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

Fabio Camilli

("Sapienza" Università di Roma) joint work with

I.Birindelli ("Sapienza") I.Capuzzo Dolcetta ("Sapienza")

Linear operator in divergence form

Consider the eigenvalue problem

( Lϕ + λϕ = 0 in Ω,

ϕ =0 on ∂Ω,

whereLϕ(x ) = div (A(x )∇ϕ(x )) + b · Dϕ + cϕ for smooth functions A, b, c in Ω with ξ^tAξ ≥ α|ξ|²for some α > 0.

Let λ₁be theprincipal eigenvalue (i.e. the smallest eigenvalue in modulus), then

λ1is real and simple

There is an eigenfunction ϕ₁∈ H₀¹(Ω)which ispositive.

λ₁≤ Re(λ) for any other eigenvalue λ.

In the symmetric case (b = 0), λ₁is given by theRayleigh-Ritz variational formula

λ₁= inf

ϕ∈H₀¹(Ω),ϕ6≡0

− R

Ω∇ϕ^tA∇ϕ + cϕ²dx kϕk²_L2(Ω)

Linear operator in divergence form

Consider the eigenvalue problem

( Lϕ + λϕ = 0 in Ω,

ϕ =0 on ∂Ω,

whereLϕ(x ) = div (A(x )∇ϕ(x )) + b · Dϕ + cϕ for smooth functions A, b, c in Ω with ξ^tAξ ≥ α|ξ|²for some α > 0.

Let λ₁be theprincipal eigenvalue (i.e. the smallest eigenvalue in modulus), then

λ1is real and simple

There is an eigenfunction ϕ₁∈ H₀¹(Ω)which ispositive.

λ₁≤ Re(λ) for any other eigenvalue λ.

In the symmetric case (b = 0), λ₁is given by theRayleigh-Ritz variational formula

λ₁= inf

ϕ∈H₀¹(Ω),ϕ6≡0

− R

Ω∇ϕ^tA∇ϕ + cϕ²dx kϕk²_L2(Ω)

Linear operator in divergence form

Consider the eigenvalue problem

( Lϕ + λϕ = 0 in Ω,

ϕ =0 on ∂Ω,

whereLϕ(x ) = div (A(x )∇ϕ(x )) + b · Dϕ + cϕ for smooth functions A, b, c in Ω with ξ^tAξ ≥ α|ξ|²for some α > 0.

Let λ₁be theprincipal eigenvalue (i.e. the smallest eigenvalue in modulus), then

λ1is real and simple

There is an eigenfunction ϕ₁∈ H₀¹(Ω)which ispositive.

λ₁≤ Re(λ) for any other eigenvalue λ.

In the symmetric case (b = 0), λ₁is given by theRayleigh-Ritz variational formula

λ₁= inf

ϕ∈H₀¹(Ω),ϕ6≡0

− R

Ω∇ϕ^tA∇ϕ + cϕ²dx kϕk²_L2(Ω)

Linear operator in divergence form

Consider the eigenvalue problem

( Lϕ + λϕ = 0 in Ω,

ϕ =0 on ∂Ω,

whereLϕ(x ) = div (A(x )∇ϕ(x )) + b · Dϕ + cϕ for smooth functions A, b, c in Ω with ξ^tAξ ≥ α|ξ|²for some α > 0.

Let λ₁be theprincipal eigenvalue (i.e. the smallest eigenvalue in modulus), then

λ1is real and simple

There is an eigenfunction ϕ₁∈ H₀¹(Ω)which ispositive.

λ₁≤ Re(λ) for any other eigenvalue λ.

In the symmetric case (b = 0), λ₁is given by theRayleigh-Ritz variational formula

λ₁= inf

ϕ∈H₀¹(Ω),ϕ6≡0

− R

Ω∇ϕ^tA∇ϕ + cϕ²dx kϕk²_L2(Ω)

Approximation:

An approximation of λ₁can be computed via discretization of the weak formulation

Z

Ω

(−A∇ϕ∇ψ + b∇ϕ ψ + cϕψ) dx = λ₁ Z

Ω

ϕψdx ∀ψ ∈ H₀¹(Ω)

by means of standardP¹finite elementsresulting in the linear equation A_hΦ = λ^h₁B_hΦ

where

• Ahis the stiffness matrix;

• B_hthe mass matrix.

