# An Appreciation of James Serrin

Chapter

## Abstract

I am particularly happy to have the opportunity of these proceedings to write a profile of one of my distinguished teachers, with whom I have also had the good fortune to collaborate for so many years, Professor James Serrin. Before entering into my recollection of these years of close association, I would first desire to mention the tribute that Professor Clifford Truesdell wrote in the volume of papers, Analysis and Continuum Mechanics, dedicated to James Serrin on the occasion of his sixtieth birthday, published by Springer-Verlag, in gratitude for his many years of work as coeditor of the *Archive for Rational Mechanics and Analysis*.

## Keywords

Global Asymptotic Stability Quasilinear Elliptic Equation Nonlinear Diffusion Equation Linear Elliptic Equation Sixtieth Birthday
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## James Serrin’s publications

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*J. Math. and Phys.*,**29**(1950), 1–12.MathSciNetMATHGoogle Scholar - [2]Uniqueness theorems for two free boundary problems.
*Amer. J. Math.*,**74**(1952), 492–506.Google Scholar - [3]Existence theorems for some hydrodynamical free boundary problems.
*J. Rational Mech. Anal.*,**1**(1952), 1–48.Google Scholar - [4]
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*J. Rational Mech. Anal.*,**2**(1953), 563–575.Google Scholar - [7]
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*J. Rational Mech. Anal.*,**3**(1954), 395–413.Google Scholar - [10]A uniqueness theorem for the parabolic equation
*u*_{t}=*a(x)u*_{xx}+*b(x)u*_{x}+*c(x)u*,*Bull. Amer. Math. Soc.*,**60**(1954), 344.Google Scholar - [11]Uniqueness of axially symmetric subsonic flow past a finite body (with D. Gilbarg).
*J. Rational Mech. Anal.*,**4**(1955), 169–175.MathSciNetMATHGoogle Scholar - [12]A characterization of regular boundary points for second-order linear differential equations.
*Bull. Amer. Math. Soc.*,**61**(1955), 224.Google Scholar - [13]On the Harnack inequality for linear elliptic equations.
*J. Analyse Math.*,**4**(1956), 292–308.Google Scholar - [14]On isolated singularities of solutions of second-order linear elliptic equations (with D. Gilbarg).
*J. Analyse Math.*,**4**(1956), 309–340.MathSciNetMATHCrossRefGoogle Scholar - [15]A note on harmonic functions defined in a half-plane.
*Duke Math. J.*,**24**(1956), 523–526.Google Scholar - [16]On the Wilder continuity of quasi-conformal and elliptic mappings (with R. Finn).
*Trans. Amer. Math. Soc.*,**89**(1958), 1–15.MathSciNetMATHGoogle Scholar - [17]
*Mathematical Principles of Classical Fluid Mechanics*(Monograph). Handbuch der Physik,**VIII/1**(1959), pp. 125–263. Russian translation: Foreign Literature Publishing House, Moscow, 1963, 265 pages.Google Scholar - [18]
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*Arch. Rational Mech. Anal.*,**3**(1959), 120–122.Google Scholar - [20]On the uniqueness of compressible fluid motions.
*Arch. Rational Mech. Anal.*,**3**(1959), 271–288.Google Scholar - [21]On the derivation of stress-deformation relations for a Stokesian fluid.
*J. Math. Mech.*,**8**(1959), 459–470.Google Scholar - [22]Poiseuille and Couette flow of non-Newtonian fluids.
*Z. Angli. Math. Mech.*,**39**(1959), 295–299.Google Scholar - [23]
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*Acta Math.*,**102**(1959), 23–32.Google Scholar - [25]The exterior Dirichlet problem for second-order elliptic equations (with N. Meyers).
*J. Math. Mech.*,**9**(1960), 513–538.MathSciNetMATHGoogle Scholar - [26]
- [27]On the differentiability of functions of several variables.
*Arch. Rational Mech. Anal.*,**7**(1961), 359–372.Google Scholar - [28]On the definition and properties of certain variational integrals.
*Trans. Amer. Math. Soc.*,**101**(1961), 139–167.Google Scholar - [29]On the entropy change through a shock layer (with Y.C. Whang).
*J. Aerospace Sci.*,**28**(1961), 990–991.MathSciNetMATHGoogle Scholar - [30]
*Dirichlet’s Principle in the Calculus of Variations*. Proc. Symposia in Pure Math., vol. 4. American Mathematical Society, Providence, RI, 1961, pp. 17–22.Google Scholar - [31]Interior estimates for solutions of the Navier—Stokes equations. In
*Partial Differential Equations and Continuum Mechanics*, (R. Langer, ed. ). University of Wisconsin Press, 1961, pp. 376–378.Google Scholar - [32]On the interior regularity of weak solutions of the Navier—Stokes equations.
