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[1] Adams R.A., Sobolev Spaces. Academic Press (New York) 1975.

[2] Apostol T.M., Mathematical Analysis: A Modern Approach to Ad­vanced Calculus. Addison-Wesley (Reading, Mass.) 1957.

[3] Babuska 1. and Aziz A.K., Survey Lectures on the Mathematical Foun­dations of the Finite Element Method, in The Mathematical Founda­tions of the Finite Element Method with Applications to Partial Dif­ferential Equations (ed. A.K. Aziz). Academic Press (New York) 1972.

[4] Baiocchi C. and Capelo A., Variational and Quasi- Variational Inequal­ities. Wiley (New York) 1984.

[5] Becker E.B., Carey G.F. and Oden J.T., Finite Elements, Volume 1: An Introduction. Prentice-Hall (Englewood Cliffs, N.J.) 1981.

[6] Binmore K.G., Mathematical Analysis: A Straightforward Approach. Cambridge University Press (Cambridge) 1977.

[7] Binmore K.G., The Foundations of Analysis: A Straightforward Intro­duction. Book 2: Topologicalldeas. Cambridge University Press (Cam­bridge) 1981.

[8] Brenner S. and Scott L.R., The Mathematical Theory of Finite Ele­ment Methods. Springer-Verlag (New York) 1994.

[9] Burnett D.S., Finite Element Analysis. Addison-Wesley (Reading, Mass.) 1987.

436 References

[10] Carey G.F. and Oden J.T., Finite Elements, Vol. 2; A Seeond Course. Prentice-Hall (Englewood Cliffs, N.J.) 1983.

[11] Ciarlet P.G., The Finite Element Method for Elliptie Problems. North­Holland (Amsterdam) 1978.

[12] Ciarlet P.G and Raviart P.-A., Interpolation theory over curved ele­ments with applications to finite element methods. Computer Methods in Applied Meehanies and Engineering 1 (1972) 217-249.

[13] Dautray R. and Lions, J.-L., Mathematieal Analysis and Numerieal Methods for Seienee and Teehnology, Vol. 2; Functional and Varia­tional Methods. Springer-Verlag (Berlin) 1988.

[14] Dhatt G. and Touzot G., The Finite Element Method Displayed. Wiley (New York) 1984.

[15] Duvaut G. and Lions J.L., Inequalities in Meehanies and Physies. Springer-Verlag (Berlin) 1976.

[16] Glowinski R., Numerieal Methods for Nonlinear Variational Problems. Springer-Verlag (Berlin) 1984.

[17] Grisvard P., Elliptic Problems in Nonsmooth Domains. Pitman (Lon­don) 1985.

[18] Halmos P., Finite Dimensional Veetor Spaees. Van Nostrand Reinhold (New York) 1958.

[19] Hewitt E. and Stromberg K.R., Real and Abstmct Analysis; A Modern Treatment of the Theory of Funetions of a Real Variable. Springer­Verlag (New York) 1965.

[20] Hoffman K. and Künze R., Linear Algebm. Addison-Wesley (Reading, Mass.) 1973.

[21] Horgan C.O., Kom's inequalities and their applications in continuum mechanics, SIAM Review 37 (1995) 491-511.

[22] Hughes T.J.R., The Finite Element Method; Linear Statie and Dy­namie Analysis. Prentice-Hall (Englewood Cliffs, N.J.) 1987.

[23] Johnson C., Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University "Press (Cambridge) 1987.

[24] Kardestuncer H. (ed.), Finite Element Handbook. McGraw-Hill (New York) 1987.

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[25] Kolmogorov A.N. and Fomin S.V., Elements of the Theory of Fune­tions and Funetional Analysis. Volume 1: Metrie and Normed Spaees. Graylock Press (Rochester, N.Y.) 1957.

[26] Kolmogorov A.N. and Fomin S.V., Elements of the Theory of Fune­tions and Functional Analysis. Volume 2: Measure, Lebesgue Integrals and Hilbert Spaee. Academic Press (New York) 1961.

[27] Kreyszig E., Introduetory Functional Analysis with Applications. Wi­ley (New York) 1978.

[28] Lang S., Introduetion to Linear Algebra. 2nd edition. Springer-Verlag (New York) 1986.

[29] Lang S., Undergraduate Analysis. Springer-Verlag (Ncw York) 1983.

[30] Lions J.L. and Magenes E., Non-Homogeneous Boundary- Value Prob­lems and Applieations, Volume 1. Springer-Verlag (New York) 1972.

[31] Lipschutz S., Set Theory and Related Topies. Schaum Outline Series. McGraw-Hill (New York) 1964.

[32] Loula A.F.D., Hughes T.J.R. and Franca L.P., Petrov-Galerkin for­mulations of the Timoshenko beam problem. Computer Methods in Applied Meehanies and Engineering 63 (1987) 115-132.

[33] Naylor A.W. and Seil G.R., Linear Operator Theory in Engineering and Seienee. Springer-Verlag (Berlin) 1982.

[34] Necas .I., Les Methodes Direetes en Theorie des Equations Elliptiques. Masson (Paris) 1967.

[35] Noble B., Applied Linear Algebra. Prentice-Hall (Englewood Cliffs, N.J.) 1969.

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[38] Oden J.T. and Reddy J.N., An Introduetion to the Mathematical The­ory of Finite Elements. Wiley (Ncw York) 1976.

[39] Raviart P.-A. and Thomas J.M., Introduetion a l'Analyse Numerique des Equations aux Derivees Partielles. Masson (Paris) 1983.

[40] Reed M. and Simon B., Methods 0/ Modern Mathematical Physics I: Functional Analysis. Acadcmic Press (New York) 1980.

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[42) Roman P., Some Modern Mathematics for Physicists and Other Out­siders, Volume 1: Algebra, Topology and Measure Theory. Pergamon (Oxford) 1975.

[43) Roman P., Some Modern Mathematics for Physicists and Other Out­siders, Volume 2: Functional Analysis with Applications. Pergamon (Oxford) 1975.

[44) Royden H.L., Real Analysis. 3rd edition. Collier-Macmillan (London) (1988).

[45) Rudin W., Real and Complex Analysis. 2nd edition. McGraw-Hill (New York) 1974.

[46) Schwartz L., Theorie des Distributions. Hermann (Paris) 1950.

[47) Schwartz L., Mathematics for the Physical Sciences. Hermann (Paris) 1966.

[48) Showalter R.E., Hilbert Space Methods for Partial Differential Equa­tions. Pitman (Boston) 1977.

[49) Smirnov V.L, A Course of High er Mathematics, Volume 5: Integration and Functional Analysis. Pergamon (Oxford) 1964.

[50) Strang G., Linear Algebra and its Applications. Academic Press (New York) 1976.

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[52) Zauderer E., Partial Differential Equations of Applied Mathematics. 2nd edition. Wiley (New York) 1989.

[53) Zeidler E., Nonlinear Functional Analysis and Its Applications. Vol­ume IIA: Linear Monotone Operators. Springer-Verlag (Berlin) 1990.

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[55) Zeidler E., Applied Functional Analysis: Main Principles and Their Applications. Springer-Verlag (Berlin) 1995.

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References 439

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Solutions to Exercises

Chapter 1

1.1. A = {-2,3}, B = {-3,-2,-I,O,I,2,3}. AUB = B; AnB = A; AnZ+=3, A-Z+={-2}.

1.2. Au C = {I, 2, 9} so B x (A U C) = {(7, 1), (7,2), (7,9), (8, 1), (8, 2), (8,9)}; An C = {I} so (A n C) x B = {(I, 7), (1, 8)}.

1.3. Let x E An (B U C). Then x E A and x E B or C; Le., x E A and xE B, or xE A and x E C. Hence x E (AnB) U (AnC). The second identity is proved in a similar way.

1.4. n(A U B U C) = n(A) + n(B) + n(C) - n(A n B n C) - n(A nB­C) - n(B n C - A) - n(C nA - B).

1.5. P(A) = {A,0,{I},{2},{3},{1,2},{2,3},{1,3}}; P(B) = {B, 0, {{I, 2}}, {3}}.

1.6. Consider the table

1/1 1/2 1/3 1/4 1/5 .. . 2/1 2/2 2/3 2/4 2/5 .. . 3/1 3/2 3/3 3/4 3/5 .. . 4/1 ...

The rationals can be listed by writing down the numbers in the pre­ceding table in the order shown, omitting those already listed (e.g.,

442 Solutions

omit 2/2 = 1). This then gives a listing of all rationals whose nu­merator and denominator add up to 2, then 3, and so on. In this way all positive rationals are covered. Multiply by -1 to get negative rationals.

1.7. (i) [a, b]; (ii) lR; (iii) [0,1].

1.8. (i) Not open: A = {±I/n7r, n = I, ... } and for every x E A there is a nhd N(x) such that N(x) - {x} ri A. Not closed: 0 ri Ais a point of accumulation. (ii) Neither open nor closed. (iii) Open, not closed since {±I/n7r} are points of accumulation, but are not in A.

1.9. Assurne I is closed. Let x E 1'; since x ri I, the distance from x to I is finite. Hence we can set up a neighborhood of radius E < d about x that lies entirely in 1'. Hence l' is open. Conversely, assurne l' is open. We always have I c 1, so we want to show that 1 c I. Let x E 1 and assurne x ri I. Then x is in 1'. Since l' is open, there is a neighborhood N of x with N nI = 0, which is a contradiction. Thus x E I and so 1 EI.

1.10. Points of accumulation: A = {z: x 2 - y2 = I}. A is open.

1.11. (i) -1,1/2, -1/6, 1/24, ... ; (ii) 1,0,1,0,1 ... ; (iii) -3,6/7,9/13,12/19, ...

1.12. (a) Converges to -3/2; (b) not convergent; (c) converges to 1.

1.13. 1(3n + 2)/(n - 1) - 31 = 15/(n - 1)1 < 0.001. Assurne n > 1, so that 5 < O.OOI(n - 1) => n > 5001. Take n = 5001.

1.14. Suppose U n is monotone increasing, with sup = m. For any E > 0 there exists N such that IUn - ml < E far n > N, so Un -> m. The same reasoning applies if U n is monotone decreasing.

1.15. (i) maxA = 1 = supA,minA is undefined, inf A = O. (ii) maxA, minA undefined; sup A = 1, inf A = -1. (iii) min A, max A do not exist, inf A = -00; supA = c. (iv) Iz2 + 11 :<::: Iz2 1 + 1 = Izl 2 + 1 :<::: 2. Maximum achieved at z = ±1. Minimum is achieved at z = ±i.

