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GNGA FOR GENERAL REGIONS:

SEMILINEAR ELLIPTIC PDE

AND CROSSING EIGENVALUES

JAY L. HINEMAN AND JOHN M. NEUBERGER

Abstract. We consider the semilinear elliptic PDE u+ f(, u) = 0 withthe zero-Dirichlet boundary condition on a family of regions, namely stadions.Linear problems on such regions have been widely studied in the past. Weseek to observe the corresponding phenomena in our nonlinear setting. Usingthe Gradient Newton Galerkin Algorithm (GNGA) of Neuberger and Swift, wedocument bifurcation, nodal structure, and symmetry of solutions. This paperprovides the first published instance where the GNGA is applied to generalregions. Our investigation involves both the dimension of the stadions andthe value as parameters. We find that the so-called crossings and avoidedcrossings of eigenvalues as the dimension of the stadions vary influences thesymmetry and variational structure of nonlinear solutions in a natural way.

1. Introduction

We are interested in the connections between the linear problem

u+u = 0 in

u = 0 on (1)

and superlinear elliptic zero-Dirichlet boundary value problems of the form

u+f(, u) = 0 in

u = 0 on,(2)

where is the Laplacian operator, RN is in general a piece-wise smoothbounded region, and f satisfies certain hypotheses detailed in Section 2. In par-ticular, we take f : R R R to be defined by f(, u) = u+ u3, where is areal parameter, and = r [0, 1] [0, 1] to be a stadionas per . Precisely, weinvestigate (1) and (2) on a discrete collection of such regions r (also referredto in the literature as stadia) defined by

(3)

r = Br((r, r))

((r, 1

r)

(0, 2r))

Br((1

r, r))

= {r :r {0.10, 0.11, . . . , 0.30}} .(see Figure 1). One may consider such regions as examples, whereby our resultsdemonstrate the applicability of the Gradient Newton Galerkin Algorithm (GNGA)to so-called general regions.

The linear problem on the family of regions r has been widely investigated,beginning with the seminal papers  and . Subsequently, interesting phenom-ena relating to the region and/or the eigenvalues of the linear problem (1) on theseregions has been documented. The most noteworthy physical phenomena is the

1

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2 JAY L. HINEMAN AND JOHN M. NEUBERGER

r 1r 1x

r

2r

y

r

Figure 1. A region in the family of stadions. The parameterr (0, 12

] determines the radius of the endcap semicircles, andthus the dimensions of the connecting rectangle. For our numericalexperiments,r takes on finitely many equally spaced values (3).

relationship between stochastic behavior of the quantum mechanical system andthe chaotic and quasi-periodic behavior of the classical system (for a concise sum-mary of these issues see ). In this article we are interested in the persistence oflinear properties in the nonlinear case. In particular, we observe and report theinfluence of the crossings and avoided crossings of eigenvalues on the bifurcationdiagram for PDE (2). Correspondingly, for our nonlinear problem we demonstratethat symmetry swapping and non-swapping can occur at multiple eigenvalues and,

respectively, at near-multiple eigenvalues.In , the GNGA was developed to investigate existence, multiplicity, nodal

structure, bifurcation, and symmetry of problems of the form (2). In that work, theregion was taken to be the unit square; here we perform analogous experimentson stadions. Since our implementation of the GNGA requires an orthonormal basisof eigenfunctions of the Laplacian as input, experiments on the square are easilyperformed using Fourier series with a basis of sine functions. In this article, weface the considerable challenge of first obtaining eigenfunctions (solutions to thelinear problem (1)) numerically. In , this was done using essentially the inversepower method with deflation. Using ARPACK, we recreate and perhaps improveupon the results in  and are successful in obtaining a sufficiently large basis ofsuch functions. This algorithmic variant of the Arnoldi process called the ImplicitlyRestarted Arnoldi Method (see ) works well on large sparse matrices such asthose associated with the discretization of the Laplacian, requiring only a user-provided subroutine giving the action of the linear map.

