International Journal of Mathematical Analysis Vol. 9, 2015, no. 42, 2053 - 2059 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.56164

Almost Everywhere Convex Functions in Generalized Sobolev Spaces Matloob Anwar and Hina Urooj School of Natural Sciences National University of Sciences and Technology Islamabad, Pakistan c 2015 Matloob Anwar and Hina Urooj. This article is distributed under the Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper we characterize almost every where convex functions using gener1,p(x) alized Sobolev space Wloc (Ω) under some condition imposed on the generalized weak gradient. We assume in this paper that the variable exponent p(x) is bounded on Ω, an open convex subset of Rn .

Keywords: Sobolev Spaces, Generalized Sobolev Spaces, mollification 1. Introduction Lebesgue and Sobolev spaces are most important spaces in mathematical analysis and these spaces have several interesting properties. We refer [2] and [4] for the basic theory of the variable exponent Lebesgue and Sobolev spaces, but give the main definitions for the reader’s convenience. Let p(x) be a measurable function on Ω ⊆ Rn with values in [1, ∞). Denote p+ = ess sup p(x) and p− = ess inf p(x). x∈Ω

x∈Ω

The generalized Lebesgue space Lp(x) (Ω) is the space of all measurable functions f : Ω −→ R such that Z Ip (f ) := | f (x) |p(x) dx < ∞. Ω

With respect to the norm given below this is a Banach space:

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kf kLp(x) = inf{λ > 0 : Ip (f /λ) ≤ 1}. 0

0

We say that Ω is compactly contained in Ω denoted as Ω ⊂⊂ Ω if there 0 p(x) exists a compact set K such that Ω ⊂ K ⊂ Ω. Lloc (Ω) is the space of 0 0 functions f such that f ∈ Lp(x) (Ω ) for all Ω ⊂⊂ Ω. The generalized Sobolev space W 1,p(x) (Ω) is the space of functions in Lp(x) (Ω) whose generalized weak gradient ∇f exists and lies in Lp(x) (Ω), with the norm kf kW 1,p(x) = kf kLp(x) + k∇f kLp(x) . 1,p(x)

p(x)

The space Wloc (Ω) is defined in the same manner as Lloc (Ω). La Torre [5] has proved the inclusion of convex functions in the usual Sobolev 1,p space Wloc (Ω) for all p ≥ 1, he also proved the converse for a.e.convex functions; see ref. [1] for the basic theory in the usual Sobolev space. This paper 1,p(x) investigates the results in [5] for generalized Sobolev space Wloc (Ω). 2. Main Results We assume, throughout the paper, that the variable exponent p(x) is bounded that is p+ < ∞. Ω is an open convex subset in Rn . Definition 2.1. A function defined on Ω f : Ω → R is said to be locally Lipschitz if for each x0 ∈ Ω there exists a neibourhood B(x0 , r) of x0 and a positive constant C such that |f (y) − f (x)| ≤ Cky − xk for all y, x ∈ B(x0 , r). C is called a Lipschitz constant, which will vary with x0 . Definition 2.2. A function f on Ω is said to be locally bounded if for each x0 ∈ Ω, there exists a neighbourhood U of x0 and a positive constant M such that |f (x)| ≤ M for all x ∈ U . Lemma 2.3. Let f : Ω → R be a locally Lipschitz function then each of its partial derivatives, if exists, is locally bounded. Proof. For x0 ∈ Ω, let B(x0 , r) be a neighbourhood of x0 . For x in B(x0 , r), choose h > 0 and the unit vectors ei = (0, 0, .., 1, ..0, 0, 0) (1 at the ith position, i = 1, , , , n ) in the direction of each coordinate axes, such that x + hei ∈ B(x0 , r). We can choose such h for each x: for instance, h with h < dist(x, ∂B(x0 , r)) will work. Given f is locally Lipschitz with Lipschitz constant say C, we have |f (x + hei ) − f (x)| ≤ C|h| for all x ∈ B(x0 , r). Dividing by h and applying limit h → 0, lim |f (x + hei ) − f (x)/h| ≤ C for all x ∈ B(x0 , r).