1.16. Y = inf A => a :<::: y :<::: x for all x E A and lower bounds a. Thus -x :<::: -y :<::: -a so that -y is the least upper bound of -A.

1.17. Take A = (-1,0) and B = (-2,0); then a = b = O. But C = (0,2) so that sup C = 2 =I ab.

1.18. (i) Lct p = supI; then x:<::: p for any x E I. Let J = {ax: xE I}; since a > 0, ax :<::: ap. Hence J is bounded above by ap. Let the supremum of J be q (we must prove that q = ap). Since ap is an upper bound for J and q is the least upper bound, q :<::: ap. But for

Solutions 443

any y E J we have y ::::; q =} a-1y ::::; a-1q. But 1= {a-1y : y E J}, hence a-1q is an upper bound for I. Thus p ::::; a-1q or ap ::::; q. Since q:::: ap also, we have ap = q.

1.19. (i) Closed. (ii) Open. Set of limit points is nU {x : x2 + y2 + z2 = a2, z > O} U {x: x2 + y2 < a2, z = O}.

1.20. (a) V2j (b) 2a.

1.21. (a) Not an equivalence relation, but a partial ordering; (b) equivalence relation, not a partial orderingj (c) neither an equivalence relation nor a partial orderingj (d) not an equivalence relation, but a partial ordering.

1.22. {(2,2), (3,3), (4,4), (5,5), (6,6), (2,5), (5,2), (3,6), (6,3)}.

1.23. Take c E A a n Ab. Then c'" a implies that a '" c. Also, c'" b. Thus a '" b by transitivity, and b '" a by reflexivityj hence a E Ab and b E A a . Take any x E A a : x '" aj hence x '" b, so x E Ab. Thus A a C Ab· Similarly show that Ab C A a ·

1.24. A a is the set of ordered pairs of integers lying on the "diamond" {z: lxi + lyl = const} on which a is located.

Chapter 2

2.1. (a) not continuous at x = ±1; (b) continuous on (-oo,Ol.

2.2. (a) Supposethat Ix-yl < o. Then Ip(x)-p(y)1 = la1(x-y)+a2(x2 -

y2) + ... + ak(xk - yk)1 ::::; Ix - yl[la11 + la211x +yl + ... + lakllxk-1 + xk - 2 y + ... + yk- 1ll < oC since term in square brackets is bounded above. Set 0 = E/C. (b) For ° :::: y ::::; :z: we have..jY::::; ,fi =} 2y ::::; 2yXfj =} x-2yXfj+y ::::; x - y or (,fi - Vy)2 :::: x -y. If Ix - yl < 0, then Iu(x) - u(y)1 < 01/2 .

For given E set 0 = E2 •

2.3. (b) Ij(x) - j(y)1 = Ix-1 - y-11 = Iy - xl/lxyl. But x > a, y > a, so xy > x2 or 1/xy < 1/a2. Hence Ij(x) - j(y)1 < a-2lx - yl.

2.4. Ij(x) - j(x)1 = l(x2 + 2y) - (x 2 + 2y)1 = l(x2 - x2 ) + 2(y - y)1 :::: Ix2-x2 1+2IY-YI. Supposethat lx-xl< Oj i.e., (X_x)2+(y_y)2 < 02. Thenlx2 - x21 = Ix - xlix + xl < O· C. Also, Iy - yl < o. Hence Ij(x) - j(x)1 < (C + 2)0. Set 0 = E/(C + 2).

2.5. Set je) = d(-, E). Then Ij(x) - j(y)1 = I infzEA Ix - zl- infzEA Iy­zil :::: Ilx - Yl + inf Iy - zi - inf Iy - zll = Ix - yl· Given E > 0, choose 0 = E.

444 Solutions

2.6. If(xo) - f(x)1 < E whenever Ixo - xl < 8, Le., for xE (xo - 8, Xo + 8). Pick any such x: either 0 < f(xo) - fex) < E in which case fex) > f(xo) - E or 0< fex) - f(xo) < Ein which case fex) < f(xo) + E. For the first case choose E smaller than I(xo) so that fex) is positive. For the seeond ease fex) > f(xo) > o.

2.7. Assume that f(a) < 0, f(b) > o. Sinee f(a) < 0, there is an interval [a, c] in which fex) < o. Let the l.u.b. of such points c be e; then fee) ~ o. We cannot have fee) < 0 sinee we would then be able to find an interval about e for whieh fex) < 0, which would imply that e is not a l.u.b. Hence fee) = o. A similar argument applies if f(a) > 0 and I(b) < O.

2.8. (a) U E G(-I, 1); (b) U E Goo([O, 71"] X [0,1]); (e) U E Gl[O, I].

2.9. Iu(x) - u(y)1 = Ilxl - lyll ~ Ix - Yl sinee lxi = Ix - Y + Yl ~ Ix - Yl + lyl·

2.10. Choose 8 = E/ L in the definition of continuity.

2.11. I = IQ U I', where IQ and I' are the subsets of rationals and irra­tionals. J.L(I') = J.L(I) - J.L(IQ) = J.L(I).

2.12. Let M be an arbitrary measurable set in IR. If 1 E M, 0 f/. M, then XE/(M) = E; 1 f/. M, 0 E M => XE/(M) = E'; 1 f/. M, 0 f/. M => XE/(M) = 0; 1 E M, 0 E M => XE/(M) = dom XE. Thus XIi/(M) is a measurable set. Conversely, if E is not measurable, then XE cannot be measurable.

2.13. Put 8n = 2-n . For eaeh n and for every x there is an integer kn

such that kn8n ~ x < (kn + 1)8n . Set ifJn(x) = kn(x)8n if 0 ~ x < n, ifJn(x) = 0 for n ~ x. Then x - 8n < ifJn(x) ~ x if 0 ~ x ~ n; furthermore 0 ~ ifJl ~ ifJ2 ~ ... ~ x and ifJn(x) ----> x as n ----> 00, for x E [0,00]. Set Sn = ifJn 0 f.

2n 1 2.14. First caleulate JlR Sk(X) dx = I:k=~ (k/2n )(I/2n) = (1/22n ) I: k.

Then use the formula I:~l k = m(m - 1)/2.

215 f+( ) = {I, 0 ~ x ~ 1 r() = {I, -1 ~ x < 0 •• X 0 otherwise,' x 0 otherwise.

,{jRf+ dx = JlRf- dx = 1, so JlRf dx = O. JlRg+ dx = +00, JlRg- dx = 1, so JlRg dx = +00.

2.16. Use the fact that III = f+ + 1-, and that integrability of f implies that of f+ and f-. For the converse use f = f+ - f-· Show that - J r - J f- ~ J f+ - J f- ~ J r + r·

2.17. (a) ap> -1; (b) ap< -1.

Solutions 445

2.18. All real a exeept a = -~, -~.

2.19. Consider 0< In lu(x) - av(x)1 2 dx for any a E R Expand and then ehoose a = In uv dx/ In Ivl2 dx.

Chapter 3

3.1. (a) Veetor spaee; (b) not a veetor spaee; (e) not a veetor spaee; (d) veetor space; (e) not a veetor space.

3.2. (a) Subspaee; (h) not a subspace: 0 ft V.

3.3. (a) Subspace; (b) not a subspace; (e) subspace; (d) subspaee.

3.4. Suppose that U = V EB W, and let u = VI + WI = V2 + W2 for VI, V2 E V and WI, W2 E W. Then VI -V2 = WI -W2. But VI -V2 E V and WI -W2 E W, so that VI -V2 = WI -W2 = 0, or VI = V2, Wl = W2.

Conversely, suppose that u = V + W for V E V, W E W with V and W uniquely defined. If V n W f {O}, then there exists z E V n W with z f o. Henee we ean write u = (V + z) + (w - z) so that the deeomposition of u is not unique, a eontradietion.

3.5. For any u E e[O,l],u(x) = v(x) + w(x), where v(x) = Hu(x) + u(-x)) and w(x) = Hu(x) - u(-x)). Thus V E V and W E W. Also, vn W = {v: V is even and odd} = {O}.

3.6. aß -::; areaA + areaB, henee aß -::; aP /p+ ßq /q sinee A = Ioo. xp- I dx = aP /p, ete. The proof now follows easily from the hints given.

3.7. (u,w) = (v,w) => (u - v,w) = 0 for aB w. Set w = u - v.

3.8. (u,v)o = 0; (u,vh = (u,v)o + (u',v')o f O.

3.9. Ilull = Ilu - V + vII :s: Ilu - vii + IIvll. Repeat with u.

3.10. Ilu + vl12 + lIu - vll 2 = (u + v,u + v) + (u - v,u - v). Expand and rearrange.

3.11. If v = au, then lIu + vii = (u + au,u + au)1/2 = (1 + a)lIull. But Ilull+llvll = (1+a)llull· Conversely, assumethat Ilu+vll = Ilull+llvll· Then Ilu + vll 2 = IIull2 + IIvl1 2 + 2(u, v) = (Ilull + IIvl1)2 = lIull2 + IIvll2 + 211ullllvll. Hence lIullllvll = (u,v) or (u,v) = 1, where u = uillull, fJ = v/llvll. Suppose v f u; then fJ = u + w, and 1 = (u,u + w) = 1 + (u, w) => (u, w) = O. Also, IIfJII 2 = 1 = 1 + IIwll 2 + 2(u, w); Le., IIwll = 0 => w = O. Hence fJ = u or v = au for some a.

3.12. Assurne that IIx - ylllly - zll = Ilx - zll. Square and rearrange to get (a, b) = 1, where a = a/llall, a = x - y, b = Y - z. Thus b = a which gives Y = ax + (1- a)z, where a = lIy - zli/(llx - yll + IIY - zll). The converse is straightforward.

446 Solutions

3.13. Verify that (. , .) defined by (u, v) = J: u' Vi dx is an inner product on X.

3.14. No.

3.15. Expand the right-hand side and simplify.

3.16. lIau + (1 - a)vll ::; allull + (1 - a)llvll ::; 1.

3.17. IIul1 2 + 2a(u,v) + a211vl1 2 = IIul12 - 2a(u,v) + a211v112. The result follows from this.

3.18. (i) (y'5 - 1) /2; (ii) 1.

3.20.

p=1 p=2 p= 00

3.21. Ilxll§ = x 2 + y2 = (lxi + lyl)2 - 21x lyl ::; (lxi + lyl? = Ilxlli- Ilxlli = x2 + y2 + 21xyI ::; 2(x2 + y2) = Ilxll~.