Convergence:

It can be proved that λ^h₁→ λ₁as h → 0 and the convergence is of order h².

Note that is Lu(x ) = div (A(x )∇u(x )), then λ^h₁is given by

λ^h₁= inf

x ∈R^Nh,x 6=0

−x^tA_hx kxk²

which is the finite dimensional analogous of theRayleigh-Ritz variational formula

λ₁= inf

ϕ∈H₀¹(Ω),ϕ6≡0

− R

Ω∇ϕ^tA∇ϕ dx kϕk²

L²(Ω)

Some references:

• Weinberger, Variational methods for eigenvalue approximation, CBMS, 15

• Babuska-Osborn, Math. Comp.,1989

• Boffi, Acta Numerica 2010

• Huang, J. Comput. Phys., 2014

Nonlinear operator

Let F [u] := F (x , u, Du, D²u) be auniformly elliptic operator, positive homogenous of degree 1(i.e. F [tu] = tF [u] for t ≥ 0). Then the principal eigenvalue for F is defined by means of the formula

λ₁:=sup{λ ∈ R : ∃ ϕ > 0 in Ω, F [ϕ] + λϕ ≤ 0 };

(from now on all the (in)equalities have to be intended in viscosity sense).

There is an eigenfunction ϕ₁which ispositive, i.e. a solution of ( F [ϕ] + λ₁ϕ =0 in Ω,

ϕ =0 on ∂Ω,

ϕ₁is the only eigenfunction that does not change sign.

λ₁issimple.

For any λ < λ₁themaximum principleholds for F [·] + λ, i.e.

If u is such that F [u] + λu ≥ 0 in Ω, u ≤ 0 on ∂Ω then u ≤ 0 in Ω.

Nonlinear operator

Let F [u] := F (x , u, Du, D²u) be auniformly elliptic operator, positive homogenous of degree 1(i.e. F [tu] = tF [u] for t ≥ 0). Then the principal eigenvalue for F is defined by means of the formula

λ₁:=sup{λ ∈ R : ∃ ϕ > 0 in Ω, F [ϕ] + λϕ ≤ 0 };

(from now on all the (in)equalities have to be intended in viscosity sense).

There is an eigenfunction ϕ₁which ispositive, i.e. a solution of ( F [ϕ] + λ₁ϕ =0 in Ω,

ϕ =0 on ∂Ω,

ϕ₁is the only eigenfunction that does not change sign.

λ₁issimple.

For any λ < λ₁themaximum principleholds for F [·] + λ, i.e.

If u is such that F [u] + λu ≥ 0 in Ω, u ≤ 0 on ∂Ω then u ≤ 0 in Ω.

Nonlinear operator

Let F [u] := F (x , u, Du, D²u) be auniformly elliptic operator, positive homogenous of degree 1(i.e. F [tu] = tF [u] for t ≥ 0). Then the principal eigenvalue for F is defined by means of the formula

λ₁:=sup{λ ∈ R : ∃ ϕ > 0 in Ω, F [ϕ] + λϕ ≤ 0 };

(from now on all the (in)equalities have to be intended in viscosity sense).

There is an eigenfunction ϕ₁which ispositive, i.e. a solution of ( F [ϕ] + λ₁ϕ =0 in Ω,

ϕ =0 on ∂Ω,

ϕ₁is the only eigenfunction that does not change sign.

λ₁issimple.

For any λ < λ₁themaximum principleholds for F [·] + λ, i.e.

If u is such that F [u] + λu ≥ 0 in Ω, u ≤ 0 on ∂Ω then u ≤ 0 in Ω.

Nonlinear operator

Let F [u] := F (x , u, Du, D²u) be auniformly elliptic operator, positive homogenous of degree 1(i.e. F [tu] = tF [u] for t ≥ 0). Then the principal eigenvalue for F is defined by means of the formula

λ₁:=sup{λ ∈ R : ∃ ϕ > 0 in Ω, F [ϕ] + λϕ ≤ 0 };

(from now on all the (in)equalities have to be intended in viscosity sense).

