Abstract
We study the real, Euclidean, classical field equation
where φ: ℝd→ℝ is suitably small at infinity. We study existence and regularity assuming that λ≧0,F∈C ∞(ℝ), andaF(a)≧0∀a∈∝. These hypotheses allow strongly nonlinearF and nonunique solutions forf≠0. WhenF′≧0, we prove uniqueness, various contractivity properties, analytic dependence on the coupling constant λ, and differentiability in the external sourcef. For applications in the loop expansion in quantum field theory, it is useful to know that φ is in the Schwartz classL wheneverf is, and we provide a proof of this fact. The technical innovations of the problem lie in treating the noncompactness of Rd, the strong nonlinearity ofF, and the polynomial weights in the seminorms definingL.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boulware, D., Brown, L.: Tree graphs and classical fields. Phys. Rev.172, 1628–1631 (1968)
Williams, D.N.: The Euclidean loop expansion for massive λΦ 44 : Through one loop. Commun. math. Phys.54, 193–218 (1977)
Jackiw, R.: Quantum meaning of classical field theory. Rev. Mod. Phys.49, 681–706 (1977)
Hopf, E.: Über den funktionalen, insbesondere den analytischen Charakter der Lösungen elliptischer Differentialgleichungen zweiter Ordnung. Math. Z.34, 194–233 (1931)
Moser, J.: A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math.13, 457–468 (1960)
Brézis, H., Strauss, W.A.: Semi-linear second-order elliptic equations inL 1. J. Math. Soc. Japan25, 566–590 (1973)
Schwartz, J.T.: Nonlinear functional analysis. New York: Gordon and Breach 1969
Strauss, W.: On weak solutions of semilinear hyperbolic equations. An. Acad. Brasil, Ci.42, 645–651 (1970)
Treves, F.: Basic linear partial differential equations. New York: Academic Press 1975
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977
Strichartz, R.: Multipliers on fractional Sobolev spaces. J. Math. Mech.16, 1031–1060 (1967)
Mizohata, S.: Lectures on the Cauchy problem. Bombay: Tata Institute Lecture Notes 1965
Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1969
Lions, J.L., Magenes, E.: Non homogeneous boundary value problems and applications, Vol. I (translated by P. Kenneth). New York: Springer 1972
Author information
Authors and Affiliations
Additional information
Communicated by J. Glimm
Work supported in part by the NSF under Grant No. MCS 7701748
Rights and permissions
About this article
Cite this article
Rauch, J., Williams, D.N. Euclidean nonlinear classical field equations with unique vacuum. Commun.Math. Phys. 63, 13–29 (1978). https://doi.org/10.1007/BF02156127
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02156127