3.22. J IUTVT I dx ::; [f luIT(p/Tlr/p[f Ivlr(q/rlt/q· Take rth roots of both sides.

3.23. Follow the argument of Example 20.

Chapter 4

4.1. 2.

4.2. I(un,vn) - (u,v)1 = I(un - U,Vn - v) + (u,vn - v) + (v,un - u)l ::; Ilun - ullllvn - vii + llullllvn - vii + llvlillun - ull -t 0 as n ---> 00.

Set Vn = v (Le. the sequence v, v, ... ) to get (un , v) -t (u, v). Finally, l(un,v) - (u,v)l ::; l(un - u,vll ::; llun - ullllvll, hence (un,v) -t

(u, V l. Set V n = U n to get the final result.

4.3. llu - wll = llu - U n + U n - wll ::; Ilu - unll + Ilun - wll < E + a. The inequality follows from the arbitrariness of a.

4.4. (a) (-1,1]; (b) (-00,00).

4.5. (a) un(x) -t 0 pointwise. But Ilun - ulli2 = J12/:: n2 dx = n -t 00

as n ---> 00; (b) un(x) -t pointwise since un(x) = n3 / 2x/ exp(n2x2 ) =

Solutions 447

n 3/ 2x/[1 + n2x 2 + ~n4x4 + ... ] -> 0 as n -> 00. But Ilun - ulli,2 = f'n y2 exp( _2y2) dy (setting y = nx) = -H[yexp( -2y2)]~n

- J:'n exp(-2y2)dy = -~(O + )(7f/2)) as n -> 00.

4.6. sup lun(x)1 = 1/2at x = l/n. Thus in [0,1], un(x) - .. 0 pointwise but Ilun - ulloo = 1/2, so convergence is not uniform. But convergence is uniform in (a,l] (a > 0) : sup lun(x)1 = na/(l + n 2 a2 ) at x = a for n > l/a (check this by sketching un(x)) and sup lun(x)1 -> 0 as n -> 00.

b 4.7. sup Iun(x) - u(x)1 < E for n > N. Hence Ja Iun(x) - u(x)IP dx :s;

(sup Iun(x) - u(x)I)P' (b - a) < (b - a)EP.

4.8. Ilull = 0 does not imply that u = 0; 111 . 111 is also not a norm.

4.9. Ilu - U 11 2 = ~ - 2mn + ~ = 2 (m_n)2 Nu-n m L2 n+2 mn+m+n m+2 (m+2)(n+2)(mn+m+n)' merator (m - n)2 :s; (m + n)2. Now show that Ilun - um lli2 -> 0 as n,m -> 00.

4.lO. Ilun - umllu = Jo1 Ixn - xml dx = n~1 - m~1 (taking m > n)

= (n+0C';-:+I) :s; (n+1)(m+l) -> 0 as n,m -> 00. Hence {un } is a Cauchy sequence.

4.11. {un} is Cauchy, so suplun(x) - um(x)1 < E for m,n > N. For any Xo, Iun(xo) - um(xo)1 < E, so {un(xo)} is a Cauchy sequence of real numbers. IR is complete, so un(xo) -> u(xo), say, which defines a function u( x). Thc rest of the proof follows easily from the hints given.

4.12. Let {x k } be a Cauchy sequence in IRn : Ilxk - xIII< E for k, I > N; i.e., 2:; IXki - xlilP < EP • Hence IXki - Xlil P < EP for each i. But IR is complete so xki -> Xi, say. Hence X -> X in IRn.

4.13. Assume {un } convergent: Ilun - ull < E for n > N. Also Ilum - ull < E'

for m > Nt Hence Ilun - Um 11 = II(un - U) + (um - u)11 :s; Ilun - ull + Ilum - ull < E + E' for n, m > N (assume N > N').

11 11 2 Jl/2+1/n[ ( 1) ( 1 )]2 d J 1/2+1/m[ 4.14. Un - Um = 1/2 n X - '2 - m X - '2 :r + 1/2+1/n 1-m(x - ~)J2 dx. Show that this -> 0 as m, n -> (Xl, SO that {Un } is

Cauchy. Also, Ilun _u11 2 = Jllg+l/n[n(x-~) -1]2 dx -> 0 as n -> 00.

So u n -> u in L 2 .

4.15. Take V n E Y with Vn -> v. It is required to show that v E Y. From Exercises 3.9 and 3.22, Illvn Ilu -Ilvii u I :s; Ilvn - vllu :s; cllvn - vllp· Thus 11-llvllul < E so that v E Y.

448 Solutions

4.16. Let v(x) E C[~I, 1] be defined by v(x) = I/E', ~E ::; x::; E, { ~1 ~1::;x<~E,

+1, E<X::;1.

We have Ilu ~ vlli2 = J~« ~1 ~ C I)2 dx + Jo«1 ~ C I)2 dx = E3 /3 ~ E2 + E. Hence v can be made arbitrarily elose to u by choosing E small enough.

4.17. Ilu ~ vll oo = supll ~ v(x)l, where Iv(x)1 < 1 and v(O) = O. Hence Ilu~vlloo = 1; neighborhoods ofu ofradius less than 1 do not contain members of V, so u is not a point of accumulation.

4.18. v E B(Uo,T) ~ Iluo ~ vll oo ::; T; Le., suplsin21rx ~ COS21rTI ::; T. sup lUD ~ vi = v'2 (at x = 3/8) so we require T ~ 3/8.

4.19. Cf. solution to Exercise 1.9.

4.20. Assume that Y is complete, and let v be a point of accumulation of Y. Then each open ball B(v, l/n), n = 1,2, ... , contains a point vn , say, in Y. The sequence {un } is convergent, hence Cauchy, in Y. Since Y is complete, v E V. Hence Y contains all its points of accumulation, and is elosed. Conversely, assume that Y is elosed, and let {vn } be a Cauchy sequence in Y. Then {vn } is a Cauchy sequence in X, and so converges to v in X. From Theorem 3 of Chapter 4, v is in Y also, so Y is complete.

4.21. W dense in X ~ for any v EX there is a w E W such that Ilw~vll < E. Similarly, for any u E Y there is v E X such that Ilv ~ ull < E.

Hence Ilu ~ wll ::; Ilu ~ vii + Ilv ~ wll < 2t:, so W is dense in Y.

4.22. Take i E LP. For given E > 0 choose a bounded function 9 in LP, where 9 has compact support, for example, Igl ::; M in [a, b] and 9 = 0 otherwise. Select 9 so that Ili ~ gllp < E. Bounded functions with compact support are dense in LP, so we can find ihn} in Co such that 9 = limhn a.e. Assume that Ihni::; M in [a,b] and 0 otherwise. Then Ig ~ hnlP ::; (2M)P on [a, b] and Iig ~ hnll p --> 0 from the Dominated Convergence Theorem. Choose n so that Ilg~hnll ::; E

and use the Minkowski inequality.

4.23. Suppose that there are two points vo, vb such that Ilu - Vo II = d. Then w = (vo + vb)/2 is in M hence, using the parallelogram law, it can be shown that d2 ::; IluD ~ wl1 2 < ~llu ~ vol1 2 + ~llu ~ v'II 2 =~, a contradiction.

4.24. Consider {un } C y.l with limit UD in X. We must show that UD E y.l also. By definition (un, v) = 0 for any v E Y; thus 0 = limn->oo(un, v) = (limn~ooun,v)=(uo,v)=O~uo E V.l.

Solutions 449

4.25. Theorem 7(b), which requires completeness of H, is used in Lemma 1.

4.26. Let u E X and w E Y 1.. Then u E Y also, so (u, w) == 0. u is arbitrary; hence w E Xl. =} yl. C Xl..

Chapter 5

5.1. (i) R(M) = points on the upper unit semicircle, N(M) = 0; (ii) R(K) = [0, 00), N(K) = {al; (iii) R(f) = (0, 00), N(f) = 0.

5.2. N(S) = {al; N(T) = {a( -8,4, In.

5.3. (i) One-to-one, not surjective; (ii) one-to-one, surjective (T is a re­fiection about a line at 45° through the origin).

5.5. (i) ST(x) = S(x,-y) = (-2y,x); TS(x) = T(2y,x) = (2y,-x); (ii) ST(x) = S(sinx) = sin2 x-I, TS(x) = T(x2 - 1) = sin(x2 - 1).

5.6. S-l : V --t U and T-1 : W --t V exist. Clearly TS : U --t W is one-to-one onto W, so (TS)-l exists. Furt hermore , (TS)u = w =}

u = (TS)-l W . But (TS)u = T(Su) = w, so Su == T-1w and u = S-lT-1w. Hence (TS)-l = S-IT-1.

5.7. (i) linear; (ii) linear; (iii) nonlinear.

( -5 -1) ( 4 ) . 5.8. Tx = -3 -5 x + 5 ,assummg that {(O,O), (1,0), (0, In

go to {(4,5), (-1,2), (3,On.

5.9. Let TUI = VI, TU2 = V2. Then T(aul + ßU2) = O~Vl + ßV2 by the linearity of T. Hence T-l(av1 + ßV2) = aUI + ßU:2. Eut aT-lvI =

aUl, aT-1v2 = aU2 =} T-l(aul + ßU2) = aT-lv + ßT-lV2.

5.10. No; e.g., d(x, B) + d(y, B) =1= d(x + y, B) in general. Null space is the set B.

5.11. For u =1= 0, IITII = sup(IITull/llull) = sup IIT(u/llulDIl (T is linear) = sup 11 Tu 11 , lIull = 1. To prove the second result, consider IITull ::; IITlillull. For every E > 0, there is a Uo such that IITuoli > (IITII -E)lIuoll· If lIull ::; 1, then IIAull ::; IIAlillull ::; 11 All =} sup 11 Au 11 ::;

IIAII, lIull ::; 1. Eut ifwe put U1 = uo/iluoll, then IIAutil = lIuo 1I-11IAuo 11

> IIAII-E, so for lIull ::; 1, sup IIAul1 2- IIAutli > IIAII-E or sup 11 Au 11 ::;

IIAII·

5.12. IIAxll= = maxl:S:i:S:n 12::7=1 AijXjl ::; maX1:S:i:S:n 2=7=1 IAijlllxjl ::;

maxl:S:i:S:n 2::7=1 lAi] I maxl:S:j:S:n IXjl = maxl:S:i:S:n 'L7=1 IAijlllxll=· Hence IIAII = sup(IIAxll=/lIxll=) ::; maXi:S:i:S:n 'L7=1IAij l. Suppose maximum occurs for i = k. Then for x such that Xj = +1 if Akj 2-0, Xj = -1 if A kj < ° we have IIAxll=/llxll oo = 2::7=1 IAijl·

450 Solutions

5.13. For x =f. 0, (IIAxll/llxllf = (a + b)2 - 2ab(x - y)2/(x2 + y2). Take the supremum (at y = x) to find IIAI12.

5.14. Illull = Ilull; I is bounded. Consider u(x) = sin nx: lIullv = 1 but Illullw = 1 + n which cannot be bounded.

5.15. IIST(u)11 = IIS(Tu)11 ~ IISllllTul1 ~ IISIIIITlillull·

5.16. Let {un} C N(T) with limit u in U. Then TUn = O. Thus 0 = limn -+oo TUn = T(limn -+oo un) = Tu =} u E N(T).

5.17. T is one-to-one since, if TUI = TU2 = v, then IITul - TU211 = 0 ~ Kllul - u211. IIT-1vll = Ilull ~ K-1 11Tull = K- 1 I1vll·

5.18. u(x) = I; u'(s) dx ~ sUPoSxsllu'(x)1 = IIDull. Take sup of both sides.