There is an eigenfunction ϕ₁which ispositive, i.e. a solution of ( F [ϕ] + λ₁ϕ =0 in Ω,

ϕ =0 on ∂Ω,

ϕ₁is the only eigenfunction that does not change sign.

λ₁issimple.

For any λ < λ₁themaximum principleholds for F [·] + λ, i.e.

If u is such that F [u] + λu ≥ 0 in Ω, u ≤ 0 on ∂Ω then u ≤ 0 in Ω.

Observe that by definition for anyλ ≤ λ1there exists ϕ > 0 such that F [ϕ](x ) + λϕ(x ) ≤ 0 ∀x ∈ Ω

Hence

λ ≤ inf

x ∈Ω−F [ϕ](x )

ϕ(x ) = −sup

x ∈Ω

F [ϕ](x ) ϕ(x ) and therefore

λ ≤sup

ϕ>0



− sup

x ∈Ω

F [ϕ](x ) ϕ(x )



= − inf

ϕ>0sup

x ∈Ω

F [ϕ](x ) ϕ(x ) .

For λ = λ₁the equality holds in the previous formula and therefore λ₁= − inf

ϕ>0sup

x ∈Ω

F [ϕ](x ) ϕ(x )

The previous formula is similar to an identity characterizing the effective Hamiltonian, i.e.

H(P) = inf

ϕ∈C_per¹

sup

x ∈Tⁿ

H(P + D_xϕ,x )

This formula was used byGomes-Oberman (Sicon 2004) to get the following approximation of ¯H

H_h(P) = inf

ϕ∈C(Th)sup

x ∈Th

H(P + D_xϕ,x )

where T_h is a triangulation of Tⁿwith cells of diameter smaller than h and C(T_h)is the collection of continuous piecewise linear grid

functions which interpolate given nodal values.

The computation of H_h(P) is given by afinite dimensional optimization problem.

The previous formula is similar to an identity characterizing the effective Hamiltonian, i.e.

H(P) = inf

ϕ∈C_per¹

sup

x ∈Tⁿ

H(P + D_xϕ,x )

This formula was used byGomes-Oberman (Sicon 2004) to get the following approximation of ¯H

H_h(P) = inf

ϕ∈C(Th)sup

x ∈Th

H(P + D_xϕ,x )

where T_h is a triangulation of Tⁿwith cells of diameter smaller than h and C(T_h)is the collection of continuous piecewise linear grid

functions which interpolate given nodal values.

The computation of H_h(P) is given by afinite dimensional optimization problem.

A general class of difference operators

Let hZⁿbe the orthogonal lattice in Rⁿwhere h > 0 is a discretization parameter and C_hthe space of the mesh functions defined on

Ω_h= Ω ⊂ Zⁿ_h. Consider a discrete operator F_hdefined by F_h[u](x ) := F_h(x , u(x ), [u]x)

where

h > 0 is the discretization parameter (h is meant to tend to 0), x ∈ Ω_his a grid point

u ∈ C_h

[·]x represents the stencil of the scheme, i.e. the points in Ω_h\{x}

where the value of u are computed for writing the scheme at the point x (we assume that [w ]_x is independent of w (y ) for

|x − y | > Mh for some fixed M ∈ N).

FollowingKuo-Trudinger (Siam J.Num.Analysis 1992) we introduce some basic assumptions for the difference operator F_h:

(i) The operator F_his ofpositive type, i.e. for all x ∈ Ω_h, z, τ ∈ R, u, η ∈ C_hsatisfying 0 ≤ η(y ) ≤ τ for each y ∈ Ω_h, then

F_h(x , z, [u + η]_x) ≥F_h(x , z, [u]_x) ≥F_h(x , z + τ, [u + η]_x) (ii) The operator F_hispositive homogeneous, i.e. for all x ∈ Ω_h,

z ∈ R, u ∈ Chand t ≥ 0, then

F_h(x , tz, [tu]x) =tF_h(x , z, [u]x).

(iii) The family of operator {F_h,0 < h ≤ h₀}, where h₀is a positive constant, isconsistentwith operator F on the domain Ω ⊂ Rⁿ, i.e. for each u ∈ C²(Ω)

sup

Ωh

F (x , u(x ), Du(x ), D²u(x )) − F_h(x , u(x ), [u]x)  

→ 0 as h → 0, uniformly on compact subset of Ω.

