Some Applications to Lebesgue Points in Variable Exponent Lebesgue Spaces

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1 Çankaya University Journal of Science and Engineering Volume 7 (200), No. 2, 05 3 Some Applications to Lebesgue Points in Variable Exponent Lebesgue Spaces Rabil A. Mashiyev Dicle University, Department of Mathematics, 2280 Diyarbakır, Turkey mrabil@dicle.edu.tr Özet. Dağılım parametreli sistemler için optimal kontrol teorisinde kullanışlı olan, Lebesgue regüler noktalarının bazı sonuçları ispatlanmıştır. Anahtar Kelimeler. Değişken üslü Lebesgue uzayı, regüler Lebesgue noktaları, optimal kontrol. Abstract. Some corollaries of Lebesgue s regular points which are useful in the theory of optimal control for distributed parameter systems are proved. Keywords. Variable exponent Lebesgue spaces, Lebesgue regular points, optimal control.. Introduction In recent years, stimulated by some problems in fluid dynamics and differential equations, a great interest has arisen in spaces with variable exponents and in the extension of the results of classical Harmonic Analysis to the variable exponent setting. The mathematical applications of these problems have been carried out by the variational integrals with non-standard growth [, 7]. Let p : [, ) be a measurable bounded function which is called the variable exponent on R n, and write p + = sup t p(t) and p = inf t p(t). The variable exponent Lebesgue space L p( ) () consists of all measurable functions f : R such that the modular is finite. If p + <, then I(f) := f p( ) = inf f(t) p(t) dt { λ > 0 : I Received February, 200; accepted October 25, 200. ( ) } f λ ISSN c 200 Çankaya University

2 06 Mashiyev defines a Luxemburg norm on L p( ) (). This makes L p( ) () a Banach space. If p is a constant function, then the variable exponent Lebesgue space L p( ) () coincides with the classical Lebesgue space. The basic properties of these spaces can be found in [3, 4, 8]. Proposition. [3, 8] Let p be a measurable function such that < p p(t) p + <, and be a measurable set in R n. Then, Let f Lp( ) () > = f p L p( ) () I(f) f p+ L p( ) (), f Lp( ) () < = f p+ L p( ) () I(f) f p L p( ) (). (a, b) {t R n : a i < t i < b i, i =, 2,..., n} be an open parallelepiped in R n, and Q(τ, r) {t R n : τ i r 2 t i τ i + r } 2, i =, 2,..., n be a closed cube with its centre at τ and its edges parallel to coordinate axes with their length equal to r > 0. The classical Lebesgue theorem on regular points accepts various generalizations and may be used in many different applications including in the calculation of variations of functionals of distributed problems in the theory of optimization. The non-trivial Lebesgue Theorem on regular points in variable exponent Lebesgue spaces (see [6]) and its consequences will be used here to calculate the functional s variations. Moreover, it may be applied in various problems of optimal control theory. 2. The Main Results Theorem. (i) Assume that R n is a bounded measurable set, f(s, y) : R l R is a Carathéodory function (i.e the map y f(s, y) is continuous for a.e. s, while s f(s, y) is measurable on for all y R l ), and the condition f(s, y) a(s) + C y holds, where s, C = const. > 0, and a( ) L p( ) (), i.e. a( ) Lp( ) () A; (ii) y 0 (s) and {y h (s) : h {τ, r} [0, )} L l p( ) (), such that y h ( ) p( ) dt B, y h y 0 L ı p( ) () 0,

3 CUJSE 7 (200), No as r 0 uniformly for τ ; (iii) < p p(t) p + < and f L p( ) (). Then, for almost everywhere, lim f(t, y r 0 r n h ) f(τ, y 0 ) p(t) dt = 0, τ, uniformly in y 0, y h, where (τ) = {t : t [τ, τ + r]}, i.e. τ is known as a Lebesgue point. Proof. Let {r i } i= be countable, and dense everywhere in [0, ). Since f L p( ) () and p + <, then f(t, y h ) r i p(t) L loc ( t) uniformly. Indeed, with fixed i, we havḙ f(t, y h ) r i p(t) dt 2 p+ f(t, y h ) r i p(t) dt 2 p+ 2 p+ [ )] r i p(t) + 2 p+ ( a(t) p(t) + C p+ y h p(t) dt [ ( r i p+ + 2 p+ A p+ + C p+ B p+)] dt <. ( ) By the Lebesgue theorem, for each i N, a subset E i, E = E i i= Lebesque measure zero is available, and with t \ E, we obtain lim r 0 r n f(t, y h ) r i p(t) dt = f(τ, y h ) r i p(τ). of Assume that 0 < ε <, 0 < ε < 4, 0 < ε 2 < 4, and τ /E. Choose r i such that and f(, y h ) r i p( ) < ε 2 p+ f(, y 0 ) r i p( ) < ε 2 with all and any y 0, y h L l p( ) (). Thus, we get 2 p+

