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GRIFFITH CONRAD EVANS May 11, 1887-December 8, 1973 BY CHARLES B. MORREY GRIFFITH CONRAD EVANS was born in Boston, Massachu- setts on May ~ I, 1887 and diecl on December 8, 1973. He receiver! his A.B. degree in 1907, his M.A. in 1908, and his Ph.D. in 191~0, all from Harvard University. After receiv- ing his Ph.D., he studied from ~ 9 ~ 0 through ~ 9 ~ 2 at the University of Rome on a Sheldon Traveling Fellowship from Harvard. He began his teaching career in 1912 as assistant professor of mathematics at the newly established Rice In- stitute, now Rice University, in Houston, Texas. He became professor there in 1916 and remained with the Institute until 1934. While he was at Rice, he was able to attract outstanding mathematicians, such as Professor Mandelbrojt of the Uni- versity of Paris, and young mathematicians, such as Tibor Rado and Car! Menger, to Rice as visiting professors. Long before Evans left Rice it was internationally known as a center of mathematical research. Evans was brought to the University of California at Berkeley in 1934 as a result of a nationwide search; he ar- rived with a mandate to build up the Department of Mathe- matics in the same way that Gilbert Lewis had already built the chemistry faculty. Evans struggled with himself to effect the necessary changes with justice. His innate sense of fair- ness, modesty, and tact, as well as his stature as a scientist, 127

128 BIOGRAPHICAL MEMOIRS brought eminent success. By the time he retired in 1954, he had had the satisfaction of seeing the department evolve into one of the country's major centers of mathematical activity. His retirement did not diminish his interest in science nor subtract from his pleasure at seeing others achieve goals he cherished. A few years before World War Il. Professor Evans and others on the Berkeley campus recognized the importance of the fields of probability and statistics, and Professor Jerzy Neyman was brought to that campus by Evans in 1939 to organize the Statistical Laboratory. A period of rapid growth followed; by the close of World War IT the Laboratory had transformed Berkeley into one of the three principal centers of probability and statistics in the country. The size and im- portance of the Laboratory continued to grow, and a separate Department of Statistics was established in 1955. Shortly after coming to Berkeley, Professor Evans inau- gurated a seminar in mathematical economics, which he gra- ciously held in his home once a week. This seminar became internationally known, providing an inspirational educa- tional activity and establishing a tradition of mathematical economics on the Berkeley campus that continues to the pres- ent. The seminar was attended by both students and faculty and promoted a friendly atmosphere in the department. FUNCTIONAL ANALYSIS In the first decade of the century, while Evans was a student, functional analysis was beginning to attract the inter- est of the mathematical community. Classical analysis was concerned with functions of real and complex variables, while functional analysis was concerned with functionals, that is, functions of"variables" that may themselves be ordinary functions or other mathematical entities. For example, iff

GRIFFITH CONRAD EVANS denotes any ordinary function continuous for O c may clefine a functional F by the equation rl F(f) = | f (x)dx. Jo 129 xc 1,we Evans began his career as a research scientist before he received the Ph.D. degree. He published his first paper in 1909. During the ensuing ten years, he contributed a great deal to the development of the general field of integral equa- tions and more general functional equations. His principal results concerned certain integro-ctifferential equations and integral equations with singular kernels. His interest in this field had been greatly stimulated by his contact with Profes- sor Vito Volterra at the University of Rome. He received early recognition for this important work in 1916 when he was invited to cleliver the prestigious Colloquium Lectures before the American Mathematical Society on the subject "Functionals and their Applications" (see bibliography, 1918). POTENTIAL THEORY IN TWO DIMENSIONS In 1920 Professor Evans published the first of his famous research papers on potential theory. He was among the first to apply the new general notions of measure and integration to the study of classical problems. In the course of this re- search, he introduced many icleas and tools that have proven to be of the utmost importance in other branches of mathe- matics, such as the calculus of variations, partial differential equations, and differential geometry; for example, he used certain classes of functions that are now known as "Sobolev spaces."

130 B IOGRAPHICAL MEMOI RS Introduction to Potential Theory. The central iclea in poten- tial theory is the notion of the potential of a distribution in R 3. Given a distribution of mass g, we define its potential U by the equation ~ I) U(`M) = J |MP|-ig(P)r1P w (W =R3,M = (x~y~z),P = (671~)) whenever this is clefinecI. In case g is Hoelcler continuous ~ for all P and vanishes outside a compact set, then U is of class C 2 and its second derivatives are Hoelcler continuous.2 In this case: (2) U (M)Uxx (x,y,z) + US (x,y,z) + Uzz (x,y,z) = - 4~g(M), M = (x,y,z). A solution that satisfies (2) with AU (M) = 0 on some domain is sail! to be "harmonic" on that clomain. Such a function has derivatives of all orders. The funciamental problem in potential theory is the DirichIet problem. Roughly speaking, this consists in proving the existence and uniqueness of the function U that satisfies Laplaces equation on a given domain G. is continuous on G (the closure of G), and takes on given continuous boundary values on the boundary FIG of G. Another problem, the Neumann problem, is to show the existence (and uniqueness except for an arbitrary acIditive constant) of a function V that satisfies Laplaces equation on G. is continuously differentiable on G. and for which the iA function g is Hoelder continuous on a set S if, and only if, I¢(P) - ¢(Q) I C L |PQ|" for some constants L and ,u, with 0 < ,u < 1 and all P and Q are both on S. See Oliver Dimon Kellogg, Foundations of Potential Theory (New York: Dover, 1929), p. 38 or 152, for instance: "This could be called a 'classical result.'"

