝑘∶=inf_{𝑋′⊂𝑋′dim𝑋=𝑘}max_{𝑢∈𝒮∩𝑋′}𝐹(𝑢) (16) under the assumptions that 𝐽,𝑆 are linear operators, 𝑋 is a Hilbert space and (𝐽𝑢,𝑢)_𝑋 defines an equivalent inner product on 𝑋 (cf. problem (11) with 𝑝=2).

Let 𝑋^′⊂𝑋, dim𝑋^′=𝑘. Then, 𝐴=𝒮∩𝑋′ is homeomorphic to 𝑆^𝑘−1, that is, 𝐴∈ℱ_𝑘 and hence 𝜆_𝑘≤𝑐_𝑘. In order to prove the opposite inequality we need the following assertion.

Lemma 1 (cf. [5]). Any compact and symmetric subset 𝐴 of a Hilbert space, such that 𝛾(𝐴)=𝑘 contains at least 𝑘 mutually orthogonal vectors 𝑢₁,…,𝑢_𝑘.

Since 𝛾(𝐴)=𝑘for 𝐴∈ℱ_𝑘, there exist 𝑢₁,…,𝑢_𝑘∈𝐴 such that (𝐽𝑢_𝑖,𝑢_𝑗)_𝑋=0 for 𝑖≠𝑗. Set 𝑋^′=Lin{𝑢₁,…,𝑢_𝑘}. It follows from the linearity of 𝐽 that max_{𝒮∩𝑋′}𝐹(𝑢)≤max_𝐴𝐹(𝑢), which implies 𝑐_𝑘≤𝜆_𝑘. Together with the opposite inequality proved above we thus get𝜆_𝑘=𝑐_𝑘. (17) Similarly, one proves that also 𝜆_𝑘=𝑐_𝑘.

In particular, it follows from our discussion that the second Ljusternik-Schnirelmann eigenvalue of (1) can be characterized as follows:𝜆₂=inf_{𝛾∈𝒞sym}max_𝑢∈𝛾𝐹(𝑢), (18) where 𝒞_sym is the family of all continuous symmetric and closed curves in 𝒮.

The idea of construction of an eigenvalue that is not of Ljusternik-Schnirelmann type comes from the following heuristics.

Set Φ(𝛾)∶=max_𝑢∈𝛾𝐹(𝑢). Then, 𝜆₂=inf_{𝛾∈𝒞sym}Φ(𝛾), that is, 𝜆₂ is a “global minimum” of Φ over the set 𝒞_sym (cf. [5]). The idea is to find a “local minimum” of Φ over 𝒞_sym, say 𝜇₂, which satisfies 𝜆₂<𝜇₂<𝜆₃, that is, 𝜇₂∉{𝜆_𝑘}^∞_𝑘=1. For this purpose we find a subset of curves 𝒞⊂𝒞_sym, which is invariant with respect to any continuous odd semiflow and which satisfies𝜆₂=inf_{𝛾∈𝒞sym}max_𝑢∈𝛾𝐹(𝑢)<inf_𝛾∈𝒞max_𝑢∈𝛾𝐹(𝑢)=𝜇₂<𝜆₃. (19) The invariance of 𝒞 then guarantees that 𝜇₂ is a critical point of 𝐹|_𝒮.

Below we present a concrete example. The technical details and precise estimates can be found in [3]. We set 𝑋=𝑊^1,𝑝_per(−𝜋_𝑝,𝜋_𝑝), the Sobolev space of all (2𝜋_𝑝)-periodic functions 𝑓∶ℝ→ℝ such that the restriction of 𝑓 to the interval [−𝜋_𝑝,𝜋_𝑝] belongs to 𝑊^1,𝑝(−𝜋_𝑝,𝜋_𝑝) and 𝑓(−𝜋_𝑝)=𝑓(𝜋_𝑝). This space is endowed with the norm‖𝑓‖∶=𝑓′^𝑝_{𝐿𝑝−𝜋𝑝,𝜋𝑝}+𝑓^𝑝_{𝐿𝑝−𝜋𝑝,𝜋𝑝}^1/𝑝<∞. (20) Here,𝜋_𝑝∶=2¹₀d𝑠(1−𝑠^𝑝)^1/𝑝=2𝜋.𝑝sin𝜋/𝑝 (21) We also set𝒮∶=𝑓∈𝑋∶‖𝑓‖_{𝐿𝑝(−𝜋𝑝,𝜋𝑝)}=1. (22) Let 𝜀>0 be a fixed number, and let 𝑞∶ℝ→ℝ be a given continuous (2𝜋_𝑝)-periodic function of 𝑥∈ℝ.