h→0

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Using continuity of modulus function and definition of partial derivative, we finally have ∂f (x) ∂xi ≤ C for all x ∈ B(x0 , r).  Theorem 2.4. Let the function f : Ω → R be a locally Lipschitz. Then f ∈ 1,p(x) Wloc (Ω). Proof. Since the given function f is a locally Lipschitz function, by Rademacher’s theorem f is almost everywhere differentiable and hence the gradient ∇f exists a.e in Ω. The Lipschitz continuity of f also assures the local boundedness of f and ∂f∂x(x) . Considering local boundedness of f , we have for every open set i U ⊂⊂ Ω there exists a constant mU such that |f (x)| ≤ mU for all x ∈ U , and hence we have Z

p(t)

|f (t)|

Ip (f ) =

Z dt ≤

U

Thus f ∈

p+ (mU )p(x) dx ≤ max{mp− U , mU }µ(U ) < ∞.

U

p(x) Lloc (Ω) and that ∂f∂x(x) ∈ i

on applying the same argument as above on p(x)

∂f (x) , ∂xi th

we conclude Lloc (Ω). Next we shall show that the classical i partial derivative of f is indeed the weak ith partial derivative of f . For an arbitrary ϕ ∈ C0∞ (Ω) and h sufficiently small we have Z Z f (x + hei ) − f (x) φ(x) − φ(x − hei ) φ(x)dx = − f (x)dx. h h Ω

Ω p(x)

Above integrals are defined since the product of a function in Lloc (Ω) with a function in C0∞ (Ω) is an integrable function. Applying limit h → 0 and using the Lebesgue convergence theorem for integrable functions, we have Z Z ∂f ∂φ φ(x)dx = − f (x)dx. ∂xi ∂xi Ω

Ω

Since φ was arbitrary we have our desired result.



Remark 2.5. Since convex functions on an open set are locally Lipschitz, above theorem also holds for convex functions. 1,p(x)

Theorem 2.6. Let f : Ω → R be a function in Wloc (Ω) satisfies f (y) − f (x) ≥< ∇f (x), y − x > almost everywhere y, x ∈ Ω then < y − x, ∇f (y) − ∇f (x) >≥ 0 almost everywhere y, x ∈ Ω. ∇f denotes the weak gradient of f . Proof. The proof follows by adding the given inequality with the inequality obtained from it by interchanging x and y. 

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Theorem 2.7. [6] Let the function f : Ω → R be a Gˆateaux differentiable. Then f is convex iff < ∇f (x) − ∇f (y), x − y >≥ 0 for all x, y ∈ Ω. Theorem 2.8. [6] Suppose f is Gˆateaux differentiable function on Ω then f is convex if and only if f (x) − f (y) ≥< ∇f (y), x − y > holds for all x, y ∈ Ω. Definition 2.9. The function ( 2 −1 Ce(kxk −1) φ(x) = 0

if kxk < 1, if kxk ≥ 1

belongs to C0∞ (Rn ) for any constant C, which can be chosen such that

R

φ(x)dx = 1.

Rn

For any  > 0, the function φ(x) generates a sequence of smooth functions φ
ε on Rn called the standard sequence of mollifiers, defined as φε (x) = ε1n φ xε , R satisfying φε (x)dx = 1 and having compact support B(0, ε). Rn

Definition 2.10. Let f ∈ L1loc (Ω), its mollification f : Ω → R is defined using the following convolution as Z f (x) = (φε ∗ f )(x) = φ(x − t)f (t)dt, Ω

where Ω = {x ∈ Ω : dist(x, ∂Ω) > }, that is for x ∈ Ω , B(x, ) ⊂ Ω and for each x, y in B(x, ) , the vector x − y lies in B(0, ). 1,p(x)

Theorem 2.11. Let the function f : Ω → R belongs to Wloc (Ω) and satisfies < y − x, ∇f (y) − ∇f (x) >≥ 0 for a.e. y, x ∈ Ω. Then for any compact set K in Ω there exists 0 > 0 such that the mollifications f are convex for all  such that 0 <  < 0 . Proof. Let K be a compact set in Ω then we can find some 0 > 0 such that K ⊂ Ω0 ⊂ Ω. This can be done by choosing 0 such that 0 < min{dist(x, ∂Ω)}. x∈K

