Maximal monotone operators on hilbert spaces and applications

Thesis Overview

<p> </p><p>Hilbert Spaces and Sobolev Spaces<br>The aim of this chapter is to recall some results on Lp spaces, distributions and<br>Sobolev spaces that we use in the next chapter.<br>1.1 Hilbert spaces<br>A normed vector space is closed under vector addition and scalar multiplication.<br>The norm defined on such a space generalises the elementary concept of the length<br>of a vector. However, it is not always possible to obtain an analogue of the dot<br>product, namely<br>a:b = a1b1 + a2b2 + a3b3<br>which yields<br>jaj =<br>p<br>a:a<br>which is an important tool in many applications. Hence, the question arises whether<br>the dot product can be generalised to arbitrary vectors spaces. In fact, this can be<br>done and leads to inner product spaces and complete inner product spaces, called<br>Hilbert spaces.<br>Definition 1.1. Let H be a linear space. An inner product on H is a function<br>h:; :i : H H ! R<br>1<br>defined on H H with values in R such that the following conditions are satisfied.<br>For x; y; z 2 H; ; 2 R<br>a) hx; xi 0 and hx; xi = 0 if and only if x = 0<br>b) hx; yi = hy; xi<br>c) hx + y; zi = hx; zi + hy; zi<br>The pair (H; h:; :i) is called an inner product space. A Hilbert space, H is a complete<br>inner product space ( complete in the metric defined by the inner product ).<br>1.1.1 Examples<br>1. Euclidean space Rn.<br>The space Rn is a Hilbert space with inner product defined by<br>hx; yi =<br>Xn<br>i=0<br>xiyi<br>where,<br>x = (x1; x2; :::; xn) and y = (y1; y2; :::; yn)<br>We obtain<br>jjxjj =<br>p<br>hx; xi = (<br>Xn<br>i=0<br>x2i<br>)<br>1<br>2<br>2. Space L2(<br>):<br>L2(<br>) := ff :<br>! R : f is measurable and<br>R</p><p>f2dx &lt; 1g, where<br>is an open<br>set in Rn; is a Hilbert space with the inner product defined<br>hf; gi =<br>Z</p><p>f(x)g(x)dx<br>and<br>jjfjj = (<br>Z</p><p>jf(x)jdx)<br>1<br>2<br>3. Hilbert sequence space l2.<br>l2 := f(xn)n0 R :<br>1P<br>i=0<br>jxij2 &lt; 1g is a Hilbert space with inner product<br>defined by<br>hx; yi =<br>X1<br>i=0<br>xiyi<br>2<br>Convergence of this series follows from Cauchy-Schwar’z inequality and the fact that<br>x; y 2 l2, by assumption.<br>The norm is defined by<br>jjxjj = (<br>X1<br>i=0<br>jxij2)<br>1<br>2<br>An inner product on H defines a norm on H given by<br>jjxjj =<br>p<br>hx; xi<br>and a metric on H given by<br>d(x; y) = jjx ô€€€ yjj =<br>p<br>hx ô€€€ y; x ô€€€ yi<br>Hence, inner products are normed spaces and Hilbert spaces are Banach space.<br>A norm on an inner product space satisfies the important parallelogram equality<br>jjx + yjj2 + jjx ô€€€ yjj2 = 2(jjxjj2 + jjyjj2) for all x; y 2 H<br>Not all normed spaces are inner product spaces.<br>4. Space lp.<br>Let 1 p &lt; 1 be a fixed real number, we define lp space as<br>lp = f(xn)n0 R :<br>X1<br>i=0<br>jxijp &lt; 1g:<br>When p 6= 2, lp is not a Hilbert space.