5.19. (I - P)(l - P) = 12 - PI - IP + p 2 = 1- P.Range : R(l - P) = N(P), R(P) = N(I - P).

5.20. From Theorem 8, IlPuli ~ Ilull. Thus IIPII ~ 1. But for u E R(P) we have Pu = u, so IIPul1 = Ilull. Hence IIPII = 1.

5.21. Take, for example, the map on lR2 that takes a point x to the point in B(O, 1) dosest to x. This is a projection, but the map is not ho­mogeneous.

5.22. Let u E N(P). By definition (u,v) = 0 for v E R(P). Hence N(P) C R(P).L. Let u E R(P).L. Then (u, z) = 0 for z E R(P). By Theorem 9, u = v+w for v E R(P), w E N(P), so Pu = Pv + Pw = Pv = v. Also, 0 = (u,z) = (v,z) + (w,z) = (v,z); hence v = o. Thus Pu = o =} u E N(P).

5.23. T is a projection since T is linear and T 2u = Tv (where v = u(x) if lxi< 1 and 0 otherwise) = v = Tu. R(T) = {u E L2(lR): u(x) = 0 for lxi ~ I}, N(T) = {u E L2 (lR): u(x) = 0 for lxi< I}.

5.24. v(y) = Pu(y) = I~1 exp(i(y - z))u(z) dz; show that Pv(x) == p 2u(x) = Pu(x). Pis an orthogonal projection.

5.25. (i) x satisfies Ax = 1 where 1 = (1, ... ,1); (ii) x satisfies Ax = 0 = (1,0, ... ,0).

5.26. u(x) = e3~1 (_e3 - 2X + eX) - 2x + 2, (l, n = I~ u(x) dx.

e3~1 (~e - e3 ) = 101 g(x)2x dx; so 9 satisfies Jo1(gj - u)dx = O.

5.28. Let {Pn} be a Cauchy sequence in X'. Then for any u E X, I(Pn,u)­(Pm, u)1 :S IIPn - Pmlillull -> 0 as m, n -> 0 so {(Cn, u)} is a Cauchy sequence in lR, with limit (C, u), say. Complete the proof by showing that P is bounded and linear, and Cn -> C in X'.

Solutions 451

5.29. In the use of the projection theorem.

5.30. If there are two elements Ul, U2 such that (Ul,V) = (U2'V) = (P,u), then (Ul - U2,V) = o. Set v = Ul - U2: IIUI - u211 2 = Oor Ul = U2· II Pli = sup(I(P,v)I/lIvID (for v =I- 0) = sup((u,v)/llvID :'S sup(llullllvll/llvll) = Ilull· Also, I(P,u)1 = (u,u) = IIul1 2 :'S IIPllllul1 so IIPII ~ lIull· Hence IIPII = Ilull·

5.31. Take I = Iglq-l sgng; then I/IP = Iglq, so I E LP, and II/lb = IlgIl1:;-1. Then show that (Pg , I) = II/lbl!gl!Lq.

5.32. P = 0 <=} (P, v) = 0 far all v E X. Given v E X there exists a sequence {vn } in Y such that Vn -+ v. Thus (P, v) == (P, limn--->oo vn ) = limn--->oo(P, vn ) = o.

5.33. la(u, v)1 2 :'S [llu'lIlIv'll +Ktllullllvll]2 :'S (K~ IIul1 2 + lIu'1I 2 )(llvI1 2 + IIv'11 2 ),

using Cauchy-Schwarz.

5.34. cf Exercise 4.2.

5.35. I(P,v)1 = If~(-1-4x)v(x)dxl = 1(-1-4x,v)ul:'S 11-1-4xllullvllu :'S kllvllHl. la(u,v)1 :'S 21fo u'v' dxl :'S 21lu'llullv'llL2 :'S 211ullH1 llvllHl, hence continuous. a(v, v) ~ fo1(v')2 dx. Now IIv'I17J2 ~ C-211vlli2 so

(C-2 +1)llv'lIi2 ~ C-21Ivllt,. fo1(-1-4x)vdx = fo1(x+l)u'v' dx = [(x + l)u'v]Ö - fo1(u' + (x + l)u") dx =? fo1 {(x + l)u" + u' - (1 + 4x)}v dx = o.

5.36. lii(u, v)1 :'S la(u, v)1 + I(u, Kv)ul :'S Kilullllvil +K'llullllvll where K' = sup IK(X)I. ii(v, v) = a(v, v) + (v, KV) ~ allvl1 2 + ß(v, v) ~ allvl1 2

where ß = inf K(X).

Chapter 6

6.1. (a) Linearly dependent; (b) linearly independent.

6.2. L;=1 akeikx = 0 =? L;=1 ak cos kx = 0 and L;=1 ak sin kx = 0 which halds only for all ak = O. Hence {eik"'} is linearly independent.

6.3. If u, v E X, then (au + ßv)" - 2(au + ßv)' + (au + ßv) = a(u" -2u' + u) + ß(v" - 2v' + v) = 0, hence au + ßv EX. dim X = 2. Basis for Xis {Ul(X) = e"', U2(X) = xe"'}.

6.4. dimM = 9, dimK = 4.

6.5. Let dimV = m with basis {Vl, ... ,Vm } and dimW = n with basis {Wl' ... , wn }. Every u E V ffi W is of the form u ,= v + W for some v E V, W E W. But v = Li aiVi and W = Lj ßjWj so u = Li aivi+ Lj ßjWj. Hence B = {VI, ... , Vm ,Wl,·· .,Wn } spans VffiW. It remains to show that B is linearly independent.

452 Solutions

6.7. ct>l = (1/V2)(1,0, 1), ct>2 = (1/V2)(1,0, -1), ct>3 = (0,1,0).

6.8. cPo(x) = Vlfi, cPl(X) = ..j3fix, cP2(X) = hß72 (3x 2-1), cP3(X) = ~..Jf72 (5x 3 - 3x).

6.9. All = ~(e2-1), A l2 = A 2l = ~(I-e-2), A22 = i(1-e-6 ). detA =I 0.

6.10. Consider I : Xl ---> X 2 : IIIul12 = IIUl12 S kllulh (show this us­ing Lemma 1; see also Theorem 4). Similarly, Ilulll S Kllul12 if we consider I: X 2 ---> Xl.

6.11. Tl2 = 2, T23 = 6, others zero.

6.12. Tll = 27r, T22 = cosx, others zero.

6.13. (b, c) = (Ta, c) = (a, TT c) = ° if c E N(TT). Let d E R(T)l... Then (d, Tu) = ° = (TT d, u) => d E N(TT). Conversely, if d E

N(TT), then if Tu = v we have (TT d,u) = ° = (d,v) => d E R(T)l... Hence N(TT) = R(T)l.. => N(TT)l.. = R(T). N(TT) =

{(1,1,-1)}, b=(a,ß,a+ß).

6.14. (0'.2, -0'.1,0), (0'.3,0, -ad·

6.15. Let BI = {el, ... ,en } and B2 = {h, ... ,fn} be orthonormal bases of X and ffi:n, respectively. For any u E X we have u = L uiei, Ui = (u, eil. Define the map T: X ---> IRn by T(u) = (Ul,"" un ). Then T is an isomorphism (show this) and Ilulli = (u,u) = (Luiei, LUjej) =

LU; = IITullffi.n.

6.16. 111:'11 = max lail·

6.17. (i) u(x) = y'2;(I/y'2;); (ii) u(x) = L;;'=1(2/k)(I- (-I)k)sinkx.

6.18. Uo = -V2/4, Ul = 5V3/6V2, U2 = ,;5/8V2.

6.19. Ck = ~(U2k - iU2k-l) for k = 1,2, ... , Ck = ~(U2k + iU2k-l) for k = -1, -2, ... , Co = uo/V2.

6.20. ° S Ilu- L;':1 (u, cPi)cPi 11 2 = lIu11 2 - L;':1 (u, cPi)2, hence L;':1 (u, cPi)2 S Ilu11 2 . Since sum is bounded, we can let N ---> 00.

6.21. Use the property PcPk = cPk to show that p 2u = Pu. Clearly R(P) c V. Conversely, if v E V, show that Pv = v so that R(P) = V. Orthogonality: take v E R(P) and W E N(P); then (w, v) = (w, Pu). Use this to show that (w,v) = 0.

6.22. See Exercise 6.8. Pu = L!=o(U,4>k)cPk = ..J275cPo + (8/35)-/572cP2'

Solutions 453

6.23. (a) Set u(r, e) = R(r)8(e) to get (8' sine)' + '\8sine = O. Set E = cos e to get Legendre's equation. General solution is u( r, e) = 2:~=o[anrn + bnr-(n+1)]Pn(cose). (b) an = (2n + 1)/2 Jo71: f(e)Pn(cose) deo

6.24. Eigenvalues satisfy.J>: cos( .;x;;C) + ßsin( .;x;;C) = O. vdx) = [(C/2)+ (1/2ß) cos2( .;x;;C)]-1/2 sin( .;x;;C). Heat equation: u(O, t) = 0, (Bu/Bx + ßu)(C, t) = O.

6.25. Use integration by parts and the boundary conditions to show that (Lu, u) 2: O. Nonnegativity ofthe eigenvalues follows from 0 :S (Lu, u) :S A(U,U). Since L2 is separable there is at most a countable number of nonzero mutually orthogonal vectors.