We define 𝐽_𝜀,𝑆∶𝑋→𝑋^∗ by 𝐽_𝜀(𝑢),𝑣∶=^𝜋𝑝_−𝜋𝑝𝑢^′𝑝−2𝑢^′𝑣^′d𝑥+𝜀^𝜋𝑝_−𝜋𝑝𝑞|𝑢|^𝑝−2⟨𝑆(𝑢)⟩𝑢𝑣d𝑥,,𝑣∶=^𝜋𝑝_−𝜋𝑝|𝑢|^𝑝−2𝑢𝑣d𝑥 (23) for 𝑢,𝑣∈𝑋. We also set𝐹_𝜀(𝑢)∶=^𝜋𝑝_−𝜋𝑝𝑢^′𝑝d𝑥+𝜀^𝜋𝑝_−𝜋𝑝𝑞|𝑢|^𝑝d𝑥. (24) The operator equation𝐽_𝜀(𝑢)−𝜆𝑆(𝑢)=0 (25) corresponds to the weak formulation of the following eigenvalue problem: −𝑢^′𝑝−2𝑢^′′+𝜀𝑞(𝑥)|𝑢|^𝑝−2𝑢=𝜆|𝑢|^𝑝−2𝑢inℝ,𝑢is2𝜋_𝑝-periodicinℝ. (26) For 𝜀=0 the set of all eigenvalues of (26) is given by the sequence𝜆₀=0,𝜆_2𝑛−1=𝜆_2𝑛=𝑝−1𝑛^𝑝,𝑛=1,2,…, (27) and it is proved in [2] that each eigenvalue 𝜆_𝑘(𝑘=0,1,2,…) has a Ljusternik-Schnirelmann variational characterization:𝜆_𝑘=inf_{𝐴∈ℱ𝑘+1}max_𝑢∈𝐴^𝜋𝑝_−𝜋𝑝𝑢^′(𝑥)^𝑝d𝑥. (28) The eigenvalue 𝜆₀ is simple, and the corresponding eigenfunction is a constant (and thus does not change the sign in [−𝜋_𝑝,𝜋_𝑝]). For any 𝑛=1,2,…, we have that 𝜆_2𝑛−1=𝜆_2𝑛 is not a simple eigenvalue. Given any fixed 𝑡∈ℝ, every function sin_𝑝(𝑛𝑥+𝑡) of 𝑥∈ℝ is an eigenfunction associated with 𝜆_2𝑛−1=𝜆_2𝑛. Here, sin_𝑝𝑥 for 𝑥∈[0,𝜋_𝑝/2] is defined implicitly by the formula𝑥=^sin𝑝𝑥₀1−𝑠^{𝑝−1/𝑝}d𝑠 (29) and then extended to an odd (about 0), (2𝜋_𝑝)-periodic function on ℝ.

Remark 2. The reader should notice that only in the linear case 𝑝=2 do the corresponding eigenfunctions associated with 𝜆_2𝑛−1=𝜆_2𝑛 form a linear space of dimension two. In contrast, if 𝑝≠2, then the multiplicity should be understood in the sense of the Ljusternik-Schnirelmann variational characterization (28). In fact, in this case the set of all eigenfunctions associated with each eigenvalue 𝜆_2𝑛−1=𝜆_2𝑛 is a two-dimensional manifold, the linear hull of which is infinite-dimensional (cf. [2]). If these eigenfunctions are normalized by a scalar multiplication factor in order to belong to 𝒮, they form a one-dimensional submanifold in 𝐿^𝑝(−𝜋_𝑝,𝜋_𝑝), which is diffeomorphic to a circle, the linear hull of which is again infinite-dimensional.

For 𝜀>0,𝑞≠0 the problem (26) possesses a sequence of eigenvalues tending to infinity and given by𝜆^𝜀_𝑘=inf_{𝐴∈ℱ𝑘+1}max_𝑢∈𝐴𝐹_𝜀(𝑢) (30) for 𝑘=0,1,2,…. The first eigenvalue is simple, and the corresponding eigenfunction does not change sign. For each 𝑛=1,2,3,… we have𝜆^𝜀_2𝑛−1≤𝜆^𝜀_2𝑛<𝜆^𝜀_2𝑛+1≤𝜆^𝜀_2𝑛+2, (31) where equalities may possibly occur. Eigenfunctions associated with 𝜆^𝜀_2𝑛−1 and 𝜆^𝜀_2𝑛 have precisely 2𝑛 zeros in (−𝜋_𝑝,𝜋_𝑝], and, for each 𝑘=0,1,2,…, we havelim_𝜀→0+𝜆^𝜀_𝑘=𝜆_𝑘 (32) (see [2]). It is also proved in [2] that for any integers 𝑚,𝑛≥1 and for any 𝜀>0, small enough, there exists a special (2𝜋_𝑝)-periodic function 𝑞∈𝐶¹(ℝ) such that 𝜆^𝜀_2𝑛−1<𝜆^𝜀_2𝑛 and the open interval (𝜆^𝜀_2𝑛−1,𝜆^𝜀_2𝑛) contains at least 𝑚 eigenvalues of (26). In particular, these eigenvalues do not allow the variational characterization (30). In the paper [3] we have shown that in contrast with the eigenvalues of Ljusternik-Schnirelmann type that are “globally variational,” some of the above mentioned “new” eigenvalues from (𝜆_2𝑛−1,𝜆_2𝑛) have a variational characterization that has a “local character,” meaning that the minimum part (inf) of the minimax formula (30) is taken only locally in the sense mentioned above. In order to illustrate our method we focus on 𝑛=1, that is, we find “local variational” eigenvalue of (26) in the interval (𝜆^𝜀₁,𝜆^𝜀₂), provided 𝜀 is small enough.