GRIFFITH CONRAD EVANS 13 outer normal derivative dV/6n takes on given continuous values on dG. The function u (r, B), defined by3 (3) u (r, B) = ~ 2~ Jo f(ff), (l_r2) f(~)d¢, r < 1 l+r2 - 2 cos(¢B) if r = 1, is the solution of the DirichIet problem in the case where G is the unit circular disc in R.,. In case (4) ~ g (by d ~ = 0 and ~ V ( 1,0) d ~ = 0. ~ BIG ~ At, the solution of the Neumann problem with boundary values g (~)-on dG proceeds as follows. Let (5) v (r,H) = J1 27r log [1 +r2 - Or cos (¢'B) ] go) d ¢. AT 0 It is easy to see that rvr is harmonic on G. and (6) rvr (r,8) = 27r To 1 + r 2 - Or cos (+H) g (I) d 1 J 27J The first term on the right in (6) is the solution of the Dirich- let problem with boundary values g (I). If (4) holds, the sec- ond term is zero and V is one of the desired solutions. Among Evans' first results were those concerning the function 3 this is Poissonts Integral Formula.

32 BIOGRAPHICAL MEMOIRS (7) ~ (r, B) = 21 ( 1r 2) [ 1 + r 2 - 2r cos (+B) ] - 1 dF (¢ ), where F(O is of boundecT variation ant] periodic. Evans prover! the following: · the function u (r, fit) is harmonic in G ran ,] |u(r,0) |a,Cis bouncled for r ~ 1; o · u (r, B) = u 1 (r,0)u 2 (r,8) each u ~ being harmonic and non-negative on dG; · if P = (1,0 is a point on dG,whereF(¢~) is continuous anc]F'(¢~) exists and F'(O =f (O. then u (r,8) If (A as (r,0) ~ (1,¢~) "in the wide sense"; i.e., (r,H) > (1,¢~) remaining In any angle with vertex at (1,¢~). · If F and F' are continuous, then (7) recluces to the solution of the DirichIet problem with continuous boundary valuesf(¢)). . , the unit disc in R2; Conversely, if we assume that u = u ~u 2 where each u iO and is harmonic on G. then u is given by (7). Early Discussion of the Dirichlet Problem. The first attempt to solve the DirichIet problem was made by Green in 1828.4 His method] was to show the existence of a Green's function of the form G (Q,P ) = - + V ( Q. P ), r = (P,Q) This function is the Green's function for the region R and the pole P. In terms of this Green's function we have U(P) = - 4 ji U(Q) ~ G(Q,P) dS, s 4See Kellogg, Foundations of Potential Theory, p. 38.

GRIFFITH CONRAD EVANS 133 where S is the boundary of R. This development is based, however, on the existence and differentiability of G (Q,P), which is obtained using physical considerations and so is not logically suitable for a mathematical derivation. In 1913 Lebesgue gave an example of the impossibility of the solution of the DirichIet problem.5 The region R can be obtained by revolving about the x-axis the area bounded by the curves y =e-"x, y = 0 and x = I. This type of region is called a Lebesgue spine. It can be shown that the region obtained by revolving about the x-axis the area bouncled by the curves y = An, y = 0, X = 1, n > 1 is a regular region; i.e., the DirichIet problem is always solvable. The Logarithmic Potential Function. A similar theory holds for the two-dimensional situations. One considers the loga- rithmic potential function in R 2, defined by (~) U(M)= JW [ g(MP\)] g`P'~P M=(xy) W=R2 whenever this is defined. If g is Hoelcler-continuous for alIP ant! vanishes outside a compact set, then U is of class C 2, and its second derivatives are Hoelder-continuous everywhere. In this case, (9) AU (M) = Uxx fx,y) + UP ~x,y) = - 2 g (M), M = (x,y). A solution of (9) that satisfies AU (M) = 0 on some domain is find., p. 285, 334.