6.26. Let the minimizer be u, and set w = u + EV; then consider R(w) = R(E) over all w that satisfy (w,el) = (w,e2) = ... = (w,en-l) = O. Set [dR/dE]<=o = 0; expand and differentiate to find that A = R(u) and u = en .

6.27. (Lsn, rn ) = (L 2:~=1 ukq;k, rn) = (2:~=1 ukAkq;k, rn ) = 0 since (rn, q;k) = o (Proof of Theorem 6.12).

6.28. Return to (6.34): for symmetry of L, (a) p(x) ----> I) as x ----> ±oo; (b) p( -L) = p(L).

6.29. (c) Show that H~(x) = 2xHn(x)-Hn+l (x). Set f(x) = exp( _x2 ) and show that f(n+1)+2xf(n)+2nf(n-l) = 0; multiply by (_l)n+l exp(x2 )

to get Hn+1 - 2xHn + 2nHn- 1 = O.

Chapter 7

7.1. 10:1 = 0 =? 0: = (0,0), (x'" / o:!)D'" f(O) = f(O). 10:1= 1 =?

xlyO D(l,O) f(O) + XOyl D(O,l) f(O) = x Bf I +- Y Bf I etc 1!0! 0!1! 8x 0 8y 0 .

7.2. J~a 8(x)q;(x) dx :S Cl J~a 8(x) dx since supq;a(x) = e- l . If 8 were

locally integrable, then lima--->o J~l 8(x) dx = O. But left-hand side = q;(0) = e- l .

7.3. f(x)q;(x) E C(O). Assume f # 0, but J N dx = O. In particular, if f(xo) # 0, then f(x) # 0 for all x E (xo - h, Xo + h) for some h. Choose arbitrary rp with compact support inside (:r:o - h, Xo + h); can always find q; such that J fq; dx # 0, a contradiction.

7.4. Consider 0 C lR?, for example; for 10:1 = m, JoJDau)v dx = In(8rnu/ 8xk8yrn-k)v dx, where 0 :S k :S m. Use Green's theorem repeatedly.

454 Solutions

7.5. ((sgn)',4» = -(sgn,4>') = - J~I(-l)4>' dx - Jo1(+I)4>' dx = [4>]~1 -

[4>]6 = 24>(0) = 2(8,4».

7.6. ((sin ax . H(x))", 4» = (sin ax . H(x), 4>") = (H(x), 4>" sin ax)

= Jo1 4>" sinax dx = W sinax - a4>cosax16 - J01 a2 4>sinax dx a4>(O) - a2 (sin ax . H(x), 4».

7.7. (1',4» = -(1,4>') = - J~1 x4>'(x) dx- Jo1(x +c)4>'(x) dx = -[X4>]~1 +

J~1 4>(x) dx - [(x + c)4>16 + Jol 4>(x) dx = c4>(O) + J~11 . 4>(x) dx = (c8,4» + (1,4».

7.8. Set A = {x: -1 < x < 0, -1 < y < O}, B = {x : 0 < x < 1, 0 < y< I}, C = Au B, with boundaries ßA, ßB. Then

D(1,I) (1, 4» = (1, DU,I) 4» = r xy ß24> dx dy Je ßxay

= r XYVx ~4> ds + r XYVx ~4> ds _ r y ~4> dx dy JöA uy JöB uy Je uy

= - iy~~ dxdy = i 4> dx dy.

7.9. Solution of homogenous equation is u(x) = e- X • Now (u' + u, 4» =

-(u, 4>') + (u,4» = -(H, /4>') + (H,/4» (usingu = Hf) = (8,4» after

integrating. Left-hand side = 1(0)4>(0) + JoIU' + f)4> dx =? fex) = e-X • Hence u(x) = (c + H(x))c x .

7.lO. (a) u E H 2(0, 3); (b) u E HI((O, 1) x (0,2)).

7.1l. u..L v in HI(O, 2).

7.13. D"'u E L 2 (fJ) far lai = 2; so m = 2 > n/2 = l.

7.14. Consider {un }, {vn } C Cl (f!) such that Un -> u and Vn -> u in the Hl-norm with u, v E HI(fJ) (H I is the closure of Cl). Then DO:Un ->

D"'u, DO:vn -> DO:v in L2, for lai ::::: l. Also, Vn -> v and Un -> U in L2(r). Thus, for example, (ßUn/ßxi,vnh2(o.) = (un,vnvih2(r) -(un , aVn/ßXi)P(o.). Take limn~oo.

7.15. Assume 0, c lR?; then left-hand side is 10. (~:~ + ~:~) (~:~ + ~:~) dx.

f ö2u ö2v f (ö2u ÖV ö3u) f ö4u d Now Jo. öx2 öx2 dx = Jr öx2 öx - öx3V Vx ds+ Jo. öx4V dx. Procee

in this manner; use ß/av = Vla/ßX + V2a/ßy.

7.16. Let {vn } be a sequence in 'D(fJ) with limit v E H{j(fJ). We have Ilvnllp ::::: clvnlHl; V n -> v in H I implies that IIvn llL2 --+ IIvllL2 and IvnlHJ -> IvlHl. I·IHJ is positive-definite since lviI = 0 implies that J l'Vvl 2 dx = 0, so that v = const = 0, given the boundary value of v.

Solutions 455

7.17. Show that (u,v) == InLlal=mDauDav dx is an inner product. In particular, (u, u) = 0 =} In(Da u)2 dx = 0 for Inl = m, hence Dau = o for Inl = m. But u E HO'(n); so u = O.

7.18. IIV2v lli2 = In [(~:~r + 2~:~ ~~ + (~:~r] dx.

f ( {)2 V ) 2 f {)3v {)v f {)2V {)2V

But in {)x {)y dx = - in {)x2 {)y {)x dx = in {)y2 {)x2 dx.

7.19. Require sup I (8, v) I to be defined, Le., v continuous. Hence m > n/2. For example, 8: HJ(n) --+ R is not defined for 0. C R2 .

7.20. u E HJ(n).L =} (U,V)Hl = 0 for all v E HJ(n); Le., 0 = In(uv + Llal=l DOtuDOtv) dx. Set v = </J E D(n): 0 = I(u. - V 2 u)</J dx using Green's theorem =} V 2u = u. Since D(n) is dense in HJ, we can extend this result in the usual way. u E HJ(n).L for 0. = (0,1) =}

u" - u = o. Basis for HJ(n).L is {eX,e-X}.

7.21. I(lnx?dx = x(lnx)2 - 2xlnx + 2x. Then use Theorem 9.

Chapter 8

8.1. (a) Second order, nonlinear, 0. = upper unit semicircle. (b) Fourth­order, linear, 0. = triangle with vertices at (0,0), (1,0), (0, I).

8.2. (a) 1t In' pv dx = In' Q dx + fr" t ds. Use Cauchy's law t = ern and the divergence theorem to rewrite the surface integral as In' diver dx. The left-hand side equals In' p{)2u /{)t2 dx. Regroup and invoke the arbitrariness of 0.' to obtain (8.5). (b) (J"ij = 'x(divu)Iij + 2J-tEij(U). Substitute in (8.5).

8.3. The argument is as in Example 2 of the Introduction: simply replace f by f - ku, ku being the force of the foundation.

8.4. (a) Lj {)(J"aj/{)Xj = Lß {)(J"Otß/{)xß + {)(J"Ot3/{)Z, where Z = X3. Inte­grate with respect to z and use the definitions of Sa. and Maß. (b) Fol­lows as part (a). (c) Differentiate (8.14h, with respect to x"' sum on n, and use (8.14h to eliminate Sa:; this gives La,ß {)2 M aß/ {)xa{)xß = -q. Next, use (8.13). This gives La,ß{)2Ma:ß/{)xa{)xß = -D[VLa [j2(V2w)/{)x; + (1 - v) La:,ß {)4w/{)x;{)x~1·

8.5. (a) w' = 0, Will = 0; (b) w = 0, w" = O.

8.6. Elliptic in A = {x: x > l,y > I} U {x: X < l,y < I}; strongly elliptic in any open subset of A.

8.7. LIOtI, IßI=l aa:ßE,Ot+ß = -(1 + x2)e + 3.,,2 + 2(1 + X2)(2 = 0 at any Xo for any E, such that e = [3.,,2 + 2(1 + z~)(2l1(1 + x6).

456 Solutions

8.8. Li,j,k,l Cijkl~i"'j~k"'l = JLle121771 2 + (>. + JL)(e· 77)2 = JLle1 21771.12 + (>. + 2JL) (e . 77)2, where 771. is the component of 77 orthogonal to e. The result follows from the independence of.,,1. and (e· 77). Pointwise sta­bility: f == Li,j,k,l CijklMijMkl = (3)'+2JL)IMs I2+2JLIMD I2 [Ms = ~(tr M)I and MD = M -MsJ. Show that IMI2 = IMs I2 + IMDI2: then f ~ cIMI2 Hf 3>' + 2{t > ko and {t > {to.

[)2U [)2U [)2U 8.9. -[) 2 V~ + 2-8 [) VI V2 + -8 2 V~ = g. Have to check Llal=2 baaa =

Xl Xl X2 X 2

v?a~+2VIV2ala2+V2a2 = (VIai +V2a2)2 = (v?+vi)2 f= 0 if a = v.

8.10. Use (8.13) and (8.14). The BC can be rewritten as ~ +V a~~~2 = o. With 10:1 = 3, Lbava = b(3,0)V? + b(I,2)VIVi = VI [V? + vviJ f= 0 along X = L, for which VI = 1, V2 = o.

8.11. Irl + Iß - 31 f= o.

8.12. ku·v-t·v = O. n = 2j bn = k, b22 = -Cn = Ij all other components are zero. So (8.33) is satisfied.

8.13. u· v = 0, t· s = {tt· v with t = UV, and u = Ce(v).

8.14. [u"'v - u"v' + u'v" - uvlllM = [-BIUSiv - BouSov + SIuBiv + SouBovM·

8.15. Bo = 8/[)v, So = -SQ = 1.

8.16. Set 8vif8xj = eijj then since u is symmetrie, Li,j O'ijeij = Li,j O'jieij (i)j also, by swapping indices, Li,j O'ijeij = Li,~ O'jieji (ii)j add (i) and (ii) to get desired result. To obtain (8.49), use the fact that Jn Lk,l O'klfkl(U) dx = Jn Lk,l O'kl(äuk/8xt} dx = Jr Lk,l O'klVIUk ds­Jn Lk,I([)O'k!/[)XI)Uk dx. Set u = Ce(v).