As in the continuous case, we define a principal eigenvalue for F_h by means of the formula

λ^h₁:=sup{λ ∈ R : ∃ ϕ ∈ Ch, ϕ >0 in Ω_h, F_h[ϕ] + λϕ ≤0 };

There is an eigenfunction ϕ^h₁which ispositive, i.e. a solution of ( F [ϕ] + λ^h₁ϕ =0 in Ω_h,

ϕ =0 on Ω^C_h,

For any λ < λ₁themaximum principleholds for F_h+ λ, i.e.

If u is such that F_h[u] + λu ≥ 0 in Ω_hand u ≤ 0 on Ω^C_h, then u ≤ 0 in Ω_h.

λ^h₁is given by the finite dimensional optimization problem

λ^h₁= − inf

ϕ∈Ch, ϕ>0sup

x ∈Ω_h

F_h[ϕ](x ) ϕ(x )

As in the continuous case, we define a principal eigenvalue for F_h by means of the formula

λ^h₁:=sup{λ ∈ R : ∃ ϕ ∈ Ch, ϕ >0 in Ω_h, F_h[ϕ] + λϕ ≤0 };

There is an eigenfunction ϕ^h₁which ispositive, i.e. a solution of ( F [ϕ] + λ^h₁ϕ =0 in Ω_h,

ϕ =0 on Ω^C_h,

For any λ < λ₁themaximum principleholds for F_h+ λ, i.e.

If u is such that F_h[u] + λu ≥ 0 in Ω_hand u ≤ 0 on Ω^C_h, then u ≤ 0 in Ω_h.

λ^h₁is given by the finite dimensional optimization problem λ^h₁= − inf

ϕ∈Ch, ϕ>0sup

x ∈Ω_h

F_h[ϕ](x ) ϕ(x )

As in the continuous case, we define a principal eigenvalue for F_h by means of the formula

λ^h₁:=sup{λ ∈ R : ∃ ϕ ∈ Ch, ϕ >0 in Ω_h, F_h[ϕ] + λϕ ≤0 };

There is an eigenfunction ϕ^h₁which ispositive, i.e. a solution of ( F [ϕ] + λ^h₁ϕ =0 in Ω_h,

ϕ =0 on Ω^C_h,

For any λ < λ₁themaximum principleholds for F_h+ λ, i.e.

If u is such that F_h[u] + λu ≥ 0 in Ω_hand u ≤ 0 on Ω^C_h, then u ≤ 0 in Ω_h.

λ^h₁is given by the finite dimensional optimization problem λ^h₁= − inf

ϕ∈Ch, ϕ>0sup

x ∈Ω_h

F_h[ϕ](x ) ϕ(x )

Convergence of λ

to λ

for h → 0

The convergence result cannot rely on standard stability results in viscosity solution theory (Barles-Souganidis’method) since the limit eigenvalue problem

( F [ϕ] + λ₁ϕ =0 in Ω,

ϕ =0 on ∂Ω,

does not satisfy a comparison principle (ϕ ≡ 0 and the principal eigenfunction ϕ₁are two distinct solutions of the problem).

So for the approximating operators, we consider a specific class of finite difference schemesF_h introduced by Kuo-Trudinger which satisfy some crucial pointwise estimates which are the discrete analogues of those valid for a general class of fully nonlinear, uniformly elliptic equations.

Convergence of λ

to λ

for h → 0

The convergence result cannot rely on standard stability results in viscosity solution theory (Barles-Souganidis’method) since the limit eigenvalue problem

( F [ϕ] + λ₁ϕ =0 in Ω,

ϕ =0 on ∂Ω,

does not satisfy a comparison principle (ϕ ≡ 0 and the principal eigenfunction ϕ₁are two distinct solutions of the problem).

So for the approximating operators, we consider a specific class of finite difference schemesF_h introduced by Kuo-Trudinger which satisfy some crucial pointwise estimates which are the discrete analogues of those valid for a general class of fully nonlinear, uniformly elliptic equations.