The first critical level of Ljusternik-Schnirelmann type (L.S. type for short) for 𝐹₀ on 𝒮, 0=𝜆₀=min_𝑢∈𝒮𝐹₀(𝑢) is attained at ±𝜑₀(𝜑₀≡constant>0). The second critical level of L.S. type, 𝜆₁=𝑝−1, is defined by the minimax formula𝜆₁=min_𝛾∈𝒞1max_{[]𝜏∈−1,1}𝐹₀𝛾(𝜏), (33) where 𝒞₁ is the set of all continuous curves in 𝒮 that connect the “south pole”-𝜑₀ with the “north pole” 𝜑₀. The level 𝜆₁ is attained at the point 𝑢=𝛾_𝑡(0) on the curve 𝛾_𝑡∈𝒞₁(𝑡∈ℝ) defined by𝛾_𝑡(𝜏)∶=ℎ_𝑡(𝜏)_𝐿^𝑝_perℎ_𝑡(𝜏),𝜏∈[−1,1], (34) whereℎ_𝑡(𝜏)=𝜏𝜑₀+1−𝜏²𝑒(𝑡)(𝑥),𝑥∈ℝ, (35) and 𝑒(𝑡)(𝑥)∶=(2𝜋_𝑝/𝑝)^−1/𝑝sin_𝑝(𝑥+𝑡) is a normalized eigenfunction associated with 𝜆₁=𝜆₂ (see [3]). Notice that the curves 𝛾_𝑡 correspond to the “meridians” on 𝒮 with the points of “longitude” 𝑡 and the “latitude” 𝜏.

For construction of a special function 𝑞=𝑞(𝑥) the following observation plays the key role: the function sin_𝑝∶𝑥→sin_𝑝𝑥, from ℝ to ℝ, is real analytic in the open set ℝ⧵{(1/2)𝑘𝜋_𝑝∶𝑘∈ℤ} and, furthermore, it is not 𝐶³ at the point 𝑘𝜋_𝑝 (𝑘∈ℤ) if 1<𝑝<2 and it is not 𝐶³ at the point (𝑘+1/2)𝜋_𝑝 if (𝑘∈ℤ) 2<𝑝<∞. Due to these facts a function 𝑞 can be found that controls the “splitting process” that involves the eigenvalues 𝜆^𝜀₁ and 𝜆^𝜀₂. In particular, 𝑞 is constructed in such a way that the function of the independent variable 𝑡,𝐹𝑡;𝑞=^𝜋𝑝_−𝜋𝑝𝑞(𝑥)sin_𝑝(𝑥+𝑡)^𝑝d𝑥, (36) has a global minimizer 𝑡_min∈[0,𝜋_𝑝] and a local minimizer 𝑡_min∈(0,𝜋_𝑝) such that𝐹𝑡_min;𝑞<𝐹𝑡_min;𝑞. (37) This property together with fine estimates for 𝑝>2 (see [3]) allows us to prove the local hyperbolic geometry of the functional 𝐹_𝜀 on 𝒮. Namely, thanks to (37), we can "localize" the set of curves 𝒞₂⊊𝒞₁ that pass through a suitable open neighborhood of the function 𝑒(𝑡_min) in the topology of 𝐿^𝑝(−𝜋_𝑝,𝜋_𝑝). The “inf” in the variational characterization of 𝜆^𝜀₁ is then approached “away from” curves that belong to 𝒞₂. The family of curves 𝒞₂ is then proved to be invariant with respect to continuous odd semiflow on 𝒮 and𝜆^𝜀₁=inf_𝛾∈𝒞1max_{[]𝜏∈−1,1}𝐹_𝜀𝛾(𝜏)<inf_𝛾∈𝒞2max_{[]𝜏∈−1,1}𝐹_𝜀𝛾(𝜏)<𝜆^𝜀₂ (38) holds for 𝜀>0 small enough. Then,𝜆^𝜀₁<𝜆^𝜀₁=inf_𝛾∈𝒞2max_{[]𝜏∈−1,1}𝐹_𝜀𝛾(𝜏)<𝜆^𝜀₂ (39) is the desired variational eigenvalue that is not of the L.S. type.

This construction is not possible in the linear case, 𝑝=2. Besides the fact that all eigenvalues of the linear problem are of L.S. type, this follows from the analyticity of the function sin𝑥. For this reason the functional 𝐹 with desired properties does not exist in the case 𝑝=2, which illustrates the striking difference between the linear and nonlinear eigenvalue problem.

Acknowledgment

This work was supported by the European Regional Development Fund (ERDF), project “NTIS—New Technologies for Information Society,” European Center of Excellence, CZ.1.05/1.1.00/02.0090 and project KONTAKT ME09109.