Also it follows from the definition of Ω that Ω0 ⊂ Ω for 0 <  < 0 and hence for such , K ⊂ Ω . We have for all x, y ∈ K and  ∈ (0, 0 ) :   n X ∂fε ∂fε < x − y, ∇f (x) − ∇f (y) >= (xi − yi ) (x) − (y) ∂xi ∂xi i=1 =

n X

(xi − yi )

i=1

=

n X i=1

 Z 

Ω

 (xi − yi )

1  εn



Z φ Ω

(φε (x − z) − φε (y − z))

x−z ε



 f (z)dz − φ

 

∂f (z)dz  ∂xi

y−z ε



 f (z)dz 

Almost everywhere convex functions

=−

n X

 

Z  (xi − yi ) 

i=1

φ(s)

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  ∂f ∂f  (x − εs) − (y − εs) ds ∂xi ∂xi

B(0,1)

Z φ(s) < x − y, ∇f (x) − ∇f (y) > ds ≥ 0.

= B(0,1)

Hence f are convex on K, follows from Theorem 2.7.



Corollary 2.12. If f satisfies f (y) − f (x) ≥< ∇f (x), y − x > almost everywhere y, x ∈ Ω and is continuous then f is convex. 1,p(x)

Corollary 2.13. Let the function f : Ω → R belongs to Wloc (Ω) and satisfies < x − y, ∇f (x) − ∇f (y) >≥ 0 almost everywhere in Ω. Then f satisfies f (x) − f (y) ≥< ∇f (y), x − y > almost everywhere in Ω. 1,p(x)

Proof. The given function f belongs to Wloc (Ω), hence being locally integrable function, f and its weak gradient ∇f can be approximated by the mollified functions f and ∇f almost everywhere in Ω, that is f → f and ∇f → ∇f everywhere in Ω except a set A of measure zero. Also we have given < x − y, ∇f (x) − ∇f (y) >≥ 0 for all x, y ∈ Ω\A. Let x, y ∈ Ω\A. Let K be the compact set containing x and y. For this compact set K, by using Theorem 2.11, we can find a number 0 > 0 such that the f are convex for all  ∈ (0, 0 ). The functions f being convex on K, for all  ∈ (0, 0 ), and differentiable will satisfy f (x) − f (y) ≥< ∇f (y), x − y >. On applying limit  → 0 to this inequality we get the desired result.  Theorem 2.14. Let f : Ω → R be an almost everywhere convex function then 1,p(x) f belongs to Wloc (Ω) and satisfies f (x) − f (y) ≥< ∇f (y), x − y > almost everywhere in Ω . Proof. The given function f is almost everywhere convex, hence can be written as a sum of two functions g and h, that is, f = g + h, where g is a convex function while h is a zero function almost everywhere in Ω. The function g being a convex function is locally Lipschitz and hence almost everywhere differentiable on Ω. Since g is differentiable on Ω\A, where µ(A) is zero, using Theorem 2.7 we have g(x) − g(y) ≥< ∇g(y), x − y > for all x, y ∈ Ω\A, where ∇g(y) is the weak gradient of g. Since f and g are equivalent the weak 1,p(x) gradient of f is the same as for g, hence f ∈ Wloc (Ω) and above equation can be written as f (x) − f (y) ≥< ∇g(y), x − y > for all x, y ∈ Ω\A. 

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Definition 2.15. A function f defined on an open set Ω of Rn is said to be differentiable at c if there exists a linear functional df (c; v), defined for all v in Rn such that for all δ > 0 satisfies the following: |f (x) − f (c) − df (c; x − c)| ≤ δkx − ck for x ∈ Ω. 1,p(x)