*Arch. Rational Mech. Anal.*,**9**(1962), 187–195.Google Scholar - [33]
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*Nonlinear Problems*(R.E. Langer, ed. ). University of Wisconsin Press, 1963, pp. 69–98.Google Scholar - [35]Variational problems of minimal surface type, I (with H. Jenkins).
*Arch. Rational Mech. Anal.*,**12**(1963), 185–212. See also entries [50], [59].MathSciNetADSMATHCrossRefGoogle Scholar - [36]A Harnack inequality for non-linear equations.
*Bull. Amer. Math. Soc.*,**69**(1963), 481–486.Google Scholar - [37]Comparison and averaging methods in mathematical physics. In
*Proprietà di Media e Teoremi di Confronte in Fisica Matematica*. Centro Internazionale Matematico Estivo, Rome, Edizioni Cremonese, 1965, pp. 1–87.Google Scholar - [38]A priori estimates for solutions of the minimal surface equation.
*Arch. Rational Mech. Anal.*,**14**(1963), 376–383. See also entry [57].Google Scholar - [39]Mathematical Aspects of Boundary Layer Theory. Notes taken by H.K. Wilson, Department of Mathematics, University of Minnesota, 1963 (131 pages, multiplied typescript).Google Scholar
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*Duke Math. J.*,**31**(1964), 159–178.MathSciNetMATHCrossRefGoogle Scholar - [41]Local behavior of solutions of quasi-linear equations. Acta Math.,
**111**(1964), 247–302.Google Scholar - [42]
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*Arch. Rational Mech. Anal.*,**17**(1964), 67–78. See also entry [48].Google Scholar - [44]Pathological solutions of elliptic differential equations.
*Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Math.*,**18**(1964), 385–387.Google Scholar - [45]Singularities of Solutions of Nonlinear Equations.
*Proc. Symposia in Pure Math.*, vol.**17**, pp. 68–88. American Mathematical Society, Providence, RI, 1965.Google Scholar - [46]Isolated singularities of solutions of quasi-linear equations.
*Acta Math.*,**113**(1965), 219–240.Google Scholar - [47]Theory of differentiation. Notes taken by T. Hatcher, Department of Mathematics, University of Minnesota, 1965 (135 pages, multiplied typescript).Google Scholar
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*Arch. Rational Mech. Anal.*,**20**(1965), 163–169.Google Scholar - [49]The Dirichlet problem for the minimal surface equation with infinite data (with H. Jenkins).
*Bull. Amer. Math. Soc.*,**72**(1966), 102–106. See also entry [59].MathSciNetMATHCrossRefGoogle Scholar - [50]Variational problems of minimal surface type, II: Boundary value problems for the minimal surface equation (with H. Jenkins).
*Arch. Rational Mech. Anal.*,**21**(1966), 321–342.MathSciNetADSMATHCrossRefGoogle Scholar - [51]Isolated singularities of solutions of linear elliptic equations (with H. Weinberger).
*Amer. J. Math.*,**88**(1966), 258–272.MathSciNetMATHCrossRefGoogle Scholar - [52]Local behavior of solutions of quasi-linear parabolic equations (with D.G. Aronson).
*Arch. Rational Mech. Anal.*,**25**(1967), 81–122.MathSciNetADSMATHCrossRefGoogle Scholar - [53]A maximum principle for nonlinear parabolic equations (with D.G. Aronson),
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*Proc. Nat. Acad. Science*,**58**(1967), 1829–1835. See also entry [64].Google Scholar - [55]On the asymptotic behavior of velocity profiles in the Prandtl boundary layer theory,
*Proc. Roy. Soc. London Set: A*,**299**(1967), 491–507.Google Scholar - [56]The Dirichlet problem for the minimal surface equation in higher dimensions (with H. Jenkins)
*J. Reine Angli. Math.*,**223**(1968), 170–187.MathSciNetCrossRefGoogle Scholar - [57]Addendum to: A piori estimates for solutions of the minimal surface equation.
*Arch. Rational Mech. Anal.*,**28**(1968), 149–154.Google Scholar - [58]On the mathematical basis of Prandtl’s boundary layer theory: An example.
*Arch. Rational Mech. Anal.*,**28**(1968), 217–225.Google Scholar - [59]Variational problems of minimal surface type, III: The Dirichlet problem with infinite data (with H. Jenkins).