<br>5. Space C([a; b];R).<br>The space C([a; b];R) provided with supremum norm is not a Hilbert space.<br>Proposition 1.2. Let (H; h:; :i) be an inner product space. Then, for all x; y 2 H<br>a. jhx; yij jjxjjjjyjj (Schwar0z inequality) where the equality holds if and<br>only if x,y are linearly dependent.<br>b. jjx + yjj jjxjj + jjyjj (triangle inequality) where the equality holds if<br>and only if x=cy (c 0)<br>Proposition 1.3. (Continuity of inner product). Let (xn)n0; (yn)n0 be sequences<br>in H, such that xn ! x and yn ! y, then<br>hxn; yni ! hx; yi:<br>3<br>1.2 Function Spaces<br>Here, we recall the definitions of functions spaces used in this thesis.<br>1.2.1 Lp Spaces<br>Definition 1.4. Let<br>be a nonempty open set in Rn, for 1 p &lt; 1, we define<br>Lp(<br>) := ff :<br>! R : f is measurable and<br>Z</p><p>jf(x)jpdx &lt; 1g<br>Remark 1.5. We say two functions f and g are equivalent if f = g almost everywhere.<br>Then we define Lp(<br>) spaces as the equivalent classes for this relationship.<br>The space Lp(<br>) can be seen as a space of functions. We do however, need to be<br>careful sometimes. For example, saying that f 2 Lp(<br>) is continuous means that f<br>is equivalent to a continuous function. Now, for f 2 Lp(<br>), we define<br>jjfjjp = (<br>Z</p><p>jf(x)jpdx)<br>1<br>p ; 1 p &lt; 1<br>The Lp(<br>) is a Banach space.<br>1.2.2 Test functions<br>Definition 1.6. Let f :<br>! R be a continuous function. The support is<br>supp(f) := fx 2<br>: f(x) 6= 0g<br>The function is said to be of compact support on<br>if the support is a compact set<br>contained inside<br>.<br>Definition 1.7. The space of test functions in<br>, denoted by D(<br>) is the space of<br>all C1 functions defined on<br>which have compact supports in<br>.<br>C1(<br>) denotes the space of all real-valued functions on<br>of class C1.<br>= (1; 2; :::; n) 2 Nn is called multi-index with length jj =<br>Pn<br>i=1<br>i.<br>Let x = (x1; x2; :::; xn) 2 Rn. We write D = @jj<br>@<br>1<br>x1 :::@n<br>xn<br>and it acts on the space<br>C1(<br>). Thus, for f 2 C1(<br>), Df = @jjf<br>@<br>1<br>x1 :::@n<br>xn<br>is it partial derivatives of order jj.<br>Definition 1.8. Let f ngn0 be a sequence in D(<br>) and 2 D(<br>).<br>n ! in D(<br>) if<br>1. 9 a compact set K<br>: supp( ); supp( n) K; for all n 1<br>2. D n ! D uniformly on K; 8 2 Nn:<br>4<br>1.2.3 Distributions<br>Definition 1.9. A distribution on<br>is any continuous linear mapping T : D(<br>) !<br>R. The set of all distributions is denoted by D0(<br>).<br>Remark 1.10. By linearity, to show that T is continuous, it is enough to show that,<br>if n ! 0 in D(<br>), then it is enough to show that (T; n) ! 