On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators

On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators

Isabeau Birindelli Dip. di Matematica, “Sapienza” Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy isabeau@mat.uniroma1.it.    Fabio Camilli Dip. di Scienze di Base e Applicate per l’Ingegneria, “Sapienza” Università di Roma, via Scarpa 16, 00161 Roma, Italy, (e-mail:camilli@dmmm.uniroma1.it)    Italo Capuzzo Dolcetta Dip. di Matematica, “Sapienza” Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy capuzzo@mat.uniroma1.it.
version: July 3, 2019
Abstract

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators.
The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method.

MSC 2000:

35J60, 35P30, 65M06.

Keywords:

Principal eigenvalue, nonlinear elliptic operators, finite difference schemes, convergence.

1 Introduction

Consider the elliptic self-adjoint operator

(1.1)

where are smooth functions in , a smooth bounded open subset of , satisfying for some . It is well-known that the minimum value in the Rayleigh-Ritz variational formula

is attained at some function satisfying

The number is usually referred to as the principal eigenvalue of in and is the corresponding principal eigenfunction. For operators of the form (1.1) and also more general linear operator in divergence form there is a vast literature on computational methods for the principal eigenvalue, see for example [2], [10], [14], [22].

General non-divergence type elliptic operators, namely

(1.2)

are not self-adjoint and the spectral theory is then much more involved: in particular, the Rayleigh-Ritz variational formula is not available anymore. In the seminal paper [12] by M.D. Donsker and S.R.S. Varadhan, a min-max formula for the principal eigenvalue of a class of elliptic operators including (1.2) was proved, namely

(1.3)

In that papers other representation formulas for were also proposed in terms of large deviations and of the average long run time behavior of the positive semigroup generated by . A further crucial step in that direction is the paper [6] by H. Berestycki, L. Nirenberg and S.R.S. Varadhan, where the validity of formula (1.3) is proved under mild smoothness assumptions ( a bounded open set and , , ). Moreover it is proved that (1.3) is equivalent to

Following this path of ideas, notions of principal eigenvalue for fully nonlinear uniformly elliptic operators of the form

have been introduced and analyzed in [1], [5], [8], [11], [15], [20]. A by now established definition of principal eigenvalue is given by

(1.4)

where the inequality in (1.4) is intended in viscosity sense. It is possible to prove under appropriate assumptions, see (2.1)-(2.2), that there exists a viscosity solution of

(1.5)

Moreover the characterization (1.3) still holds in this nonlinear setting.

As it is well-known, the principal eigenvalue plays a key role in several respects, both in the existence theory and in the qualitative analysis of elliptic partial differential equations as well in applications to large deviations [1], [12], bifurcation issues [20], ergodic and long run average cost problems in stochastic control [4]. For linear non self-adjoint operators and, a fortiori, for nonlinear ones the principal eigenvalue can be explicitly computed only in very special cases, see e.g. [9, 21], hence the importance to devise numerical algorithms for the problem. But, apart some specific case (see [7] for the -Laplace operator), approximation schemes and computational methods are not available in the literature, at least at our present knowledge.

The aim of this paper is to define a numerical scheme for the principal eigenvalue of nonlinear uniformly elliptic operators via a finite difference approximation of formula (1.3). More precisely, denoting by the orthogonal lattice in where is a discretization parameter, we consider a discrete operator acting on functions defined on a discrete subset of and the corresponding approximated version of (1.3), namely

(1.6)

As for the approximating operators , we consider a specific class of finite difference schemes introduced in [17], [18] since they satisfy some useful properties for the convergence analysis.

We prove that if is uniformly elliptic and satisfies in addition some quite natural further conditions, then it is possible to define a finite difference scheme such that the discrete principal eigenvalues and the associated discrete eigenfunctions converge uniformly in , as the mesh step is sent to , respectively to the principal eigenvalue and to the corresponding eigenfunction for the original problem (1.5). It is worth pointing out that the proof of our main convergence result, Theorem 3.2, cannot rely on standard stability results for fully nonlinear partial differential equations, see [3], since the limit problem does not satisfy a comparison principle (see Remark 3.1 for details).

We mention that our approach is partially inspired by the paper [13] where a similar approximation scheme is proposed for the computation of effective Hamiltonians occurring in the homogenization of Hamilton-Jacobi equations which can be characterized by a formula somewhat similar to (1.3).