*Arch. Rational Mech. Anal.*,**29**(1968), 304–322.MathSciNetADSMATHCrossRefGoogle Scholar - [60]The behavior of similar solutions in a compressible boundary layer (with J.B. McLeod).
*J. Fluid Mech.*,**34**(1968), 337–342.MathSciNetADSMATHCrossRefGoogle Scholar - [61]A new proof in differentiation theory.
*Notices Amen Math. Soc.*,**15**(1968), 1036. Abstract; see also W. Rudin,*Real and Complex Analysis*, 2nd edn. McGraw-Hill, New York, 1974, pp. xii, 435, and 162–167.Google Scholar - [62]A general chain rule for derivatives and the change of variable formula for the Lebesgue integral (with D.E. Varberg).
*Amer. Math. Monthly*,**76**(1969), 514–520.MathSciNetMATHCrossRefGoogle Scholar - [63]The existence of similar solutions for some laminar boundary layer problems (with J.B. McLeod).
*Arch. Rational Mech. Anal.*,**31**(1969), 288–303.MathSciNetADSGoogle Scholar - [64]The problem of Dirichlet for quasi-linear elliptic differential equations with many independent variables.
*Philos. Trans. Roy. Soc. London Ser., A*,**264**(1969), 413–496.Google Scholar - [65]On surfaces of constant mean curvature which span a given space curve.
*Math. Z.*,**88**(1969), 77–88.Google Scholar - [66]Existence theorems for some compressible boundary layer problems. In
*Qualitative Theory of Nonlinear Differential and Integral Equations. SIAM Stud. Appl. Math.*, vol.**5**1970, pp. 35–42.Google Scholar - [67]The Dirichlet problem for surfaces of constant mean curvature.
*Proc. London Math. Soc.*,**21**(1970), pp. 361–384.Google Scholar - [68]On the strong maximum principle for nonlinear second-order differential inequalities.
*J. Funct. Anal.*,**5**(1970), 184–193.Google Scholar - [69]Boundary curvatures and the solvability of Dirichlet’s Problem.
*Proc. International Congress of Mathematicians*(Nice, 1970), vol. 2, Paris, 1970, 867–875.Google Scholar - [70]Curvature inequalities for surfaces over a disk (with H.F. Weinberger).
*In Some problems of Mathematics and Mechanics—M.A. Lavrentieff Anniversary Volume*. Nauka Leningrad, 1970, pp. 242–250. English version: Amer. Math. Soc. Translation, vol.**104**, 1976, 223–231.Google Scholar - [71]Recent developments in the mathematical aspects of boundary layer theory.
*Internat. J. Engng. Sci.*,**9**(1971), 233–240.Google Scholar - [72]Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form (with J. Douglas, Jr. and T. Dupont).
*Arch. Rational Mech. Anal.*,**42**(1971), 157–168.MathSciNetADSMATHCrossRefGoogle Scholar - [73]A symmetry problem in potential theory.
*Arch. Rational Mech. Anal.*,**43**(1971), 304–318.Google Scholar - [74]Gradient estimates for solutions of nonlinear elliptic and parabolic equations. In
*Contributions to Nonlinear Functional Analysis*(E. Zarantonello, ed. ). University of Wisconsin Press, 1971, 565–601.Google Scholar - [75]Nonlinear elliptic equations of second order. Lectures at
*Symposium on Partial Differential Equations*, Berkeley, 1971. Mimeographed notes (57 pages, multiplied typescript).Google Scholar - [76]
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*Proc. London Math. Soc.*,**24**(1972), 348–366.Google Scholar - [78]A note on the preceding paper of Amann.
*Arch. Rational Mech. Anal.*,**44**(1972), 182–186.Google Scholar - [79]Rectilinear steady flow of simple fluids (with R.L. Fosdick).
*Proc. Royal Soc. London*,**332**(1973), 311–333.MathSciNetADSMATHCrossRefGoogle Scholar - [80]Global properties of continuum thermodynamic processes (with R.L. Fosdick).
*Arch. Rational Mech. Anal.*,**59**(1975), 97–109.MathSciNetADSMATHCrossRefGoogle Scholar - [81]On the axioms of classical mechanics. Department of Mathematics, University of Minnesota, 1974 (74 pages, multiplied typescript).Google Scholar
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*Convegno Internazionale sui Metodi Valutativi nella Fisica—Matematica*, Accad. Naz. Lincei, Problemi Attuali di Scienza e di Cultura, vol.**217**, 1975, pp. 207–215.Google Scholar - [83]
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*Arch. Rational Mech. Anal.*,**66**(1977), 295–310.Google Scholar - [86]Optimal shapes for brittle beams under torsion (with H.F. Weinberger). In
*Complex Analysis and its Applications—Jubilee Volume for I.Y. Vekua*, Nauka, Moscow, 1978, pp. 88–91.Google Scholar - [87]Gradient estimates and Liouville theorems for quasi-linear elliptic equations (with L.A. Peletier).