134 BIOGRAPHICAL MEMOIRS said to be harmonic on that domains such a function has derivatives of all orders. The Dirichiet and Neumann Problems in Space. The solution of the DirichIet problem in the unit sphere S is given bye , . (,lO)u(M) = ~ ~ f (M) ~ (4~-i or [~lr2) (MP)-3f (P) dS, O ~ r ~ I, s (r = 0M), M = (x,y,z) , r = 1 (S = S1, Sr = dB<O,r)~. The solution of the Neumann problem with given values g (M) of the normal derivative is obtained as in the case of the unit circle as follows. Let ( ~ 1) v (M) = - (~4 7r) -i JJ (MP) rig (P) dS s Then it is easy to see that rear is harmonic and ~ 12) no r (M) = 4JJ ~ ~ r 2) (MP ~ -3g (P ~ dS s 4 IT JJ g The first term on the right is just the solution of the Dirich- let problem with the boundary values g(M). If rat g(`M) dS = 0, then v is a clesired function. Evans and his colleague H. E. Bray proved a necessary and sufficient condition that a function a, harmonic on the s 6This is Poisson's Integral Formula for three dimensions. l

GRIFFITH CONRAD EVANS unit ball, be given by the formula ( 13) u (M) = (41r) -1 Ji ( 1 r2) (MP)-3 dG (P) for some distribution G (e) on S. is that J~J ~u(M) HIS be bouncled for O Or c 1, s 135 or that u = u ~ u 2 where u ~ and u 2 are non-negative and harmonic on B (0,1). If F(e) is a distribution on S. and if iim (par pa) -I |F [B (P. p)] | =f (P), then u (M) Of (P ) as M > p to P in the wide sense (i.e., M remains in a cone with vertex atP). The Riesz Theorem. A function V is said to be "super- harmonic" on a domain Q if, ant] only if, (i) it is lower semi- continuous and ~ + ~ on Q. and (ii) V |M |its mean value over the surface of any sphere with center M that lies with its interior in Q. Professor Evans proved that any potential function of a positive mass is superharmonic on any clomain on which it is defined. Evans also gave the simplest proof of the following theorem due to F. Riesz: Suppose u is superharmonic on a domain Q. and D iS any domain, the closure of which is compact and lies in Q. Then U (M) = U (M) + V (M) M ~ D where U is the potential of a positive mass on D and v is harmonic on D .7 7 F. Riesz, "Sur des functions superharmoniques et leur rappaport a la theorie du potential,"Acta Math, 48 (1926):32~43; 54 (1930):321-60.

136 BIOGRAPHICAL MEMOIRS Connection with Sobolev Spaces. In addition, Evans proved the following important theorems: Suppose U is super- harmonic on a domain Q. Then U (x,y,z) is absolutely contin- uous in each variable for almost all pairs of values of the other two and retains this property under one-to-one changes in variables of class C i.8 Finally Professor Evans proved the following theorems: Suppose U is superharmonic on some domain ant! Up(M) denotes the average of U over the surface OB (M,p); then Up(M) is continuously differentiable over any domain Qpo (which consists of all M such that B (M spot c Q) anct V UO > VU in L., on any such domain. A necessary and sufficient condition for a potential U offfe) to have a finite DirichIet integral is that Jew U(M) Offer exist. In this case U must belong to the Sobolev space H ~ on interior domains. Evans proved many more similar theorems. A Sequence of Potentials (A Sweeping Out ProcessJ. Evans gave a simple proof that the limit of a non-decreasing bounded sequence of potential functions of positive mass each distributed on a fixed bounded closed set F is itself a potential of positive mass F. The limit of a non-increasing sequence of such functions, however, is not necessarily super- harmonic (since the limit of a non-increasing sequence of lower-semicontinuous functions is not necessarily lower- . . sem~cont~nuous Nevertheless, Evans showed how to associate a particular type of positive mass distribution with a particular type of non-increasing sequence of potential functions on a bounded, closed set F. To do this, Evans let U , U.,, ..., be a non-increasing sequence of potentials of positive mass dis- tributionsf~,f.,, ..., respectively on F. Let UO be the limit X See bibliography entries of 1935 for the three-dimensional case and those in 1920 for the two-dimensional case.

GRIFFITH CONRAD EVANS 137 function. Clearly UO (M) ~ O but is not necessarily super- harmonic, although it is harmonic on T where T is the infinite domain Tying in the complement of F. whose boundary t c F. Thef, are uniformly bounded. Hence there is a subsequence {in) such that {fin ~ converges weakly to a positive mass func- tionf on F (or a subset of F). Thus, limit" <t>(M)dfin (ep)=J +(M)dffep) (W=3 space) for every bounded continuous function ¢. Also,f is inde- pendent of the subsequence. Let U be the potential off, then f (e) = 0 for all Borel sets e c T. Thus we may associate the positive mass function f with the non-increasing sequence U i, U.,, ..., all the mass having been swept out of T. Since f (e) = 0 for all e c T. U must satisfy the Laplace's equation on 1. Professor Evans cliscoverecT a great variety of similar sweeping out processes.9 He appliecl this type of process to sweep out a unit mass at a point Q in a domain T containing Q. This led to a number of interesting results ant! to a for- mula for the Green's function for T with pole at Q. Capacity. The notion of the capacity of a set arises in the applications and was usec! by Evans ant! was developed at some length in the second part of his paper "Potentials of Positive Mass."~° Evans also clef~necl the idea of a regular boundary point. It turns out that a boundary point Q of a domain ~ is regular if, and only if, a barrier V (M,Q) can be constructed at Q. Such a function V (M,Q) is continuous and superharmonic in I,, which approaches ~ at Q. and has a positive lower bounce in I, outside any sphere with center at 9 See Transactions of the American Mathematical Society, 38 (1935):205-13. t°Ibid., 218-26.