We assume that the stencil [·]x of the scheme is given by x + hY where Y = {y₁, . . . ,y_k} ⊂ Zⁿis a finite set containing all the vectors of the canonical basis of Rⁿ.We consider a finite difference operator of the form

F_h[u] = F (x , u, δ_hu, δ_h²u) where F : Rⁿ× R × R^Y × R^Y → R and

δ_h,yu(x ) = u(x +hy )−u(x −hy ) 2h|y |

δ_h,y² u(x ) = u(x +hy )+u(x −hy )−2u(x ) h²|y |²

δ_hu = {δ_h,yu : y ∈ Y } δ_h²u = {δ_h,y² u : y ∈ Y }.

Moreover F satisfies the following assumptions

∂F

∂s_y −|hy | 2

∂F

∂q_y    

≥ α₀, ∂F

∂s_y ≤ a₀,    

∂F

∂q_y    

≤ b₀ (1)

where α₀, a₀, b₀are given constants

We assume that the stencil [·]x of the scheme is given by x + hY where Y = {y₁, . . . ,y_k} ⊂ Zⁿis a finite set containing all the vectors of the canonical basis of Rⁿ.We consider a finite difference operator of the form

F_h[u] = F (x , u, δ_hu, δ_h²u) where F : Rⁿ× R × R^Y × R^Y → R and

δ_h,yu(x ) = u(x +hy )−u(x −hy ) 2h|y |

δ_h,y² u(x ) = u(x +hy )+u(x −hy )−2u(x ) h²|y |²

δ_hu = {δ_h,yu : y ∈ Y } δ_h²u = {δ_h,y² u : y ∈ Y }.

Moreover F satisfies the following assumptions

∂F

∂s_y −|hy | 2

∂F

∂q_y    

≥ α₀, ∂F

∂s_y ≤ a₀,    

∂F

∂q_y    

≤ b₀ (1) where α₀, a₀, b₀are given constants

If F isuniformly elliptic, then it is always possible to find a scheme of the previous type which is ofpositive type and consistent with F. For this class of schemes we have

Theorem

Let (λ^h₁, ϕ^h₁)be the sequence of the discrete eigenvalues and of the corresponding eigenfunctions associated to F_h.

Then

λ^h₁→ λ₁, ϕ^h₁→ ϕ₁

uniformly in Ω as h → 0, where λ₁and ϕ₁are respectively the principal eigenvalue and the corresponding eigenfunction associated to F .

If F isuniformly elliptic, then it is always possible to find a scheme of the previous type which is ofpositive type and consistent with F. For this class of schemes we have

Theorem

Let (λ^h₁, ϕ^h₁)be the sequence of the discrete eigenvalues and of the corresponding eigenfunctions associated to F_h.

Then

λ^h₁→ λ₁, ϕ^h₁→ ϕ₁

uniformly in Ω as h → 0, where λ₁and ϕ₁are respectively the principal eigenvalue and the corresponding eigenfunction associated to F .

Main ingredients of the proof are

The semi-relaxed limits in viscosity solution sense;

A maximum principle for the limit problem (rather than the comparison principle);

The following local Hölder estimate proved by Kuo-Trudinger:

If u_his a solution of F_h[u] = f , then for any x , y ∈ Ω_h

|uh(x ) − u_h(y )| ≤ C|x − y |^δ R





max

B^h_R

u_h+ R α₀





 X

x ∈Ω_h

hⁿ|f (x)|ⁿ







1 n





,

where R = min{dist(x , ∂Ω_h), dist(x , ∂Ω_h)}, B_R^h =B(0, R) ∩ Ω_h, δ, α0and C are positive constants independent of h.

Main ingredients of the proof are

The semi-relaxed limits in viscosity solution sense;

A maximum principle for the limit problem (rather than the comparison principle);

The following local Hölder estimate proved by Kuo-Trudinger:

If u_his a solution of F_h[u] = f , then for any x , y ∈ Ω_h

|uh(x ) − u_h(y )| ≤ C|x − y |^δ R





max

B^h_R

u_h+ R α₀





 X

x ∈Ω_h

hⁿ|f (x)|ⁿ







1 n





,

where R = min{dist(x , ∂Ω_h), dist(x , ∂Ω_h)}, B_R^h =B(0, R) ∩ Ω_h, δ, α0and C are positive constants independent of h.