4 08 Mashiyev lim sup r 0 r n f(t, y h ) f(τ, y 0 ) p(t) dt 2 p+ lim r 0 sup r n f(t, y h ) r i p(t) dt + r n 2 p+ ( f(τ, y h ) r i p(t) + f(τ, y 0 ) r i ) 2 p+ The proof of Theorem is complete. r i f(τ, y 0 ) p(t) dt ( ε 2 + ε ) 2 < ε. p+ 2 p+ In the distributed parameters systems and optimal control theories we face the following problem [2, 5]: Let R n be a bounded measurable set and {f[m, τ]( ) L p( ) () : m N, τ } be a family of functions satisfying the following condition: Moreover, assume that f[m, τ]( ) Lp( ) () 0, as m +, uniformly in τ. F p [m] (τ) = rm n [m,τ] f[m, τ](t) p(t) dt, where [m, τ] {t : t [τ, τ + r m ]}, r m 0 as m +. Then, the question is for which subsequence {m k } N and set 0, meas( 0 ) = meas() of Lebesque measure zero will be valid. F p [m k ](τ) 0 τ 0 as k + () Let s explain why we refer to a subsequence {m k } rather than a whole sequence {m}. We consider the simple case, i.e. when subintegral functions do not relate to τ and a family of functions {f m ( ) L p( ) () : m N} such that f m ( ) Lp( ) () 0 as m +, and F p [m](τ) = rm n [m,τ] f m (t) p(t) dt.

5 CUJSE 7 (200), No We will show that, in general, convergence F p [m](τ) 0 for almost all τ as k + may not be true. Remark. Let n =, = [0, ], and define the sets [ j Q k,j = k, j ) [ j, P k,j = k k, j ) k γ k,j, where γ k,j = j2 k for j =,..., k, k = 2, 3,.... Further, let s assume P k = P k,j, k = 2, 3,... and Q 0 = j= P k. Then it is clear that meas( 0 ) > 0. Let ϕ k,j be the characteristic function of sets Q k,j, and f (t) = ϕ 2,, f 2 (t) = ϕ 2,2, f 3 (t) = ϕ 3, f 4 (t) = ϕ 3,2, f 5 (t) = ϕ 3,3,.... Similarly, let constitute the following sequence k=2 r = γ 2,, r 2 = γ 2,2, r 3 = γ 3,, r 4 = γ 3,2, r 5 = γ 3,3,.... It is obvious that f m ( ) L p( ) [0, ] and f m ( ) Lp( ) [0,] 0, r m 0 as m +, while F p [m](τ) = rm n [m,τ] f m (t) p(t) dt 0, τ 0 as m +. Indeed, if we take τ 0, then τ P k for k = 2, 3,..., and there exists j = j(k) such that τ P k,j. However, by this construction, if r m = γ k,j, then [m, τ] = [τ, τ + γ k,j ] Q k,j, and thus f m (t) = ϕ k,j (t) = for t [m, τ], and eventually F p [m](τ) =. This means the set F p [m](τ) contains a subset F p [m s ](τ) =. We will require the following definitions to obtain the main results. Definition. A system M of measurable sets containing a point ξ R n will have a compaction to the point ξ, if sets are as small in diameter as desired among system s sets and regular compaction, and a cube Q(ξ, h) e for all and any e M such that h n L meas(e), where L is a constant which is invariable with e. Definition 2. A system {[m, τ] : m N, τ } of measurable sets will have a regular compaction on a set, provided (i) τ [m, τ] m N,

6 0 Mashiyev (ii) sup diam [m, τ] 0 with m +. τ Remark 2. It is obvious that the subsystem deriving from the sets system which have regular compaction on with constant τ will have compaction to the point τ. In addition, if the condition of regularity holds we can speak about a uniform and regular compaction sets system [m, τ] {t : t [τ, τ + r m ]}, r m 0. It is obvious that a parallelepiped system will be uniformly and regularly compactable. It is also clear that the said class will extend further. Theorem 2. Assume that f(t, y) : satisfies the following conditions: R l R is a Carathéodory function and (i) Sets system {G[m, τ] : m N, τ } is uniformly and regularly compactable on, [m, τ] G[m, τ], (ii) < p, p + <, { y[m, h]( ) : m N, h G} L l p( ) (), and {b m ( ) L p( ) ()} such that b m ( ) Lp( ) () 0 as m +, and y[m, h]( ) b m ( ) m N, h G for almost all τ, (iii) y L l p( ) () we will get f(, y( )) L p( )(), where f(, y( )) Lp( ) () 0, y( ) L l p( ) () 0. Remark 3. To get the condition (ii), it is sufficient to show the following inequality holds true y[m, h](t) b m (t)n ( ) χ [m,τ] Ψ(t) Lp( ) (), m N for almost all τ, where b m ( ) L p( ) () and Ψ( ) L p( ) () are some constant functions and N( ) : R + R is a nondecreasing function such that N(β) +0 with β +0. Proof. [Remark 3] Let s point out that for all m N the set [m, τ] is contained in some closed ball of radius diam [m, τ]. Therefore, its measure does not exceed this ball s measure, i.e. diam [m, τ] C(n)[diam [m, τ]] n, C(n) = const. > 0. Proof. [Theorem 2] This can be obtained immediately from the proof of Theorem 3. Lemma. Assume S() is a space of measurable functions for almost everywhere on, l N, c( ), d( ) S l ()-measurable on, l-vector functions, c(t) d(t) for almost all t such that c, d R l : c d c j d j, j =, 2,..., l, [c, d] = [c, d ] [c 2, d 2 ]... [c l, d l ],