138 BIOGRAPHICAL MEMOIRS Q. If every boundary point is regular then the Dirichlet prob- lem is solvable. Multipite Valued Harmonic Functions in Space. In I896 Sommerfelt developed a method of using multivalued har- monic functions in three space to solve certain problems in potential theory, particularly the diffraction problem for a straight line. In 1900 ~2 Hobson used a combination of double-valued harmonic functions to obtain the conductor potential for a circular disc. Evans showed that if s is a simple closed curve of O capacity (any curve with a continuously turning tangent has capacity 0), there exists a unique surface S bounder} by s that has a minimum capacity among all such surfaces. If the part of S outside a neighborhood of s is composed of a finite number of sufficiently smooth pieces and V(M) is the conductor potential for S. then Evans showed that V must satisfy V (Q) ~ V (Q) = , Q C S an ~ ~n- where n ~ and n- are the normals to S at Q. Moreover, if this hoIcis on a smooth part of S. then S is analytic on that part. The proof of this involves "clouble valued" functions, the tract by Evans, "Lectures on Multiple-Valucct Harmonic Functions in Space" (see bibliography, 1951), presents an ex- tensive systematic development of a part of the theory of such ~ . functions. A simple example of a multiple space of a type used by Evans is the double space H. which consists of all orderect i'Mathematische Theorie der Diffraction,MathAnnalen, 47(1896):317-74 and .. Uber verzweigte Potential wie Rauma, Proceedings of the London Mathematical Society, 28(1897):395-429. 12 E. W. Hobson, "On Green's Function for a Circular Disc," Cambridge Philosophical Transactions, 18(1900):277-91.

GRIFFITH CONRAD EVANS pairs (M,m) where m = 0 or +1 or -1, and 139 ifm= - 1, thenMeR3s; if m= +1, thenMeR3s; ifm=O, thenM=s, S = {(x,y,z): x2 +y2 = I, anaz = 0~. Geometrically, we may think of H as consisting of two infinite flat, rigid, 3-dimensional sheets of the form R 3S joined together along s. ,' . . ouch a space Is a three-d~mensional analog of a Riemann surface in the complex plane, and Evans' result led him into his extensive research on multiple-valued functions, his prin- cipal interest during his later years. Since any multiple space is, by definition, a topological space, open, closed, and connected subsets of such spaces are clefinecl and the usual theorems hoIcI. Also harmonic, super- harmonic, and subharmonic functions can be clefinec! on domains T c multiple spaces. Much of the existence and uniqueness theory for harmonic and potential functions is carried over by Evans to the case of multiple-valuecI func- tions, that is, single values! functions defined on multiple spaces. For example, Evans showed that there is a unique harmonic function that takes on given continuous boundary values at all regular points. Moreover the definitions and theorems about barriers carry over. But there are many new results for infinite domains (on multiple spaces). For example, Evans proved that there is a unique function A(M) bounclecI and harmonic in T. u T2 u . . . u Tn that takes on the values ~ at infinity on the leaf T and approaches O at infinity on the other leaves. (Ti = T n Hi where Hi is the i-th leaf of H.) Let T be a bouncled domain c H. a particular space, and let A be a fixed point in T. Evans showed that there exists a unique Green's function with pole at A that has the following properties: As a function of M: (i) ~ (A,M) is harmonic in T except at A.

140 BIOGRAPHICAL MEMOIRS (ii) by (A,M) is bounded except in a neighborhood of A and fly (A,M ~~ /AM remains bounded near A. (iii) by (A,M) vanishes at all regular points of the boundary t of T. in addition, fly (B,~4 ~ = by (A,B ~ for A and B where A and B ~ T. There is a unique function K(~4,T) with the same properties, with A end M onA = TIU. . . U Tn. Also, K(A,M) ~ O asM ~ so on any leaf. Finally Evans proved that there is a unique ~ ~ . . . . surface ot minimum capacity that spans a given space curve s or a set of space curves Is I. it is the locus of the equation A (M) = I/2 where H is taken as a two-leaved space and the si are chosen as branch curves in H. Finally, a version of Green's theorem holds for domains T on multiple spaces whose boundary consists of several branch of zero capacity and several smooth curves s,, . surfaces. ~3 r MATHEMATICAL ECONOMICS Evans' work in mathematical economics was that of a pio- neer. At a time when most economists in this country dis- dained to consider mathematical treatments of economic questions, he boldly formulated several mathematical models of the total economy in terms of a few variables and drew conclusions about these variables. Some of these expositions were based on the theories of Cournot (~837) and some are found in the book Mathematical Introduction to Economics by Evans. The simplest theory is the following: It is envisaged that there is only one commodity being manufactured by one producer, and one consumer. The cost of manufacturing and marketing u units of the commodity per unit time is q (u ); this i3 For a full discussion, see Griffith Conrad Evans, "Multiply Valued Harmonic Functions. Green's Theorem," Proceedings of the National Academy of Sciences of the United States of America, 33 (1947):270-75.

