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A Brief Introduction to Weak Solutions Appendix A A Brief Introduction to Weak Solutions A.1 Weak Derivative and Sobolev Spaces Let be a bounded and open subset of <n.LetD./ be the vector space of 1 C ./-functions having a compact support S' contained in : ˚ « 1 D./ D ' 2 C ./; S' : If f 2 C 1./, then it is always possible to write Z Z Z @f @ @' 'dD .f '/ d f d: (A.1) i i i @x @x @x If the first integral on the right-hand side is transformed into an integral over the boundary @ by Gauss’s theorem and the hypothesis ' 2 D./, then the previous identity becomes Z Z @f @' 'dD f d: (A.2) i i @x @x Conversely, if for any f 2 C 1./ a function 2 C 0./ exists such that Z Z @' ' d D f d 8' 2 D./; (A.3) i @x then, subtracting (A.2) and (A.3), we have the condition Z  à @f 'dD 0 8' 2 D./; (A.4) i @x © Springer Science+Business Media New York 2014 463 A. Romano, A. Marasco, Continuum Mechanics using MathematicaR , Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-1-4939-1604-7 464 A A Brief Introduction to Weak Solutions which in turn implies that D @f = @ x i , owing to the continuity of both of these functions. All the previous considerations lead us to introduce the following definition. Definition A.1. Any f 2 L2./ is said to have a weak or generalized derivative if a function 2 L2./ exists satisfying the condition (A.3). It is possible to prove that the generalized derivative has the following properties: •iff 2 C 1./, then its weak derivative coincides with the ordinary one; • the weak derivative of f 2 L2./ is defined almost everywhere by the condition (A.3); i.e., it is unique as an element of L2./; • under certain auxiliary restrictions,1 we have that if is the weak derivative of the product f1f2, where f1;f2 2 L2./, then Z Z ' d D .1f2 C f12/ 'd; where i is the generalized derivative of fi , i D 1; 2. From now on, the weak derivative of f.x1;:::xn/ 2 L2./ with respect to xi will be denoted by @f = @ x i . Example A.1. The weak derivative of the function f.x/Djxj, x 2 Œ1; 1,isgiven by the following function of L2.Œ1; 1/: df 1; x 2 Œ1; 0/; D dx 1; x 2 .0; 1: This is proved by the following chain of identities: Z Z Z Z 1 df 1 d' 0 d' 1 d' 'dxD jxj dx D x dx x dx 1 dx 1 dx 1 dx 0 dx Z Z 0 1 0 1 D Œx'1 'dx Œx'0 C 'dx 1 0 Z Z 0 1 D 'dx C 'dx: 1 0 Example A.2. The function 0; x 2 Œ1; 0/; f.x/D 1; x 2 Œ0; 1; 1See [62], Section 109. A A Brief Introduction to Weak Solutions 465 does not have a weak derivative. In fact, we have Z Z 1 1 df d' 1 dx D f dx DŒ'0 D '.0/; 1 dx 1 dx and no function can satisfy the previous relation. Definition A.2. The vector space 1 @f W2 ./ D f W f 2 L2./; 2 L2./; i D 1;:::;n (A.5) @xi of the functions which belong to L2./, together with their first weak derivatives, is called a Sobolev space. It becomes a normed space if the following norm is introduced: ! Z Z  à 1=2 Xn @f 2 kf k D f 2 dC d : (A.6) 1;2 @x iD1 i Recalling that in L2./ we usually adopt the norm  à Z 1=2 2 kf kL2./ D f d ; we see that the relation (A.6) can also be written as Xn 2 2 2 @f kf k Dkf k C : (A.7) 1;2 L2./ @x iD1 i L2./ The following theorem is given without a proof. 1 Theorem A.1. The space W2 ./, equipped with the Sobolev norm (A.6), is a Banach space (i.e., it is complete). More precisely, it coincides with the completion 1 H2 ./ of the space 1 ff 2 C ./; kf k1;2 < 1g: 1 This theorem states that for all f 2 W2 ./, a sequence ffkg of functions fk 2 C 1./ with a finite norm (A.7) exists such that lim kf fkk1;2 D 0: (A.8) k!1 In turn, in view of (A.6), this condition can be written as ( Z Z  à ) Xn 2 2 @f @f k lim .f fk/ dC d D 0; k!1 @x @x iD1 i i 466 A A Brief Introduction to Weak Solutions or equivalently, Z 2 lim .f fk/ d D 0; !1 k Z  à @f @f 2 lim k d D 0; i D 1;:::;n: !1 k @xi @xi 1 In other words, each element f 2 W2 ./ is the limit in L2./ of a sequence ffkg 1 of C ./-functions and its weak derivatives are also the limits in L2./ of the sequences of ordinary derivatives f@f k=@xi g. The previous theorem makes it possible to define the weak derivatives of a function as the limits in L2./ of sequences of derivatives of functions belonging 1 1 to C ./ as well as to introduce the Sobolev space as the completion H2 ./ of 1 ff 2 C ./; jjf jj1;2 < 1g. 1 1 Let C0 ./ denote the space of all the C -functions having a compact support contained in and having a finite norm (A.6). Then another important functional O 1 1 space is H2 , which is the completion of C0 ./. In such a space, if the boundary @ is regular in a suitable way, the following Poincaré inequality holds: Z Z  à Xn @f 2 f 2 d Ä c d 8f 2 HO 1./; (A.9) @x 2 iD1 i where c denotes a positive constant depending on the domain . If we consider the other norm ! Z  à 1=2 Xn @f 2 jjf jj O 1 D d ; (A.10) H2 ./ @x iD1 i then it is easy to verify the existence of a constant c1 such that 1 jjf jj O 1 Äjjf jj1;2 Ä c jjf jj O 1 ; H2 ./ H2 ./ so that the norms (A.6) and (A.9) are equivalent. We conclude this section by introducing the concept of trace.Iff 2 C 1./, then 1 it is possible to consider the restriction of f over @. Conversely, if f 2 H2 ./, it is not possible to consider the restriction of f over @, since the measure of @ is zero and f is defined almost everywhere, i.e., up to a set having a vanishing measure. In order to attribute a meaning to the trace of f , it is sufficient to remember 1 1 that, if f 2 H2 , then a sequence of C ./-functions fk exists such that lim jjf fkjj1;2 D 0: (A.11) k!1 A A Brief Introduction to Weak Solutions 467 Next, consider the sequence ff kg of restrictions on @ of the functions fk and suppose that, as a consequence of (A.12), it converges to a function f 2 L2./. In this case, it could be quite natural to call f the trace of f on @. More precisely, the following theorem could be proved: Theorem A.2. A unique linear and continuous mapping 1 W H2 ./ ! L2./ (A.12) exists such that .f / coincides with the restriction on @ of any function f 1./. Moreover, O 1 1 H2 ./ Dff 2 H2 ./; .f / D 0g: (A.13) A.2 A Weak Solution of a PDE Consider the following classical boundary value problems relative to Poisson’s equation in the bounded domain <having a regular boundary @: Xn @2u D f in ; @x2 iD1 i u D 0 on @I (A.14) Xn @2u D f in ; @x2 iD1 i du Xn @u D D 0 on @I (A.15) dn @x iD1 i where f is a given C 0./-function. The previous boundary problems are called the Dirichlet boundary value problem and the Neumann boundary value problem, respectively. Both these problems admit a solution, unique for the first problemT and defined up to an arbitrary constant for the second one, in the set C 2./ C 0./. 1 By multiplying (A.14)1 for any v 2 H2 ./ and integrating over , we obtain Z Z Xn @2u v d D f v d: (A.16) @x2 iD1 i 468 A A Brief Introduction to Weak Solutions If we recall the identity  à @2u @ @u @v @u v 2 D v @xi @xi @xi @xi @xi and use Gauss’s theorem, then (A.16) can be written as Z Z Z Xn @u Xn @v @u v n d d D f v d; (A.17) @x i @x @x iD1 @ i iD1 i i where .ni / is the unit vector normal to @. In conclusion, we have: •Ifu is a smooth solution of Dirichlet’s boundary value problem (A.14) and v is O 1 any function in H./2, then the previous integral relation becomes Z Z Xn @v @u d D f v d 8v 2 HO 1./: (A.18) @x @x 2 iD1 i i •Ifu is a smooth solution of Neumann’s boundary value problem (A.15) and v is 1 any function in H./2, then from (A.17) we derive Z Z Xn @v @u d D f v d 8v 2 H 1./: (A.19) @x @x 2 iD1 i i Conversely, it is easy to verify that if u is a smooth function, then the integral relations (A.18) and (A.19) imply that u is a regular solution of the boundary value problems (A.14) and (A.15), respectively.
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