*Ann. Scuola Norm. Sup. Pisa*,**5**, Ser. IV (1978), 65–104.MathSciNetMATHGoogle Scholar - [88]The concepts of thermodynamics. In
*Continuum Mechanics and Partial Differential Equations*(G.M. de la Penha et al., eds.). North-Holland, Amsterdam, 1978, pp. 411–451.Google Scholar - [89]On the impossibility of linear Cauchy and Piola–Kirchhoff constitutive theories for stress in solids (with R.L. Fosdick).
*J. Elasticity*,**9**(1979), 83–89.MathSciNetMATHCrossRefGoogle Scholar - [90]Conceptual analysis of the classical second laws of thermodynamics.
*Arch. Rational Mech. Anal.*,**70**(1979), 254–272.Google Scholar - [91]Foundations of thermodynamics. Lecture Notes, University of Naples, 1979 (150 pages, multiplied typescript).Google Scholar
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*Recent Methods in Nonlinear Analysis and Applications*(Proc. Fourth International Meeting of SAFA), (A. Canfora et al., eds.). Liguori Editore, Naples, 1981, pp. 169–175.Google Scholar - [93]The second law of thermodynamics for systems with approximate cycles (with B.D. Coleman and D.R. Owen).
*Arch. Rational Mech. Anal.*,**77**(1981), 103–142.MathSciNetADSMATHCrossRefGoogle Scholar - [94]Uniqueness of solutions of semilinear Poisson equations (with K. McLeod).
*Proc. Nat. Acad. Sci. USA*,**78**(1981), 6592–6598. See also entry [114].MathSciNetADSMATHCrossRefGoogle Scholar - [95]Uniqueness of positive solutions of semilinear equations in ℝ
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*Quart. Appl. Math.*,**41**(1983), 357–364.Google Scholar - [97]The mechanical theory of fluid interfaces and Maxwell’s rule (with E. Aifantis).
*J. Coll. Interface Sci.*,**96**(1983), 519–547.Google Scholar - [98]The structure and laws of thermodynamics.
*Proc. International Congress of Mathematicians*(Warsaw 1983), 1717–1728.Google Scholar - [99]Applied mathematics and scientific thought. In
*Nonlinear Analysis and Optimization*. Lecture Notes in Mathematics, vol.**1107**, Springer-Verlag, New York, 1984, pp. 1927.Google Scholar - [100]One-dimensional shock layers in Korteweg fluids (with R. Hagan). In
*Phase Transformations and Material Instabilities in Solids*. Academic Press, New York, 1984, 113–128.Google Scholar - [101]Extensions of the mountain pass theorem (with P. Pucci).
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*Rend. Accad. Naz. Lincei*,*Serie 8*,**79**(1985),121–126. See also entries [119], [150].MathSciNetMATHGoogle Scholar - [104]A continuum model for chemical mixture dynamics. In
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*Suppl. Rend. Circolo Mat. Palermo*,**8**(1985), 171–185.MATHGoogle Scholar - [106]On the thermomechanics of interstitial working (with J.E. Dunn).
*Arch. Rational Mech. Anal.*,**88**(1985), 95–133.MathSciNetADSMATHCrossRefGoogle Scholar - [107]An outline of thermodynamical structure. In
*New Perspectives in Thermodynamics*,**(**J. Serrin, ed.). Springer-Verlag, New York, 1986, 3–32.Google Scholar - [108]Existence and non—existence theorems for ground states of quasi-linear partial differential equations. The anomalous case (with W.-M. Ni).
*Accad. Naz. Lincei, Atti dei Convegni*,**77**(1986), 231–257.Google Scholar - [109]Dynamic changes of phase in a van der Waals fluids (with R. Hagan). In
*New Perspectives in Thermodynamics*. ( J. Serrin, ed.). Springer-Verlag, New York, 1986, pp. 241–260.Google Scholar - [110]Uniqueness of non-negative solutions of semilinear equations in ℝ
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*Indiana Univ. Math. J.*,**35**(1986), 681–703.MathSciNetMATHCrossRefGoogle Scholar - [113]
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*Trans. Amer. Math. Soc.*,**229**(1987), 115–132.MathSciNetCrossRefGoogle Scholar - [116]Ground states for a prescribed mean curvature equation (with L.A. Peletier).
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