GRIFFITH CONRAD EVANS 141 is called the cost function. The consumer will buy y units of the commodity (per unit time) if the price is p per unit; thus y = ~ (p) is the demand function. The market is in equilib- rium if y = a, that is, if all the commodity is sold. Clearly the profit ~ made by the producer is given by ~ =pu -q(u) =p+(p) -qL+(P)] The producer is a monopolist if he can sell all he produces at any given price. In this case it would be reasonable to assume that the producer would set the price to maximize his profits. This leads to the equation: dp {P ¢(P)q L¢(P)]'J = dp = 0 In order to get a solution, we must know the functions q (a) and ¢, (p). The simplified form for q (u) is Au2 + Bu + C. C represents the overhead and should be > 0. The average . . cost per unit IS q(u) =,4u +B +- u u which may reasonably increase ultimately, so that A > 0. The "marginal unit cost" is ~ = 2Au +B . du If dqldu is 2 0 for uO. we must haveBO; we may as well assume B > 0. Clearly ¢(p) is decreasing and positive; the simplest form for ~ (p) is ap + b where a ~ O and b > 0. If the market is in equilibrium, we have y =u =ap +b u -b a u -b A=- or a MU 2BuC.

142 BIOGRAPHICAL MEMOIRS In order to maximize A, one must have al~/du = 0. This yields 2AuB =0 a a b +ba Ba + 2Aab - b u= p= 2 - 2Aa 2a ( lHa) Of course a monopolist may choose u (or p) to satisfy some other condition. As a second theory, Evans assumes that there are two producers manufacturing amounts u ~ and u 2 of the commoct- ity (per unit time). :Let us assume that the producers are subject to the same cost function q (ui) = Ad + Bu i + C and there is proclucect only what is sold; that is, the market is in equilibrium. if we assume the same clemancl function, y =U~ +U2 =ap +b, then the selling values are pat and the profits are Hi = phi(A Uz2 + Bui + C), i = I,2. Additional hypotheses are needed to fins! p and then ui. Suppose each producer tries to determine ui so as to maxi- mize the total profit, still assuming equilibrium. In this case, we say that the producers are cooperating. Then the total profit ~ = ~~ + ~r2 iS Jo =p(U ~ +U 2)A (u 2 +U~B but +u2)2C, 1T = (U ~ + U 2)A (u 2 + u 2 ~B(u ~ + u 2)2C a using (1) to determine p. Assume u ~ and u 2 are chosen to maximize A. Then we must have

GRIFFITH CONRAD EVANS d7r din O or du1 dU2 2 (u 1 +U2) - b a 2(Ul+U2)-b a 2Au 1B = 0, 2Au2B =0, 143 which determine u 1 and u 2 uniquely. As a third theory, let us suppose that a producer is subject to the same cost and demand functions but has no control over the price. Then he will choose u to maximize ~ =pu Au2Bu C for the given price; this yields ~ ~T/dU = p 2A u B to determine bB 2A This theory can be generalizes! to the case where there are n producers, each subject to a different cost and demand function, but who all set the same price p. Then the i-th producer produces ui units where the total profit is n U=Z ~i= i = I n ~ (pal i = I This will be a maximum if du; which yields d~=o or p2AiuiBiUi, pBi Hi=- a = 1~. 2Ai 7 A iUi2B in iCi). i = 1. . . , n. , n,

144 BIOGRAPHICAL MEMOIRS Similar problems can be solved in some cases where the u i end p depenc! on time. Evans solvent such a problem in which there is one producer with q =Au2+Bu +C, y =ap+b +hp', ~ >O. B >O. C >0 a Orb >0,h TO, p' =dp/dt, and A,B, C,a,b,h are all constants. The term hp' is suggested by the consideration that the de- mand is greater when the price is going up than when the price is going clown, other things being equal. This problem required more sophisticated mathematics. It is still assumed that u = y for all t and that the rate of profit is ~ =puq (u), and finally that the profit made cluring the interval (t Ott ,) is r' AT=} U(p,p,) dt I,, is a maximum over any interval (to,t~. This leads to the condition that the integral Jt I (2) {p(ap +b +hp')A(ap +b +hp')2 t,, . B (ap + b + hp')C} dt Is a maximum of any interval, where 7T (p,p ') is given by the integrand in (2). This is a standard problem in the calculus of variations.~4 From that theory we conclucle that the Euler equation (3) (arm) = REP dt resee any book on the calculus of variations, for example, G. A. Bliss, Calculus of Variations (Washington, D.C.: Mathematical Association of America, 1925).