In Section 2 we introduce the main assumptions and we investigate some issues related to the Maximum Principle for discrete operators. In Section 3 we study the approximation method for a class of finite difference schemes and we prove the convergence of the scheme. In Section 4 we show that under some additional structural assumptions on the inf-sup problem (1.6) can be transformed into a convex optimization problem on the nodes of the grid and we discuss its implementation. A few tests which show the efficiency of our method on some simple examples are reported in Section 4 as well.

2 The Maximum Principle for discrete operators

We start by fixing some notations and the assumptions on the operator . Set , where denotes the linear space of real, symmetric matrices. The function is assumed to be continuous on and locally uniformly Lipschitz continuous with respect to for each fixed . We will also suppose that the partial derivatives , , satisfy the following structure conditions:

(2.1)

for some constants , , , . A further condition is the positive homogeneity of degree , that is

(2.2)

The principal eigenvalue of problem (1.5) is defined by

where the differential inequality is meant in the viscosity sense. Under assumptions (2.1)-(2.2), there exists a viscosity solution of (1.5) and the characterization (1.3) of holds (see [8], [11]).

Remark 2.1

It is possible to define

When is not odd in its dependence on the Hessian, then in general . Of course it is possible to see as of some other operator. Hence we will only consider in this paper . For example, for the extremal Pucci operators and , since , the following holds

Remark 2.2

The assumption , i.e. the monotonicity of the differential operator in the zero-order term, could be removed. Indeed , with large, satisfies this assumption, moreover and have the same principal eigenfunction and the eigenvalues differ by .

We now describe the discrete setting that we shall consider. Given , let denote the orthogonal lattice in . Let be a discrete operator acting on functions defined in . We shall consider an approximation of (1.5) (which can be seen also as an eigenvalue problem for the discrete operator ). We look for a number and a positive function such that

(2.3)

where

  • is the discretization parameter ( is meant to tend to ),

  • is the point where (1.5) is approximated,

  • is a real valued mesh function in meant to approximate the viscosity solution of (1.5),

  • represents the stencil of the scheme, i.e. the points in where the value of is computed for writing the scheme at the point (we assume that is independent of for for some fixed ).

We denote by the space of the mesh functions defined on and we introduce some basic assumptions for the scheme (see [17], [18]).

  • The operator is of positive type, i.e. for all , , satisfying for each , then

  • The operator is positively homogeneous, i.e. for all , , and , then

  • The family of operators , where is a positive constant, is consistent with the operator on the domain , i.e. for each

    uniformly on compact subset of .

We study below some properties related to the maximum principle and a comparison result for the operator . Let us start by the following definitions:

Definition 2.1

A function is a subsolution (respectively is a supersolution) of

(2.4)

if

Definition 2.2

The Maximum Principle holds for the operator in if

(2.5)

implies in .

Proposition 2.1

Assume that is of positive type and positive homogeneous and satisfies either

(2.6)

or

(2.7)

for some positive constants . Then the Maximum Principle holds for the operator in .

Proof  Assume by contradiction that satisfies (2.5) and . Let be such that . Since on , it is not restrictive to assume that there exists such that . Hence

a contradiction. A similar proof can be done with the assumption (2.7).

Remark 2.3

The assumptions (2.6) and (2.7) correspond to the uniform ellipticity and, respectively, to the strict monotonicity of the operator with respect to the zero-order term.

The following proposition shows that, as it is known in the continuous case (see for example [6, 8]), the validity of the Maximum Principle for subsolutions of the operator is equivalent to the positivity of the principal eigenvalue for .

Proposition 2.2

Assume that the scheme is of positive type and that it is positively homogeneous. Suppose that for , there exists a nonnegative grid function with in such that . If, for , the function satisfies

then in , i.e. satisfies the Maximum Principle.

Proof  Suppose by contradiction that . Let as in the statement and set (note that the maximum is taken only with respect to the internal points). Then is continuous, decreasing, and for . Hence there exists such that . Moreover, since on , we also have . Let be such that

(2.8)

and set . Then and for some . Hence and . Since is of positive type, it follows that

Then

and therefore a contradiction to (2.8).

The following result gives a comparison principle for (2.4).

Proposition 2.3

Assume that is of positive type and it satisfies either (2.6) or (2.7). Let and be a subsolution and respectively a supersolution of (2.4) such that on . Then in .

Proof  Suppose by contradiction that and let be such that . Hence in and it is not restrictive to assume that . It follows that

and therefore a contradiction. A similar proof can be carried on under assumption (2.7).