0 in R:<br>Definition 1.11. A function f :<br>! R is locally integrable if for any compact set,<br>K<br>, we have that Z<br>K<br>jf(x))jdx &lt; 1<br>The collection of all locally integrable functionals on<br>is denoted by L1l<br>oc(<br>)<br>If f 2 C(<br>), then f 2 L1l<br>oc(<br>). For any f 2 L1l<br>oc(<br>), f gives a distribution Tf defined<br>by<br>(Tf ; ) =<br>Z</p><p>f(x) (x)dx; for all 2 D(<br>)<br>Definition 1.12. If T 2 D0(<br>) is a distribution on an open set<br>Rn, and if<br>is any multi-index, we define the distribution DT by<br>(DT; ) = (ô€€€1)jj(T;D ) (1.1)<br>and it is the th partial derivative of T.<br>So, the map D : D0(<br>) ! D0(<br>) defined in (1.1) is linear and continuous.<br>1.3 Sobolev spaces<br>Sobolev spaces are based on the concept of weak (distributional) derivatives. It gives<br>us a modern approach to the study of differential equations.<br>Definition 1.13. Let 1 p &lt; 1 and k be a non-negative integer. Then, Sobolev<br>space Wk;p(<br>) is defined by<br>Wk;p(<br>) := fu 2 LP (<br>) : Du 2 Lp(<br>); 8 0 jj kg<br>The space is equipped with the norm<br>jjujjWk;p(<br>) := (<br>X<br>0jjk<br>jjDujjp<br>LP (<br>))<br>1<br>p<br>5<br>WK;p<br>0 (<br>) = D(<br>)</p><p>Wk;p(<br>) i.e., WK;p<br>0 (<br>) is the closure of D(<br>) with respect to the<br>norm jj:jjWk;p(<br>).<br>When p=2, we write Hk(<br>) = Wk;2(<br>) and Hk<br>0 (<br>) = Wk;2<br>0 (<br>) and these are real<br>Hilbert spaces with the following inner product<br>hu; viHk(<br>) =<br>X<br>0jjk<br>Z</p><p>DuDvdx<br>and the norm<br>jjujjHk(<br>) = (<br>X<br>0jjk<br>jjDujj2<br>L2(<br>))<br>1<br>2<br>For, k=0,<br>W0;p(<br>) = LP (<br>):<br>Wk;p(<br>) are Banach spaces.<br>Given that<br>is smooth, then:<br>Wk;p<br>0 (<br>) := fu 2 Wk;p(<br>) : u = Du = ::: = Dkô€€€1u = 0 on @<br>g:<br>For p=2, we have<br>Wk;2<br>0 (<br>) := fu 2 Wk;2(<br>) : u = Du = ::: = Dkô€€€1u = 0 on @<br>g<br>For p=2,and k=1 , we have<br>W1;2<br>0 (<br>) := fu 2 W1;2(<br>) = H1(<br>) : u = 0 on @<br>g<br>and we denote it by H1<br>0(<br>)<br>For p=2, k=2, we write<br>W2;2(<br>) = H2(<br>):<br>Theorem 1.14. Let<br>be smooth and u 2 L2(<br>) such that u 2 L2(<br>). Then<br>u 2 H2(<br>):<br>6<br>Green’s Formula<br>Theorem 1.15. Let<br>be bounded and smooth. Let u 2 H2(<br>) and v 2 H1(<br>),<br>then Z</p><p>ru:rvdx =<br>Z<br>@</p><p>v<br>@u<br>@n<br>ds ô€€€<br>Z</p><p>vudx<br>where @u<br>@n denotes the normal derivative defined by @u<br>@n = ru:ô€€€!n<br>:<br>where ô€€€!n<br>denotes the normal vector.<br>if u = v, then<br>Z</p><p>jjrujj2dx =<br>Z<br>@</p><p>u<br>@u<br>@n<br>ds ô€€€<br>Z</p><p>uuds<br>=<br>Z<br>@</p><p>u<br>@u<br>@n<br>ds +<br>Z</p><p>u(ô€€€u)ds<br>Then,<br>Z</p><p>(ô€€€u)udx =<br>Z</p><p>jjrujj2dx ô€€€<br>Z<br>@</p><p>u<br>@u<br>@n<br>ds<br>Theorem 1.16. (Lax-Milgram). Let a : V V ! R be a bilinear, continuous,<br>and coercive functional. Then, for each f 2 V 9! u 2 V :<br>a(u; v) = (f; v); for all v 2 V<br>Proposition 1.17. (Poincaré’s inequality). Suppose<br>is a bounded set. Then<br>there exists a constant C(<br>) &gt; 0 such that<br>jjujjL2(<br>) C(<br>)jjrujjL2(<br>); for all u 2 W1;2<br>0 (<br>):</p><p>&nbsp;</p> <br><p></p>