GRIFFITH CONRAD EVANS 145 holds. Carrying out the differentiation with respect to t in (3), we get for Euler's equation (4) where (5) ~T, p, means means , etc. ~p ~P'P' d P2 +~P,P dP ='7Tp, 62~ (6 ')2' ~P'P means 82~ dp'dp ~T(p,p') =p(ap +b +hp')A(ap +b +hp')2B(ap + b + hp ')C and the derivatives in (5) are the indicated partial . ' as independent variables. derivatives regarding p and p Carrying out the differentiations in (5), we get (6) ~p,p, = - 2Ah2, ~p,p = h (1 - 2aA), rrp = a (2pB) +hp '(1 - 2aA) +b (1 - 2aA)3a2Ap. Setting ~Ip/dt = p ' in (4) and using (5) and (6), Euler's equa- tion becomes 2Ah 2p,, + h (1 - 2aA)p' = a (2p B) + hp'(1 - 2aA) + b ( 1 - 2aA ) - 2a 2Ap 2Ah 2p', = 2a ( 1 aA )p + b ( 1 - 2aA ) aB 2a (1aA) b (1 - 2aA)aB = > p " = - p + = M p N which is reduced to the form 2Ah2 -, i.e.,p" dt2 f(P)

46 BIOGRAPHICAL MEMOIRS This is solvable by standard methods in differential equa- tions. This is one of the simplest cases. More sophisticated theo- ries involving such things as taxes, tariffs, rent, rates of change, transfer of credit, the theory of interest, utility, the- ories of production, and problems in economic dynamics were worked out by Evans. Evans' scientific career resulted in over seventy substantial published articles, four books, and several ciassifiecl reports. It should be added, since it is such a rare occurrence among mathematicians, that he continued his productive work for many years after his retirement. He gave a number of invites! actresses in Italy and elsewhere cluring that period. Professor Evans was elected to the National Academy of Sciences in 1933 anti became a member of the American Academy of Arts and Sciences, the American Philosophical Society, the American Mathematical Society (vice president, 192~26; president, 1938-401; the Mathematical Association of America (vice president, 1934), and the American Associa- tion for the Advancement of Science. He was a fellow of the Econometric Society. Evans was invited to give addresses in connection with the Harvard Tercentenary and the Princeton Bicentennial Cele- bration. He was also asked to give the Roosevelt Lecture at Harvard in ~ 949 and was Faculty Research Lecturer in Berk- eley in 1950 and was awar(lecl an honorary degree by the University in 1956. The Griffith C. Evans Hall on the Berke- ley campus was cledicatect in 1971. During World War I, Evans served as a captain in the Signal Corps of the U.S. Army. During World War Il. he was a member of the Executive Board of the Applied Mathemat- ics Pane! anct was part-time technical consultant, Ordnance, with the War Department. He received the Distinguishecl Assistance Award from the War Department in 1946 and received a Presidential Certificate of Merit in 1948.

GRIFFITH CONRAD EVANS 147 The charming hospitality of the Evanses is remembered with pleasure by those fortunate enough to have been guests at their home. And Evans' own keen, dry sense of humor was much appreciated by his many friends and associates. Professor Evans married Isabel Mary John in 1917. They had three children, Griffith C. Evans, Jr., George William Evans, and Robert John Evans and many grancichildren.

48 BIOGRAPHICAL MEMOIRS BIBLIOGRAPHY 1909 The integral equation of the second kind, of Volterra, with singular kernel. Bull. Am. Math. Soc., 2d ser., 41:130-36. 1910 Note on Kirchoffts law. Proc. Am. Acad. Arts Sci., 46:97-106. Volterra's integral equation of the second kind, with discontinuous kernel. Trans. Am. Math. Soc., 11: 393-413. 1911 Volterra's integral equation of the second kind, with discontinuous kernel, Second paper. Trans. Am. Math. Soc., 12:429-72. Sopra l'equazione integrale di Volterra di seconda specie con un limite del'integrale infinito. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., ser. 5, 20:409-15. L'equazione integrale di Volterra di seconda specie con un limite del'integrale infinito. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., ser. 5, 20:656-62. L'equazione integrale di Volterra di seconda specie con un limite del'integrale infinito. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., ser. 5, 20:7-11. Sul calcolo del nucleo dell'equazione risolvente per una data equa- zione integrale. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., ser. 5, 20:453-60. Sopra ['algebra delle funzioni permutabili. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., ser. 5, 8:695-710. Applicazione dell'algebra delle funzioni permutabili al calcolo delle funzioni associate. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., vol. XX, ser. 5, 20:688-94. 1912 Sull'equazione integro-differenziale di tipo parabolico. Rend. R. Accad. Lincei C1. Sci. Fis., Mat. Nat., ser. 5, 21:2~31. L'algebra delle funzioni permutabili e non permutabili. Rend. Circ. Mat. Palermo, 34: 1-28.