3 Approximation of the principal eigenvalue

In this section we consider a specific class of finite difference schemes introduced in [18]. These schemes satisfy certain pointwise estimates which are the discrete analogues of those valid for a general class of fully nonlinear, uniformly elliptic equations.
We assume that for all , the stencil of the scheme is given by where is a finite set containing all the vectors of the canonical basis of . Then we consider a discrete operator in (2.3) given by a finite difference scheme written in the form

(3.1)

where and for ,

Set and denote by the generic points in . The operator given by (3.1) is of positive type if

(3.2)
(3.3)

and positively homogeneous if

Moreover if in (1.5) satisfies the assumptions (2.1), then it is always possible to find a scheme of type (3.1) which is consistent with and which, besides (3.2)-(3.3), satisfies for all , the bounds

(3.4)

where , , are constants depending on , , , in (2.1) (see [17], [18]). Note that in particular (3.4) implies (2.6).
We recall some important properties of the previous scheme (for the proof we refer to [18])

Proposition 3.1

Assume (3.2)-(3.4) and let , be two given mesh functions. Then for every sufficiently small there exists a unique solution to the Dirichlet problem

(3.5)
Proposition 3.2

Assume (3.2)-(3.4) and let be a subsolution of (3.5). Then

(3.6)

where the constant is independent of . Moreover if is a solution of (3.5), then for any

(3.7)

where , , and are positive constants independent of .

We give an example of a scheme of the form (3.1). Consider the Hamilton-Jacobi-Bellman operator

where

(3.8)

It is always possible to rewrite the operator in (3.8) in the following form (see [18])

where and is a finite set containing all the vectors of the canonical basis in . Moreover the coefficients , and satisfy the same properties of , and . Then we consider

(3.9)

where

(3.10)

For with the previous scheme reads as

3.1 The linear case

In this part we assume that the operator in (1.5) is linear, i.e. with

and we consider a scheme defined as in (3.9)–(3.10), obviously without the dependence on , .

Proposition 3.3

Under the assumption (3.4) the eigenvalue problem (2.3) has a simple eigenvalue which corresponds to a positive eigenfunction. The other eigenvalues correspond to sign changing eigenfunctions.

Proof  Choose large enough so that and set

Let be the positive cone of the nonnegative grid functions in . For a given grid function , by Proposition 3.1 and Proposition 2.3 there exists a unique solution to

Since is a finite dimensional space it follows that defined by is a compact linear operator. Moreover, if , then by Proposition 2.1 and if , .
Therefore, by the Krein-Rutman theorem [19], the spectral radius of is a simple real eigenvalue with a positive eigenfunction such that . Hence for , satisfies

The following characterization of is a simple consequence of Proposition 2.2.

Proposition 3.4

We have

(3.11)

or, equivalently,

(3.12)

Proof  Denote by the right hand side of (3.11). Clearly . If then there exist and such that . A contradiction follows immediately by Proposition 2.2 since the eigenfunction corresponding to is positive. Hence we have (3.11).
Let such that for . Hence

Consequently

We give next an upper bound for (compare with the corresponding estimate for in [6], Lemma 1.1).

Lemma 3.1

Let and assume that lies in with . Then

Proof  Given the linear operator

let , , be positive constants such that and in . Let and assume for simplicity that for some . Set and consider the grid function

Then for we have

Denote by , and the coefficients of the linear operator computed at the point . Since it follows that

(3.13)

If

then the second term in (3.13) dominates the first one and therefore

(3.14)

In the remaining part of ,

(3.15)

By (3.14) and (3.15), we get

To conclude the proof, we show that if for some positive function and , , then . For this purpose, assume that ; then in and on , while . Hence by Proposition 2.2, it follows in , a contradiction, and therefore .

3.2 The nonlinear case

We consider now a general discrete operator given by (3.1) and we study the corresponding eigenvalue problem (2.3). In analogy with formula (3.11), we define

(3.16)

We prove for each the existence of a pair satisfying (2.3) with in .

Proposition 3.5

Assume that satisfies (3.4), and . Then there exists a nonnegative solution to

(3.17)

Proof  We can assume , since for , satisfies (2.7) and therefore by Propositions 3.1 and (2.3) there exists a unique solution to problem (3.17).
Let us define by induction a sequence by setting and, for we consider the equation:

(3.18)

For any there exists a non negative solution to (3.18). For , existence follows by Proposition 3.1. Moreover since is a subsolution to (3.18), by Proposition 2.3 we get . The existence of a non negative solution at the -step is proved in a similar way; moreover the solution is non negative since .
We claim now that, for any , . For the claim is trivially true since . Assume then by induction that . Since it follows that is a subsolution of (3.18). By Proposition 2.3, we get that