Arthur G. Wasserman

Global Bifurcation of Steady-State Solutions

In this paper, we study the bifurcation of steady-state solutions of a reaction-diffusion equation in one space variable. The steady-state solutions satisfy the equation a” +f(u) = 0 on the interval -L 0, or (0, for some (nonzero) function c = c(a). For the Neumann problem, we use entirely different techniques to prove that S is never critical, for any cubic polynomial J This implies at once that the Neumann problem can have at most one nonconstant solution (having a given number of maxima or minima). This solution is necessarily strongly nondegenerate, in the sense that zero is not contained in the spectrum of the linearized operator (see [2]). For the Dirichlet problem, the situation is far more complicated, and the bifurcation diagrams undergo qualitative changes, depending on the positions of the roots off: For example, Fig. 1 shows the bifurcation diagram (for the positive solutions), for two different cubic functions of the form f(u) = (a - u)(u - b)(u - c). It is interesting to note, however, that in every case we study, there are at most three solutions to the Dirichlet problem, for each

Smooth static solutions of the Einstein/Yang-Mills equations

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Existence of infinitely many smooth, static, global solutions of the Einstein/Yang-Mills equations

We prove the existence of infinitely-many globally defined singularity-free solutions, to the EYM equations withSU(2) gauge group. The solutions are indexed by a coupling constant, have distinct winding numbers, and their corresponding Einstein metrics decay at infinity to the flat Minkowski metric. Each solution has a finite (ADM) mass; these masses are derived from the solutions, and arenot arbitrary constants.

Existence of black hole solutions for the Einstein-Yang/Mills equations

We consider the Einstein/Yang-Mills equations in 3 + 1 space time dimensions with SU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime. 1 The only static, i.e., time independent, solution to the vacuum Einstein equations for the gravitational field R¡j \Rgij = 0 is the celebrated Schwarzschild metric that is singular at r = 0 [1]. Despite this defect, this solution has applicability for large r to physical problems, e.g., the perihelion shift of Mercury. Similarly, the Yang/Mills equations d*F = 0, which unify electromagnetic and nuclear forces, have no static regular solutions on R4 [3]. Furthermore, if one couples Einsteins equations to Maxwells equations, to unify gravity and electromagnetism (1) Ru \R8ij = oTij, d*F = 0 ( Tij is the stress-energy tensor relative to the electromagnetic field Fy ), the only static solution is the Reissner-Nordström metric, which is again singular at the origin [1]. Finally, the Einstein-Yang/Mills (EYM) equations, which unify gravitational and nuclear forces, were shown in [4] to have no static regular solutions in (2+1) space time dimensions for any gauge group G. We announce here that the contrary holds in (3+1) space-time dimensions. Indeed, with SU(2) gauge group (i.e., the weak nuclear force) we prove that the EYM equations (cf. (1), where now F¡j is the su(2)-valued Yang/Mills field), admit Received by the editors November 5, 1991 and, in revised form, January 29, 1992. 1991 Mathematics Subject Classification. Primary 83C05, 83C15, 83C75, 83F05, 35Q75. The first authors research was supported in part by NSF Contract No. DMS-89-05205 and, with the second author, in part by ONR Contract No. DOD-C-N-00014-88-K-0082; the third author was supported in part by DOE Grant No. DE-FG02-88ER25065; the fourth author was supported in part by the U.K. Science and Engineering Council. ©1992 American Mathematical Society 0273-0979/92

Bifurcation and symmetry-breaking

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Symmetry-Breaking for Solutions of Semilinear Elliptic Equations with General Boundary Conditions*

.25 per page 239 240 J. SMOLLER, A. WASSERMAN, S. T. YAU, AND B. McLEOD nonsingular static solutions, whose metric is asymptotically flat, i.e., Minkowskian. (Strong numerical evidence for this conclusion was obtained by Bartnik and McKinnon [2] who also derived the relevant equations.) Thus for nonabelian gauge fields, the Yang-Mills repulsive force can balance gravitational attraction and prevent the formation of singularities in spacetime. Viewed differently from a mathematical perspective, it is the nonlinearity of the corresponding Yang/Mills equations that allows the existence of smooth solutions. The EYM equations are obtained by minimizing the actionThis paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge groupSU(2), for any event horizon. It is also demonstrated that the ADM mass of each solutions is finite, and that the corresponding Einstein metric tends to the associated Schwarzschild metric at a rate 1/r2, asr tends to infinity.

Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations

Let E and F be Banach spaces, E⊂F, and Λ an interval in R. In this paper we use bifurcation theory to study the set of solutions of an equation M(u,λ)=0, where M is a smooth operator, M:E×Λ→F

Existence of positive solutions for semilinear elliptic equations in general domains

We study the bifurcation of radially symmetric solutions of Δ+f(u)=0 onn-balls, into asymmetric ones. We show that ifu satisfies homogeneous Neumann boundary conditions, the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension. For general boundary conditions, we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions, and on the structure of the kernels of the linearized operators.

Regular solutions of the Einstein–Yang–Mills equations

We study positive solutions of the Dirichlet problem: Δu(x)+f(u(x))=0,x∈Dn,u(x)=0,x∈∂Dn, whereDn is ann-ball. We find necessary and sufficient conditions for solutions to be nondegenerate. We also give some new existence and uniqueness theorems.

Generic bifurcation of steady-state solutions

The main purpose of this paper is to prove some new existence theorems for positive solutions to the Dirichlet problem