GRIFFITH CONRAD EVANS 1913 149 Some general types of functional equations. In: Fifth International Congress of Mathematicians, Cambridge, vol. 1, pp. 389-96. Cam- bridge: Cambridge University Press. Sul calcolo delta funzione di Green per le equazioni differenziali e intergro-differenziali di tipo parabolico. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., ser. 5, 22:855-60. 1914 The Cauchy problem for integro-differential equations. Trans. Am. Math. Soc., 40:215-26. On the reduction of integro-differential equations. Trans. Am. Math. Soc., 40:477-96. 1915 Note on the derivative and the variation of a function depending o all the values of another function. Bull. Am. Math. Soc.,2d ser., 21:387-97. The non-homogeneous differential equation of parabolic type. Am. J. Math., 37:431-38. Henri Poincare. (A lecture delivered at the inauguration of the Rice Institute by Senator Vito Volterra. Translated from the French.) Rice Inst. Pam., 1 (21: 1 33~2. Review of Volterra's "Le~cons sur les fonctions de lignes." Science, 41(1050):246-48. 1916 Application of an equation in variable differences to integral equa- tions. Bull. Am. Math. Soc., 2d ser., 22:493-503. 1917 I. Aggregates of zero measure. II. Monogenic uniform non-analytic functions. (Lectures delivered at the inauguration of the Rice Institute by Emile Borel. Translated from the French.) Rice Inst. Pam., 4( 1 ): 1-52. I. The generalization of analytic functions. II. On the theory of waves and Green's method. (Lectures delivered at the inaugura- tion of the Rice Institute by Senator Vito Volterra. Translated from the Italian.) Rice Inst. Pam., 4~11:53-117.

150 BIOGRAPHICAL MEMOIRS 1918 Harvard college and university. Intesa Intellet., 1:1-11. Functionals and their Applications Selected Topics, Including Integral Equations. Amer. Math. Soc. Colloquium Lectures, vol. 5, The Cambridge Colloquium. Providence, R.I.: American Mathemat- ical Society. 136 pp. 1919 Corrections and note to the Cambridge colloquium of September, 1916. Bull. Am. Math. Soc., 2d ser., 25:461~3. Sopra un'equazione integro differenziale di tipo Bocher. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat., vol. XXVIII, ser. 5, 33:262-65. 1920 Fundamental points of potential theory. Rice Inst. Pam., 7: 252-329. 1921 Problems of potential theory. Proc. Natl. Acad. Sci. USA, 7:89-98. The physical universe of Dante. Rice Inst. Pam., 8:91-1 17. 1922 A simple theory of competition. Am. Math. Monogr., 29:371-80. 1923 A Bohr-Langmuir transformation. Proc. Natl. Acad. Sci. USA, 9:230-36. Sur l'integrale de Poisson generalisee (three notes). C. R. Seances Acad. Sci., 176: 1042-44; 176: 1368-70; 177:241-42. 1924 The dynamics of monopoly. Am. Math. Monogr., 31:77-83. 1925 I1 potenziale semplice ed it problema di Neumann. Rend. R. Accad. Naz. Lincei C1. Sci. Fis. Mat. Nat., ser. 6,2:312-15. Note on a class of harmonic functions. Bull. Am. Math. Soc., 31: 14-16.

GRIFFITH CONRAD EVANS 151 Economics and the calculus of variations. Proc. Natl. Acad. Sci. USA, 11:90-95. The mathematical theory of economics. Am. Math. Monogr., 32: 10~10. On the approximation of functions of a real variable and on quasi- analytic functions. (Lectures delivered at the Rice Institute by Charles de la Vallee Poussin. Translated from the French by G. C. Evans.) Rice Inst. Pam., 12~2~:105-72. Enriques on algebraic geometry. Bull. Am. Math. Soc., 31 :449-52. 1927 With H. E. Bray. A class of functions harmonic within the sphere. Am. I Math., 49: 153-80. Review of "Linear integral equations" by W. V. Lovitt. Am. Math. Mon., 34:142-50. The Logarithmic Potential. Am. Math. Soc. Colloquium Publications, vol. 6. Providence, R.I.: American Mathematical Society. 150 pp. 1928 Note on a theorem of Bocher. Am. I. Math., 50:123-26. The Dirichlet problem for the general finitely connected region. In: Proceedings of the International Congress of Mathematicians, Toronto, vol. 1, pp. 549-53. Toronto: University of Toronto Press. Generalized Neumann problems for the sphere. Am. I. Math., 50: 127-38. The position of the high school teacher of mathematics. Math. Teacher, 21 :357-62. 1929 Discontinuous boundary value problems of the first kind for Pois- son's equation. Am. J. Math., 51: 1-18. With E. R. C. Miles. Potentials of general masses in single and double layers. The relative boundary value problems. Proc. Natl. Acad. Sci. USA, 15: 102-8. Cournot on mathematical economics. Bull. Am. Math. Soc., 35:269-71.