8.19. N(A) = {u: u(x) = ax + b} = N(A*). Solution exists if J; fex) dx =

Jol xf(x) dx = O. Solution is unique if J; u(x) dx = Jol xu(x) dx = O.

8.20. N(A) = {u: u const.}j unique solution if Jn u(x) dx = O. N(A*) = {u: u(x) = 0:1 + 0:2(X - Y)}j solution exists if Jn f dx = Jn(x­y) f dx = O. If n = (-1, 1) x (-1, 1), then Jn f dx = 0 if f is odd in X or Yj Jn(x - y)f(x) dx = 0 if f(x,y) = f(y,x).

8.22. (b) From (a), A : N(A)1. --> R(A) is bounded. Hence, using the Banach theorem, A-I : R(A) --> N(A)1. is linear, bounded '* IIA-IVIl s Kllvll for all v E R(A), so setting v = Au we have lIull s KIlAull for u E N(A)1.. If {vn} is a Cauchy sequence in R(A) with limit v, then with Un = A-1vn we have lIum - unll s Kllvm - vnll --> 0 as m, n --> 00; so {um} is a Cauchy sequence in N(A)1.. N(A)1. is closed;

Solutions 457

SO Um ----> u in N(A)1-. Since A is continuous, Vn 0= AUn =* v = Au. Hence v E R(A) =* R(A) is closed.

8.23. Flexible foundation: N(A) = {O} and a unique solution exists. Coulomb friction: N(A) = {clel + c2ed, where Cl and C2 are con­stants. A unique solution exists if and only if h =, 12 = O. If friction is not limiting, then N(A) = {O}.

Chapter 9

9.1. V = {v E H 2(O,I): v(O) = 0, v'(O) = O}. I~ [kUli v" + du'v' + cuJ dx = I; Iv dx + ßv(l) + Qv'(l).

9.2. Let angle between T and v be ß. Boundary term in VBVP is Ir vVu· v ds, v E HI(n). But T = vcosß + ssinß (s = tangent = (-V2,VI)), or T = (VICOSß-V2sinß, V2COSß+VIsinß) =* v = (Tl cos ß + T2 sin ß, - Tl sin ß + T2 COS ß). Boundary term is thus Ir v(g cos ß - Vu· IL sin ß) ds, where IL is normal to T.

9.3. a(w,v) follows by direct substitution of (8.13).

9.4. For continuity of a, use the Sobolev Embedding Theorem to obtain Iv(l)1 ~ CllvliHl ~ CllvllH2, etc.

9.5. a( v, v) ~ 10. J-l L.i t;, t;, dx using strong ellipticity. Complete by us­ing the Poincare-Friedrichs inequality.

9.6. Use (8.13) to obtain the first part. For the second part return to Exercise 9.3: the remaining boundary term is Ir M,,(w)8v/öv ds = O. Use the identity a2 + 2vab + b2 ::>: (1- v)(a2 + b2 ) and (7.18) to show that a(v, v) ::>: (1 - v) 10. L.1<>1=2 D<>u1 2 dx. Here v is Poisson's ratio.

Use the Poincare inequality (7.18) to get Io.(~';?dx ~ c Io.[(~,;)2 +

(a~::X2?J dx, etc. Then apply the Poincare-Friedrichs inequality to obtain a similar bound on Io.v2 dx. This leads to a(v,v) ~ C(l-v)llvll~2'

9.7. a( u, v) = 101 (pu'v' +TUV) dx+p(l)u(l)v(l). V-ellipticity: use Theorem I, Chapter 7, to get a(v,v) ::>: Qllvll~" Q = min(po,To). Continuity:

a(u,v) ~ 101 (PIU'V' + TIUV) dx + Plu(l)v(l) ~ k 101 (u'v' + uv) dx + Plilull oo Ilvll oo (k = max(Pl, Tl)) ~ k(u, V)Hl + PlKIlullH1llvllw (Sobolev Embedding Theorem) ~ (k + PlK)lluIIH1IlvIIHl.

9.8. (b) VBVP is: Jol U"V" dx + [h)v(l) - gIV'(l) - hov(O) + gov'(O)J = 101 Iv dx, v E H 2 (O, 1); so P = PI (0,1). Hence Q = {v E H 2 (O, 1) :

101 v dx = Jo1 xv dx = O.} Q-ellipticity is tricky, but see Rektorys [39], Chapter 35. A unique solution exists if and only if 0 = (P.,p) =

I; fp dx + [glP'(l) - hlP(1) + hop(O) - gOp'(O)J for all P E PI(O,I).

458 Solutions

9.9. See Exercise 8.8: use Korn's inequality.

9.11. a(u + p, v + p) = a(u, v) for p = Prer + p()e() , where Pr,P() E Po(!1). Q = {v E V: In Vr dx = In v() dx = O}; a unique solution exists if f satisfies In fr dx = In f() dx = O.

9.12. (DJ(u),v) = lim()->oe-1[J(u+ev) - J(u)] by definition. Set f(e) = J(u+ev) for any given u,v. Then (DJ(u),v) = lim()->oe-1[f(e)­f(O)] = 1'(0) = (d/de)(DJ(u)v)I()=o.

9.13. äJ / äXi = lim()->o[J(x+e(O, ... , Yi, . .. ,0) )-J(x )]/(eYi) (Yi in ith slot). Multiply by Yi and sum over i to get rcsult.

9.14. J(eu + (1- e)v) = He2a(u, u) + (1- e)2a(v, v) - 2e(1 - e)a(u, v)}­e(C,u) - (1 - e)(C,v). a(u - v,u - v) > 0 since a is V-elliptic, so 2a(1L, v) < a(u, u) + a(v, v). Use this to obtain strict convexity of J.

9.15. J(eu+(I-e)v) = eJ(u)+(I-e)J(v)-~e(l-e)a(u-v,u-v). Thelast term on the right is nonnegative. To show that u is a minimizer: from convexity, J(v) - J(u) :::: e- 1 [J(u + e(v - u)) - J(u)] = (DJ(u), v) when e ---7 0 (see Example 15).

9.16. J(v) = ~ 101 [k(v"? + d(v'? + cu2] dx - 101 fv dx - ßv(l) - o:v'(I).

9.17. Since u is a minimizer, J(u) :::; J((1 - e)u + ev) for 0 < e < 1 (since V must be convex). Expand and rearrange to get a(u, v - u) - (C, v­u) + ~ea(v - u,v - u) :::: o. Let e ---7 O.

Chapter 10

10.3. Un satisfies a( u, v) = (f., v) or (un , CPk)a = (C, CPk/. Also, Un =

L~=1 (u n , CPk)aCPk = L~=l (C, CPk)CPk. Now J(u) = -~ Ilull; (show this); but J(un ) = ~llun - ull; - ~llull;; hence Ilun - ull a ---7 O. The result Ilun - uliH ---7 0 follows from continuity of a(·, .).

10.5. Ilu - uhll; = a(u - Uh, U - Uh) = Ilull; - Iluh II~ - 2a(uh, u - Uh)' The last term is zero.

10.6. a(u,v) = (f.,v) so a(u,u) = (C,u); hence J(u) = -~a(u,u).

10.8. Uh(X) = (V2/2)( -CP1(X) - CP2(X) + CP3(X)) = (V2/2)(x2 + ~x - 1).

10.9. Replace v by AVh in Green's formula (G(u,v) = 0); (10.37) gives (AVh,f) = (AVh,Auh) =?- (vh,A*f) = (vh,A*Auh)' If A = _\72 ,

then A* = _\7 2 .

10.10. (a) I;(-u~ +Uh -- sinX)Vh dx = 0, Vh E vh C L 2 (0, 1), Uh E Uh C H 2 (0, 1) n HJ(O, 1). (b) Least squares: solve MT a = F, where Mij = 10\ -CP:' +CPi)( -'ljJ~' +

Solutions 459

'l/Jj) dx and Pj = J; (sinx)( -'l/Jj' +'l/Jj) dx. Collocation: solve L:=1 (-cp% (Xi) + CPk(xd)ak = f(xi), i = 1, ... , N. (c) Solve MT a = F, where Mij = J; CPi( -'l/Jj' + 'l/Jj) dx and Pj = Jo1 f'I/Jj dx.

Chapter 11

11.1. Must show that a function v, say, exists such that Jn ViCP dx = - Jn VOCP/OXi dx. For each Oe, Jne (OV/OXi)CP dx = Jre V!/iCP ds -Jne V OcpjOXi dx since vbe E H 1(O).

11.2. Optimal B = 5.

11.4. emax exists at point x, where e' = 0; then e(:l:i) = 0 = e(x) + ~e"(z)(xi - x? => le(x)1 = ~!e"(Z)I(Xi - x? If i is node nearer to x, then lXi - xl ::::: ~h. Hence le(x)1 ::::: ~h2Ie"(x)l. Maximize ovcr all elements to get result.

11.5. f"(x) = 27rcos7rx-7r2xsin7rx.IMax.valuel = 12r.1 at X = 0,1. Hence lIell= ::::: ~h2 ·271" = 7rh2/4. See whether log lIell= ::= 2h + const.

11.7. Retain 1,~, TJ, e, ~TJ, TJ2, eTJ, ~TJ2. Then, for example, if node 5 is located at ce, TJ) = (0, -1), N5 (e, TJ) = ~Cl - e)(:l- TJ)·

11.10. One needs to solve a system of 21 equations uniquely, for any given right-hand side. Equivalently, show that any polynomial for which D"p = 0 for lai::::: 2 at the vertices, and p" = 0 at the midpoints is identically zero. See Ciarlet [11] (Theorem 2.2.11) for full details.

4' , 11.11. x = LA=1 xANA(~,TJ), whcrc NA are given by (11.27). Substitute

and use the geometry of the parallelogram to verify that x = A~ + b for suitable A and b.

11.12. j = H2d - (e + TJ)(I- d)] > 0 for all ~ E n if d > ~.

11.15. (a) a = [1 1 1 IjT, b = 2[-1 1 1 - IjT, c = 2[1 1 - 1 - I]T, d = [1 -11 _1]T. (b) C'ilN)T = [-~b+TJd ~c+~d].

11.16. Show by direct integration over reference element; for examplc, Jne (a + b1x + b2 y) dxdy = Ae(a + bTx) where b = [bI b2 jT and x = (1/6)[1 1jT, for a polynomial of degrec one.