7 CUJSE 7 (200), No. 2 and f(t, y) : R l R is measurable in t and continuous in y R l. Then the function ϕ(t) max f(t, y) will be measurable on and there exists y [c(t),d(t)] θ( ) T [c, d] {y S l () : y(t) [c(t), d(t)]} such that f(t, θ( )) = ϕ(t) for almost all t. Proof. The proof can be obtained immediately from [5, Assertion.2, p. 326 and Theorem.4, p. 327]. Theorem 3. Assume < p p(t) p + < and f(t, y) : R l R is a Carathéodory function and satisfies the following conditions: (i) y L l p( ) () we get f(, y( )) L p( )(), (ii) b m ( ) L p( ) (), m N. Then b m ( ) Lp( ) () 0 b m (t) p(t) dt 0 with y( ) L l p( ) () 0. Moreover, for sufficiently large m N, we have ϕ(t) max{f(t, y) : y R l, y b m (t)}, (2) where ϕ m ( ) L p( ) () and ϕ m ( ) Lp( ) () 0 as m +. Proof. In view of Lemma, for each m N the function ϕ m (t) is measurable on. Let ε > 0. By the condition (i), there exists a number δ(ε) > 0 such that f(t, y) p(t) dt f(t, y) Lp( ) () < ε, y L l p( )(), y(t) p(t) dt y( ) L l p( ) () < δ(ε). Since b m ( ) Lp( ) () 0 as m +, we can choose a number m = m (ε) N sufficiently large such that l b m ( ) Lp( ) () < δ(ε) with m m. By Lemma, we have θ m ( ) S l () : θ m (t) b m (t),

8 2 Mashiyev for almost all t, ϕ m (t) = f(t, θ m (t)). Thus, if we consider θ m ( ) L l p( ) () = θ m Lp( ) () l b m ( ) Lp( ) (), we will get ϕ m ( ) L p( ) (), ϕ m ( ) Lp( ) () ε, m m (ε). This implies that (2) is fulfilled. Therefore, the proof of Theorem 3 is complete. Theorem 4. Assume conditions (i), (ii), and (iii) of Theorem 2 hold with p + <. Then there exists a subsequence m k + as k +, and a set 0 with meas( 0 ) = meas(), such that τ 0 and h G f(t, y[m k, h](t)) p(t) dt 0 as k +. meas(g[m k, τ]) [m k,τ] Proof. We may assume that f(t, y) 0 for almost all t, y R l. Set Then by (2), we get ϕ m max y [ b(t),b(t)] l f(t, y). f(t, y[m, h](t)) ϕ m (t) h G, for almost all t. (3) By Theorem 3, ϕ m ( ) L p( ) () for all sufficiently large m N and ϕ m ( ) Lp( ) () 0 as m +. Thanks to the Theorem on the Convergence Regulator, a subsequence ε k +0 as k + and a function ω( ) L p( ) () can be found so that ϕ m (t) ε k ω(t) for almost all t. (4) Let s choose any closed parallelepiped [] including all sets G[m, τ], m N, τ along with a set and extend function ω( ) to this parallelepiped with zero. Then, obviously, ω( ) L p( ) (), and directly from (3) and (4) and by contractiveness

9 CUJSE 7 (200), No. 2 3 of system {G[m, τ], m N, τ } to the point τ, it follows that f(t, y[m k, h](t)) p(t) dt meas(g[m k, τ]) [m k,τ] = = ε k meas(g[m k, τ]) ε k meas(g[m k, τ]) [m k,τ] G[m k,τ] ω(t) p(t) dt for almost all τ. Hence, the proof of Theorem 4 is complete. References ω(t) p(t) dt 0 ω(τ) p(τ) = 0, [] O. M. Buhrii and R. A. Mashiyev, Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Nonlinear Anal.-Theor. 70 (2009), [2] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, American Mathematical Society, Boston, MA [3] O. Kováčik and J. Rákosník, On spaces L p(x) and W k,p(x), Czech. Math. J. 4 (99), [4] R. Mashiyev, Some properties of variable Sobolev capacity, Taiwan. J. Math. 2 (2008), [5] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems, Nauka, Moscow 974. [6] P. Harjulehto and P. Hästö, Lebesgue points in variable exponent spaces, Ann. Acad. Sci. Fenn.-M. 29 (2004), [7] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, 748, Springer-Verlag, Berlin, [8] X. Fan and D. Zhao, On the spaces L p(x) () and W m,p(x) (), J. Math. Anal. Appl. 263 (200),