In Section 2 we briefly state the hypothesis of the nonlinear problem, providesome background, and describe the GNGA. In Section 3 we describe the ARPACKimplementation used to generate the necessary basis of eigenfunctions needed toexecute GNGA. We provide some results concerning the symmetry of eigenfunc-tions and projections. Also, we summarize our linear numerical results for (1),which essentially duplicate those found in . In Section 4 we provide results aboutthe symmetry of solutions to the nonlinear PDE. Section 5 details our nonlinear

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GNGA FOR GENERAL REGIONS 3

experiments for (2), in which the phenomena revealed in the linear experimentsare reflected. In particular, we provide portions of 8 bifurcation diagrams, corre-

sponding to before and after a crossing and an avoided crossing. The associatedcontour plots demonstrate that the symmetry swapping and the nodal shape chang-ing well observed in the linear case in fact persists in the nonlinear case. We includesome numerical evidence for the analysis of symmetry done in Section 4. Section 6provides some short concluding remarks.

2. Variational Formulation and GNGA

2.1. New Challenges. Our variational algorithm requires that we have an or-thonormal collection of eigenfunctions which spans some subspace ofL2(). Recallthat the eigenvalues of with zero-Dirichlet boundary condition on any piecewisesmooth boundary satisfy

0< 1< 2

3

.

We designate the corresponding eigenfunctions by {i}iN, taken to be normalizedinL2 =L2() and of course orthogonal in both the Sobolev space H = H1,20 ()and inL2, with inner products

u, vH=

u v dx and u, v2=

u vdx,

respectively. For relevant theorems, definitions, and an explanation of this notation,see . The references  and  are also good resources for information onSobolev spaces. For our stadions = r, we can no longer use the well-knowndoubly indexed basis of sine functions. In the next section we describe how weobtain reasonable numerical approximations to basis elements on our decidedlynon-square region where there is no known closed-form solution.

2.2. Hypotheses on f. We consider specific assumptions which have lead to exis-tence theorems of sign-changing solutions (see , , ). As in , in this paperwe focus on the case where fis superlinear and subcritical, and in particular, de-fined by f(u) = u+ u3. We wish to emphasize that although infinitely manysolutions have been proven to exist for various special cases, e.g., when N = 1,f is odd, or is a ball in RN, in the general case only 3 nontrivial solutions arecurrently proven to exist (see for example ).

2.3. The Gradient Newton Galerkin AlgorithmGNGA. We provide abrief outline of the underlying variational machinery. We define the action (en-ergy) functional J :H R by

(4) J(u) = 1

2 |u

|2

F(u) ,

where F(u) =u

0 f(s) for all u H defines the primitive off. The appropriate

hypotheses imply that J is well-defined on all of H (by the Sobolev EmbeddingTheorem, see ) and twice differentiable (see ). Moreover, u is a solution of (2)if and only ifJ(u)(v) = 0 for all v H. We refer the reader to  and  for theregularity theory proving this assertion. Since we are searching for critical pointsofJwe need the following identities for J with u, v,w H:(5) J(u)(v) =

{u v f(u) v}

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4 JAY L. HINEMAN AND JOHN M. NEUBERGER

and

(6) J(u)(v, w) = {v w f(u) v w} .For more precise details about GNGA see  and .

To summarize the GNGA, one performs the following steps:

(1) Define region , nonlinearityf, and step size .(2) Obtain orthonormal basis {k}Mk=1for a sufficiently large subspace G H.(3) Choose initial coefficientsa = a0 = {ak}Mk=1, setu = u0 =

akk, and set

n= 0.(4) Loop overn until

g g ||J(u)|| is sufficiently small.

(a) Calculateg = gn+1 = (J(u)(k))Mk=1 RM (gradient vector).(b) CalculateA = An+1 = (J(u)(j , k))Mj,k=1 (Hessian matrix).

(c) Compute = n+1 =A1g by implementing least squares.(d) Set a = an+1 =an

and update u = un+1 = akk.(e) Calculate sig(A(a)) and J(a) if desired.

The signature sig(A(a)) of a solution is taken to be the number of negativeeigenvalues of the Hessian of that solution. The signature provides us with theMorse index of a solution whenever the solution is nondegenerate (has an invertibleHessian), provided thatM is sufficiently large. The parameter (0, 1] is the stepsize for damped Newtons method; generally undamped Newtons method (with= 1) suffices.

3. The Linear Problem, ARPACK, and Basis Generation

3.1. The Linear Problem. The linear problem (1) is solved using ARPACK, sincethe standard discretization of the negative Laplacian map results in a large, sparse

matrix (L). To make our system visually intuitive and to provide ARPACK withknowledge of the region, we first generate a region filewith (n + 1)2 values. A valueof1represents a point of our grid that is interior to the stadion boundary, whereasa value of 0 represents a point which is exterior. To find the eigenvalues of thematrixL, ARPACK requires a user-provided subr