Chapter 12

12.1. IIT;;-lll = sup IIT;:-I Y II/lIylI, Y f O. Set z = Pey/llyll; thcn IIT;111 = supllp;IT;;-lzll· Pick x,y in Oe such that IIx - ylI = Pe: IIT;111 = p;l sup IIT;;-l(x - b + b - y)11 = p;1 sup IIx - yll = h/ Pe.

460 Solutions

12.2. //lv//m,1l = /lv/lm,1l ~ /lv/lk+l,n since m ~ k + 1. /lrrv/lm,n = /lI:iV(Xi)~i/lmll ~ I:i/iJ(Xi)"/~i/lmll ~ CsUP/iJ(Xi)/ (C is inde-pendent of iJ). ' ,

12.3. Let the triangle have angles o:,ß,"'( with (Je = 0: ~ ß ~ "'(. Let the sides opposite 0:, ß, "'( be a, b, c, respectively. Then a ~ b ~ c and he = c. The largest cirele inscribed in the tri angle touches all sides. Draw a sketch and show that he = (Pe/2) (cot 0:/2 + cotß/2). Now 0: < 7r/2,ß < 7r/2; so cotß/2 ~ coto:/2. Hence he/Pe ~ coto:/2 ~ a if we prescribe 0: ~ Bo, so that a = cot Bo/2.

12.4. k = 2 O(h~-m), 0 ~ m ~ 3, u E H 3 (n e )

k = 3 O(h!-m), 0 ~ m ~ 4, u E H 4 (n e ).

12.5. /Iv - TIev/l;" = I:;:o Iv - TIevli ~ C2h2(k+l)[aOh~ + a 2h;;2 + ... + a2mh-2ml/v/2 < C2eh2(k+l-m)[h2m+h2m-2+···+1l/vI2 (where

e k+l - e e k+l e = max(ao, .. . ,a-2m )). Given K > 0 we can always find E > 0 such that the term in square brackets < 1 + K provided he < E.

12.6. 'OiJ(a) = I:i :;, ai = I:i,j t:, ~ai = I:i,j t:] Tjiai = 'Ov(Ta). Pro­ceed in the same way for higher derivatives. Then for k = 2, for example, ID"'iJ(x) I ~ /l'02iJ/I = sup 1'02 iJ(a, b)/ (/la/l ~ I, /lbll ~ 1)

= sup/'02v(Ta,Tb)1 = SUpl'02V(~~ 1~1)1·/lTII2. Use IITa/l ~ IIT/Iliall ~ IITII·

12.7. Any v E X h also belongs to L2 (n), so it is required to find Vi E L 2 (n) such that Jn Vicj; dx = - Jn vß/ßx; dx Vcj; E '0(0.). Use Green's theo­rem applied to the function ßW/ßXi' where w = vln e ; then sum over

all elements to get In ViCP dx = - In Ußcj;/ßXi dx = I::=1 Jarle Wcj;Vi ds; the boundary integrals vanish.

12.8. a(w,e) = /lell1,2' where e = u - Uh. Also, a(wh,e) = 0; so a(w-wh,e) = lIe/l1,2' Hence /le 11 1,2 ~ Kllw-wh//I,n/lel/I,n ~ KCh/Lllwllp,nM3I1ullr,n for w E HP(n), u E Hr(n), where J-l = min(k,p - 1) and ß =

min(k,r - 1). Since Aw = e, we have w E H 2 (n) and I/w/l2m,n ~ ellellv; so /lel/L2 ~ C1hv llull r ,n.

12.9. /lu - Uh/lU ~ C1hvllu/lr,rl, where v = min(2,r) for linear or bilinear elements.

12.10. Using the Hermite basis functions and making appropriate changes (e.g., replace C(n) by CI(n)), the estimate (12.24) remains valid.

The VBVP is: find u E Hg(O, 1) such that I; (u"v" + k(x)uv) dx = 101 fv dx for all v E Hg(O,I). We obtain an error estimate from

lIu - uhl/2,n ~ Kllu - uhll2,rl = K (I:e /Iu - uhll~,nY/2 ~ Kh;-1

Solutions 461

(Ee lul~+l,oY/2 = Khk- 1 Iu lk+1,O, provided that Pk(O) c X c H 2 (0) and Hk+l(O) C C 1 (0) and u E Hk+l(n).

12.12. This follows as in Theorem 9. In particular k"ilv . v ~ kol"ilvl2 so

that a~(vh,vh) ~ koE~=lWtl"ilVh(~l)12 = kolvhltoe' since "ilvh E [P1(ne )J2 and a rule of order three is exact for quadratic functions.

Index

additivity, 88 adjoint problem, 286 affine family, 413 affine map, 372, 384, 412

in R?, 380 affine-equivalent element, 412 affine-equivalent elements, 413 almost everywhere (a.e.), 66 assembly, 380 Aubin-Nitsche method, 433 Axiom of Choice, 45

equivalence with Zorn's Lemma, 45

balance of energy, 2, 3, 9 of forces, 9 of moment um, 9

Banach space, 115 Banach Theorem, 151 Banach-Tarski paradox, 65 basis, 177 basis function, 16, 367

finite element, 367 bending moment (M), 261

for beams, 263 bending stiffness (D), 261 Bessel's Inequality, 193, 210 Best Approximation Theorem, 192 biharmonic equation, 312 biharmonic operator C'V4 ), 262 bijective operator, 139 bilinear form, 163

V-elliptic, 165, 316 continuous, 164, 316

bilinear polynomials (Qk), 384 biological population dynamics, 256,

264 biquadratic basis, 385 Bolzano--Weierstrass Theorem, 36,

60 boundary

insulated, 5 boundary conditions (BC), 5, 264

essential, 309 for elastic plate, 268 homogeneous, 309 in elasticity, 276 natural, 309 nonhomogeneous, 298

464 Index

boundary operators, 273 boundary value problem (BVP),

5,6,13 homogeneous, 6 two-point, 264 variational, 306

bounded fUllction, 60

C(a, b) or C[a, b], 56 c(n), 56 C(ri),56

not an inner product space, 97

as a complete space, 114 cm(n),57

as a vector space, 84 COO(n),57 C8"(n),216 calculus of variations, 332 Cartesian product, 27 Cauchy sequence, 113 Cauchy's equation of motion, 258 Cauchy's law, 258 Cauchy-Schwarz inequality, 91, 96 Cea's Lemma, 348, 411 characteristic function (XE), 68 choice function, 45 closed ball, 118 closed neighborhood, 116 closed set, 31, 116, 117

in ]Rn, 40 closure of a set, 31 collocation methods, 354 compact set, 120

in]Rn,40 compactness, 37 complete space, 114 completeness, 113

equivalence with closedness, 119

of C[a, b], 129 of finite-dimensional spaces,

183 completion, 124, 146 complex conjugate, 29

complex number imaginary part, 29 modulus,29 real part, 29

complex-valued function, 77 connected set, 40 consistency error estimate, 429 constitutive equation, 2, 9 constitutive law, 259 continuity, 54

equivalence of t: - 8 and limit definitions, 111

in]Rn,56 limit definition, 111 of a function of several vari­

ables, 56 continuous dependence on data,

14, 287 for elliptic BVP, 292

continuous functions, 54 measurability of, 68 on compact sets, 58

continuous operator, 143 convergence, 17, 33

in V(n), 217 in LP, 112 in the mean, 112 of sequences, 33 of sequences of functions, 108 pointwise, 108 rate of, 18 uniform, 108

convergence of interpolates, 350 convergence of sequences, 106 convex function, 327 convex functional, 328 convex set, 103 countable additivity, 65 covering condition

of boundary operators, 278

v(n), 216 distribution (V'(n)), 217 data, 256 dense sets, 121

inLP,122

density,3 of Co in V, 123

differential equation (DE), 256 linear, 256 order of, 256 ordinary (ODE), 256 partial (PDE), 256

diffusion, 2 equation, 7 steady,2

diffusion equation, 257 dimension, 177

of domain in finite-dimensional space, 185

Dirae delta, 157, 216, 217 direet sum, 86 Diriehlet system, 283

of boundary eonditions, 274 diseonneeted set, 40 displaeement, 9, 258 distanee from a point to aset (d(x, B)),

103 distribution, 214

derivative of, 219 generated by a loeally inte­

grable funetion, 218 in H-m, 246

produet with smooth funetion, 218

regular, 217 singular, 218

distributional derivative, 220 distributional differential equation,

223, 307 divergenee theorem, 4 domain, 40, 134, 135

Lipsehitz, 226 of dass cm, 226 of Sturm-Liouville problem,

202 with eurved boundary, 427

dual spaee, 157 of LP, 161

Index 465

eigenfunetion, 199 eigenfunction expansion, 200 eigenfunctions

orthogonal, 203 eigenvalue, 199

problem, 198 elastie

bar: well-posedness ofVBVP, 319

beam, 262, 393 membrane, 2, 9, 10 elastie plate, 260

well-posedness of VBVP, 320 elastieity

isotropie, 259 operator (0), 260 elliptie, 300 strongly elliptie, 300 tensor, 259

eleetrostaties, 2, 7 elliptie operator, 270 elliptie problem, 10 embedding, 232

eontinuous, 232 energy inner produet, 344 energy norm, ~144 equivalenee dass, 43 equivalenee relation, 42 equivalent norms, 97

and eonvergenee, 107 on H'[{'(f!), 244

error, 17 error estimate, 17, 18, 406

for seeond-order problems, 423 interpolation, 351 with numerical integration, 430

error estimates for fourth-·order problems, 434 loeal interpolation, 416

essential supremum (ess sup), 94 Euler-Bernoulli hypothesis, 262 existenee, 14

of solutions, 287, 316 to minimization problem, 332 to elliptie BVP, 292

466 Index

extension of an operator, 140

family of problems, 422 finite difference method, 16 finite element mesh, 365, 367 finite element method, 16

for second-order problems, 364 finite elements

regular family, 420 finite-dimensional space, 176 formal adjoint, 352

operator, 280 formally self-adjoint operator, 280 Fourier coefficients, 191 Fourier Series Theorem, 194 Fourier's law, 4 fourth-order problems, 392 Frechet derivative, 432 functions

bounded continuous, 18 even, 153 odd, 153 positive and negative parts,

72 with compact support, 121,

216 functional, 11

Gateaux-differentiable, 328

Galerkin approximations, 364 convergence, 348 errors in, 346

Galerkin method, 340 properties of approximations,

345 Gateaux derivative, 328 Gauss quadrat ure

in one dimension, 402 Gauss's law, 7 generalized partial derivative, 220 global basis function, 367 global interpolation, 422

error estimate, 423 gradient of a functional, 328

Gram-Schmidt orthonormalization, 181

greatest lower bound (inf), 36 Green's formula, 280 Green's theorem, 219, 242

Hm(n),226 H-m(n),246 H-1-methods, 356 half-bandwidth, 406 harmonie oscillator, 211 heat capacity, 3 heat conduction, 2, 5, 15, 16, 265

one-dimensional, 257 steady, 2, 6, 257

heat equation, 4, 198, 257 unsteady,5

heat ftux, 3 he at source, 3 Heaviside step funetion, 61

generalized derivative of, 221 measurability of, 68

Hermite differential equation, 211 polynomials, 212 basis functions, 394 families of elements, 392, 394