Theorem 2.16. Suppose that f : Ω → R be a function in Wloc (Ω) which satisfies f (x) − f (y) ≥< ∇f (y), x − y >, f (x) → f (x) and ∇f (x) → ∇f (x) for all x in Ω\A, with µ(A) = 0. Then f is almost everywhere convex on Ω. Proof. To reach the desired conclusion, we will first prove that f is an element 1,∞ (Ω), that is, f and its weak gradient ∇f are essentially locally bounded of Wloc on Ω. We prove that f is essentially locally bounded on Ω by proving the local boundedness of f on some compact set in Ω. Choose a cube Q in Ω\A centered at x0 with vertices v1 , v2 , ....., v2n . Since Q is compact there is a positive number 0 such that f are convex on Q for all  in (0, 0 ). Also, it is given that f converges to f a.e in Ω\A, there exists a constant C1 such that f (xi ) ≤ C1 for all  in (0, 0 ) i = 1, 2, .....2n . Using convexity of f on Q and the fact that every x in Q is a convex combination of its vertices, we have f (x) ≤ C1 for all x in Q and for all  in (0, 0 ). Hence f is bounded above on Q. Now by symmetry of Q, for every x ∈ Q there exists y ∈ Q such that x0 = (x + y)/2. Hence f (x0 ) ≤ (f (x) + f (y))/2 which implies f (x) ≥ 2f (x0 ) − f (y) ≥ 2f (x0 ) − C1 = C2 . Hence f is bounded on Q by the constant C = max(C1 , C2 ), and there exists an open ball B(x0 , δ) such that f is bounded a.e. on B(x0 , δ), that is kf kL∞ (B(x0 ,δ)) < C To prove the essential local boundedness of ∇f we will show that for sufficiently small  > 0, f are Lipschitzian on B(x0 , δ). Since for each sufficiently small , the functions f are defined, also bounded on B(x0 , δ), we can apply the definition of differentiability at x0 to each f as follows: For some η > 0 and x ∈ B(x0 , δ), we have |f (x) − f (x0 ) − df (x0 , x − x0 )| ≤ ηkx − x0 k for all x ∈ B(x0 , δ), which implies |f (x) − f (x0 )| < |df (x0 , x − x0 )| + ηkx − x0 k.

(1)

Since df (x0 ; x − x0 ) =< ∇f (x0 ), x − x0 >, the Cauchy Schwartz inequality implies that |df (x0 ; x − x0 )| ≤ k∇f (x0 )kkx − x0 k, hence inequality (2.1) becomes 0

|f (x) − f (x0 )| ≤ k∇f (x0 )kkx − x0 k + ηkx − x0 k = C kx − x0 k, 0

for all x ∈ B(x0 , δ), where C = η +k∇f (x0 )k. Hence for  small enough, f are Lipschitz on B(x0 , δ) and therefore have gradient ∇f which are bounded by 0 C . This implies that the weak gradient ∇f is essentially locally bounded on 1,∞ B(x0 , δ). Hence it is proved that f is an element of Wloc (Ω). Since functions

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1,∞ in Wloc (Ω) can be identified as locally Lipschitz functions (see [3]), we can find a continuous function F on Ω such that F = f almost everywhere on Ω. Also, the function F satisfies the inequality given in the hypothesis almost everywhere x in Ω, hence convex by Corollary 2.12, which in turn implies that f is almost every where convex. 

Conclusion. In this paper we have discussed some interesting properties of almost every where convex functions in Generalized Sobolev spaces. These results are useful to study some further applications of convex functions in Generalized Sobolev Spaces. Like in Sobolev Spaces convex functions are used to solve differential equations. Acknowledgements. This research work is funded by School of Natural Sciences, National University of Sciences and Technology, Islamabad Pakistan. References [1] H. Brezis, Functional Analysis Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010. http://dx.doi.org/10.1007/978-0-387-70914-7 [2] L. Diening, P. Harjulehto, P. H¨ast¨o, M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer Verlag, 2011. http://dx.doi.org/10.1007/978-3-642-183638 [3] L.C. Evans, Partial differential equations, Berkeley Mathematics Lectures Notes, Berkeley, 1994. [4] O. Kov´ aˇcik, J. R´ akosn´ık, On spaces Lp(x) and W k,p(x) , Czechoslovak Mathematical Journal, 41 (1991), 552-618. [5] D. La Torre, M. Rocca, Almost everywhere convex functions on Rn and weak derivatives, Rendiconti del Circolo Matematico di Palermo, 50 (2001), 405-414. http://dx.doi.org/10.1007/bf02844421 [6] C.P. Niculescu and L.E. Persson, Convex Functions and Their Applications: A Contemporary Approach, Springer Verlag, New York, 2006. http://dx.doi.org/10.1007/0387-31077-0

Received: June 23, 2015; Published: August 12, 2015