152 BIOGRAPHICAL MEMOIRS 1930 With W. G. Smiley, Jr. The first variation of a functional. Bull. Am. Math. Soc., 36:427-33. The mixed problem for Laplace's equation in the plane discontin- uous boundary values. Proc. Natl. Acad. Sci. USA, 16:620-25. Mathematical Introduction to Economics. New York: McGraw-Hill. 177 pp. Stabilite et Dynamique de la Production dans l'Economie Politique. Me- morial fasicule no. 61. Paris: Gauthiev-Villars. 1931 With E. R. C. Miles. Potentials of general masses in single and double layers. The relative boundary value problems. Am. I. Math., 53 :493-516. Zur Dimensionsaxiomatik. Ergebnisse eines mathematischen kollo- quiums. Kolloquium, 36: 1-3. A simple theory of economic crises. Am. Stat. J., 26:61-68. Kellogg on potential. Bull. Am. Math. Soc., 37:141~4. 1932 An elliptic system corresponding to Poisson's equation. Acta Litt. Sci. Regiae Univ. Hung. Francisco-}osephinae, Sect. Sci. Math., 6~11:27-33. Complements of potential theory. Am. T. Math., 54:213-34. Note on the gradient of the Green's function. Bull. Am. Math. Soc., 38:879-86. The role of hypothesis in economic theory. Science, 75:321-24. 1933 Complements of potential theory. Part II. Am. }. Math., 60:29~9. Application of Poincare's sweeping-out process. Proc. Natl. Acad. Sci. USA, 19:457-61. 1934 Maximum production studied in a simplified economic system. Econometrica, 2: 37-50. 1935 Correction and addition to "Complements of potential theory." Am. }. Math., 62: 623-26.

GRIFFITH CONRAD EVANS 153 On potentials of positive mass Part I. Trans. Am. Math. Soc., 37: 226-53. Potentials of positive mass. Part II. Trans. Am. Math. Soc., 38: 201-36. Form and appreciation (address, annual meeting of Chapter A1- pha, Phi Beta Kappa). Counc. Teach. Math., 11:245-58. 1936 Potentials and positively infinite singularities of harmonic func- tions. Monatsch. Math. Phys., 43:419-24. 1937 Modern methods of analysis in potential theory. Bull. Am. Math. Soc., 43:481-502. Indices and the simplified system. In: Report of Third Annual Re- search Conference on Economics and Statistics, pp. 56-59. Colorado Springs, Colo.: Cowles Commission for Research in Economics. 1938 Dirichlet problems. Am. Math. Soc. Semicenten. Publ., II: 185-226. Review of "Theorie generate des fonctionelles" by V. Volterra and I. Peres. Science, 88, no. 2286, n.s., 380-81. 1939 With K. May. Stability of limited competition and cooperation. Rep. Math. Colloq., Notre Dame Univ., 2d. ser.:3-15. 1940 Surfaces of minimal capacity. Proc. Natl. Acad. Sci. USA, 26: 489-91. Surfaces of minimum capacity. Proc. Natl. Acad. Sci. USA, 26: 662-67. 1941 Continua of minimum capacity. Bull. Am. Math. Soc., 47:717-33. 1942 Two generations and the search for truth. (Twenty-eighth Annual Commencement Address, Reed College, 1942.) Reed Coll. Bull., 21:1-14.

154 BIOGRAPHICAL MEMOIRS 1947 A necessary and sufficient condition of Wiener. Am. Math. Mon. 54:151-55. Multiply valued harmonic functions. Green's theorem. Proc. Natl. Acad. Sci. USA, 33: 270-75. 1948 Kellogg's uniqueness theorem and applications. Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 95-104. New York: Wiley Interscience. 1950 Mathematics for theoretical economists. Econometrica, 18: 203-4. 1951 Lectures of multiple valued harmonic functions in space. Univ. Calif. Publ. Math., n.s. 1:281-340. 1952 Note on the velocity of circulation of money. Econometrica, 20:1. 1953 Applied mathematics in the traditional departmental structure. In: Conference on Applied Math, pp. 11-15. Mimeographed. New York: Columbia University. Metric Spaces and Function Theory. Mimeographed. Berkeley, Calif.: ASUC Store. 1954 Subjective values and value symbols in economics. In: From Symbols and Values, an Initial Study, Thirteenth Symposium of the Conference on Science, Philosophy, and Religion, p p. 745-57. New York: Harper & Row. 1957 Calculation of moments for a Cantor-Vitali function. Am. Math. Mon., 64:22-27.

GRIFFITH CONRAD EVANS 1958 155 Surface of given space curve boundary. Proc. Natl. Acad. Sci. USA, 44:766-88. Infinitely multiple valued harmonic functions in space with two branch curves of order one. Tech. Rep. 29, Contract Nonr-222 (37) (NR 041-157), pp. 1-59. Washington, D.C.: Office of Naval Research. 1959 Infinitely multiple valued harmonic functions in space with two branch curves of order one. Tech. Rep. 12, Contract Nonr-222 (62) (NR 041-214), part II, pp. 1-20. Washington, D.C.: Office of Naval Research. 1960 Infinitely multiple valued harmonic functions in space with two branch curves of order one. Tech. Rep. 12, Contract Nonr-222 (62) (NR 041-214), part III, pp. 1-75. Washington, D.C.: Office of Naval Research. 1961 Funzioni armoniche polidrome ad infiniti valor) Hello spazio, coil due curve di ramificazione di ordine uno, Rend. Math., 20:289-311. 1964 Theory of Functionals and Applications (reissued). New York: Dover Publications.