Hermitian, 88 Hilbert space, 115 Hölder inequality, 101

for sums, 103 homogeneity, 88 homogeneous medium, 257 Hooke's law, 259

identity operator, 138 image, 59, 135 image space, 134, 135 inductive limit topology, 216 infimum (inf), 35, 36 infinity 00, 32 initial boundary value problem (IBVP) ,

5,264 initial condition, 5 initial conditions (les), 264

initial value problem (IVP), 264 injective operator, 138 inner product, 87

defined by abilinear form, 344 inner product space, 87

finite-dimensional, 179 real,89

integrable function, 61, 73 integration by parts, 219 interior point, 30 interior point: in jRn, 39 interpolate, 349 interpolation error, 411

for isoparametrie elements, 427 interpolation operators

IIh ,421 (tr and IIe ), 415

interval, 29 closed,29 half-open, 29 open, 29

into, 135 inverse image, 135 inverse operator, 138 irrational numbers, 28 irrationality of v'2, 48 isometrie isomorphism, 146, 161 isometry, 145 isomorphisms, 142, 186

in finite-dimensional spaces, 187

isoparametrie elements triangular, 398 quadrilateral, 400

Jacobian matrix, 399

Kirchhoff-Love hypothesis, 260 Korn's inequality, 295, 320, 325

L 2 (n) as an inner product space, 90 as the completion of Coo(n),

231 LOO(n), 77

Index 467

LP(n), 62, 67, 75 as a vector space, 84

LP(a, b), 62 C(X, Y), 147 Lagrange bases, 373 Lame's constants, 259 Laplace's equation, 6 Laplacian, 6

in spherical coordinates, 210 operator, 137

Lax-Milgram Theorem, 166 least upper bound, 36 Lebesgue Dominated Convergence

Theorem, 74, 123 Lebesgue integral, 53, 64, 67, 69

of a measurable function, 70 of a simple function, 70

Lebesgue measure, 54, 65 Legendre polynomials, 207

and Gauss quadrature, 404 Legendre's equation, 202, 207 limit of a sequence, 33, 107 linear

combination, 176 dependence, 176 elasticity, 257 functional, 156

on finite-dimensional space, 189

independence, 176 interpolate, 377 operator, 140

bounded iff continuous, 150 on finite-dimensional space,

184 ordering, 42 space, 82

linearity, 88 Lipschitz

continuous function, 61 uniform continuity of, 80

domain,40 load vector, 370

element, 370 local basis functions, 415

468 Index

on reference element, 372 on square reference element,

385 on triangular element, 381 piecewise quadratie, 373

local numbering system, 379 locally integrable function, 217

mapping, 134 mass density (p), 258 matrix

representing linear operator, 187

maximal element, 44 maximum, 35 measurable function, 67 measurable set, 65 measurable space, 65 measure, 61 mesh parameter, 422 method of least squares, 354 method of weighted residuals, 353 metric, 98

generated by a norm, 99 metric space, 99 minimization of functionals, 326 minimization problem, 11

equivalence with VBVP, 330 minimizing sequence, 357 minimum, 35 Minkowski inequality

for integrals, 84, 10 1 for sums, 103

multi-index notation, 214

Navier's equations, 260 necessary condition, 46 neighborhood, 30, 116

in IRn , 38, 39 nodal points, 365 nonhomogeneous, 3 norm, 18,92

generated by an inner prod­uct, 95

matrix, 148

of an operator, 147 on LOO(O), 94 on LP(O), 94

normal boundary conditions, 274 normal derivative, 5 normed spaee, 18, 92, 95 norms

equivalent, 97 on IRn , 93, 103

null space, 135 numerical integration, 402

on square, 404 on triangle, 404 order, 402

one-to-one, 186 one-to-one operator, 138 onto, 135 open ball, 117 open mapping, 151 Open Mapping Theorem, 151 open neighborhood, 116 open set, 30, 116, 117

inlRn , 39 operator, 134

bijeetive, 139 bounded, 146 eontinuous, 143 differential, 136 identity, 138 injective, 138 inverse, 138 linear, 140 matrix, 136 one-to-one, 138 projection, 152 symmetrie, 203 uniformly continuous, 143

operators composition of, 137 equal, 137 product of, 137 sum of, 137

ordered n-tuples (!Rn), 38 ordered pairs (1R2 ), 37

ordered tripies (]R3), 38 orthogonal complement, 124

of HJ(fl) in H 1 (fl), 251 orthogonal projection, 154

on Hilbert spaces, 155 orthogonality, 91 orthonormal basis, 181, 190

eigenfunctions ofSturm-Liouville operator, 204

in Hilbert space, 196 orthonormal set, 180

maximal, 190

parallelogram law, 96, 102 Parseval's Formula, 193, 210 partial ordering, 42 partial sum, 191 partition, 43 Pascal triangle, 383 Petrov-Galerkin method, 355 piecewise linear function, 371 Poincare-Friedrichs Inequality, 244 Poincare Inequality, 233 point of accumulation, 31, 117

in ]Rn, 39

pointwise stable, 300 Poisson equation, 6, 257 Poisson's ratio (v), 261 positive homogeneity, 92 potential energy, 327 principal part, 270 projection, 152

orthogonal, 154 Projection Theorem, 127, 155, 194 proof by contradiction, 47

quality of approximation, 17 quintic polynomial, 397

]Rn, 37 as a complete space, 114 as a vector space, 83

ramp function, 62 generalized derivative of, 221

range, 59, 135

Index 469

Rayleigh-Ritz method, 345 rectangular elements, 383 reductio ad absurdum, 47 reference element, 371, 384, 412

triangular, 380 regular family

of isoparametric elements, 426 regularity of solutions, 287, 325 relation, 41

antisymmetric, 42 reflexive, 42 symmetrie, 42 transitive, 42

restrietion of an operator, 140 Riemann integral, 63 Riesz map, 161 Riesz Representation Theorem, 159 Riesz's Theorem, 162 rigid body displacement, 296, 324 Ritz-Galerkin method, 16

sampling points, 402 Schrödinger operator, 211 seminorm, 245 separable space, 123

Hilbert space as, 197 separation of variables, 197 sequences, 32

bounded,50 convergence of, 106 convergent, 33 finite, 32 of numbers, 32 in normed spaces, 106 infinite, 32 monotone, 50

serendipity element, 407 set, 23

complement of, 26 countable, 26, 66 elements of, 23 empty (0), 24 finite, 24 infinite, 24 linearly ordered, 42

470 Index

sets

null (0),24 of complex numbers (C), 29 of integers (Z), 24 of measure zero, 66 of natural numbers (fIT), 28 ofnonnegative integers (Z+),

24 of rational numbers (!Q), 28 of real numbers (IR), 28 partially ordered, 42 universal, 25

difference of, 25 equal, 25 intersection of, 25 of functions, 53 of numbers, 28 union of, 25

shear force (8), 261 for beams, 263

simple function, 64, 69 Sobolev Embedding Theorem, 232 Sobolev inner product (u, v) Hm,

227 Sobolev space

H m (0),226 ascompletionofCm (0),233 as completion of COO(O),

229 H()'(O) , 243 Wm,P(O), 235 alternative definition, 233 as a Hilbert space, 229

solution distributional, 225 generalized, 225 weak,225

space of admissible functions, 310 span, 177 square-integrable function, 75 steady-state, 2 stiffness matrix, 370

element, 370 strain, 259 strain energy, 326

Strang's Lemma, 429 stress, 258 strictly convex function, 327 strong convergence, 162 strongly elliptic operator, 270 Sturm-Liouville operator

positive, 204 symmetry of, 204

Sturm-Liouville problem, 201 regular, 201 singular, 202

subset, 24 proper, 24

subspace, 84 sufficient condition, 46 sum of subspaces, 85 supremum (sup), 35, 36 surjective, 135, 186

temperature, 3 test functions, 216 thermal conductivity, 4 thermal diffusivity, 6 trace,236

of a matrix, 259 trace operator

"Y,236 "YOll 241 as continuous map from H 1 (0)

into L 2 (r), 240 Trace Theorem, 240 traces

in the sense of, 242 transformation, 134 triangle inequality, 93 triangular elements, 379, 381

underintegration, 408 uniform continuity, 55, 56 uniqueness, 14

of solution, 287, 316 to elliptic BVP, 291 to minimization problem,

332 unit ball, 104

sufficient condition, 46 sum of subspaces, 85 supremum (sup), 35, 36 surjective, 135, 186

temperature, 3 test functions, 216 thermal conductivity, 4 thermal diffusivity, 6 trace, 236

of a matrix, 259 trace operator

,,236 'Oll 241 as continuous map from

H1(O) into L2 (r), 240 Trace Theorem, 240 traces

in the sense of, 242 transformation, 134 triangle inequality, 93 triangular elements, 379, 381

underintegration, 408 uniform continuity, 55, 56 uniqueness, 14

of solution, 287, 316 to elliptic BVP, 291 to minimization problem,

332 unit ball, 104 upper bound

of a partially ordered set, 44

variational boundary value problem (VBVP), 13, 306

continuous dependence on data, 316

Index 471

equivalence to classical problem, 307

existence of solution, 316 formulation, 309 uniqueness of solution, 316

variational inequality, 334 variational problem, 10 vector space, 82

Wm,P(O) as a Banach space, 235 as completion of COO(O),

235 as completion of cm(o),

235 continuous embedding in

C k (O),235 WS,P(O) for real s, 248 weak convergence, 162

in finite-dimensional spaces, 184

weak derivative, 222 weak* convergence, 162 Weierstrass Theorem, 121, 124 weighting function, 202 weights, 402 well-posedness, 14

Young's modulus (E), 261

Z~, 214 zero operator, 138 Zhl.mal's condition, 431 Zorn's Lemma, 45, 197

equivalence with Axiom of Choice,45