Main ingredients of the proof are

The semi-relaxed limits in viscosity solution sense;

A maximum principle for the limit problem (rather than the comparison principle);

The following local Hölder estimate proved by Kuo-Trudinger:

If u_his a solution of F_h[u] = f , then for any x , y ∈ Ω_h

|uh(x ) − u_h(y )| ≤ C|x − y |^δ R





max

B^h_R

u_h+ R α₀





 X

x ∈Ω_h

hⁿ|f (x)|ⁿ







1 n





,

where R = min{dist(x , ∂Ω_h), dist(x , ∂Ω_h)}, B_R^h =B(0, R) ∩ Ω_h, δ, α0and C are positive constants independent of h.

The algorithm in R

Recall the formula

λ^h₁= − inf

u∈C_h,u>0sup

x ∈Ωh

F (x, u, δ_hu, δ²_hu) u(x )

Set U_i =u(x_i). If F (x , z, q, s) is linear or more generally convex in (q, s), then the functions G_i : R^Nh → R^Nh

G_i(x , U₁, . . . ,U_Nh) = F



x_i,1,U_i+1− U_i−1

2hU_i ,U_i+1+U_i−1 h²U_i − 2

h²

 . are either linear or respectively convex in U_i, U_i+1, U_i−1and therefore G : R^Nh → R defined by

G(U1, . . . ,U_Nh) = max

i=1,...,Nh

G_i(x_i,U₁, . . . ,U_Nh) is either linear or convex.

Hence the computation of λ^h₁is equivalent to theconvex minimization problem

min

U∈R^Nh+

 max

i=1,...,Nh

G_i(x_i,U₁, . . . ,U_Nh)



and this problem can be solved by means of standard algorithms in convex optimization. Note that

The vector U attaining the minimum is unique (up to a multiplicative constant)

The map is sparse, in the sense that the value of G_i at U_i depends only on the values at U_i−1 and U_i+1.

The algorithm also computes an approximation of the eigenfunction.

Similar considerations hold for eigenvalue problem in Rⁿ.

Hence the computation of λ^h₁is equivalent to theconvex minimization problem

min

U∈R^Nh+

 max

i=1,...,Nh

G_i(x_i,U₁, . . . ,U_Nh)



and this problem can be solved by means of standard algorithms in convex optimization. Note that

The vector U attaining the minimum is unique (up to a multiplicative constant)

The map is sparse, in the sense that the value of G_i at U_i depends only on the values at U_i−1 and U_i+1.

The algorithm also computes an approximation of the eigenfunction.

Similar considerations hold for eigenvalue problem in Rⁿ.

Hence the computation of λ^h₁is equivalent to theconvex minimization problem

min

U∈R^Nh+

 max

i=1,...,Nh

G_i(x_i,U₁, . . . ,U_Nh)



and this problem can be solved by means of standard algorithms in convex optimization. Note that

The vector U attaining the minimum is unique (up to a multiplicative constant)

The map is sparse, in the sense that the value of G_i at U_i depends only on the values at U_i−1 and U_i+1.

The algorithm also computes an approximation of the eigenfunction.

Similar considerations hold for eigenvalue problem in Rⁿ.

Hence the computation of λ^h₁is equivalent to theconvex minimization problem

min

U∈R^Nh+

 max

i=1,...,Nh

G_i(x_i,U₁, . . . ,U_Nh)



and this problem can be solved by means of standard algorithms in convex optimization. Note that

The vector U attaining the minimum is unique (up to a multiplicative constant)

The map is sparse, in the sense that the value of G_i at U_i depends only on the values at U_i−1 and U_i+1.

The algorithm also computes an approximation of the eigenfunction.

Similar considerations hold for eigenvalue problem in Rⁿ.

Some (very simple) examples

Example: eigenvalue of the second order derivative ( ϕ⁰⁰+ λϕ =0 in (0, 1),

ϕ(0) = ϕ(1) = 0 In this case

λ1= π², ϕ1(x ) = sin πx

Given a discretization step h and the corresponding grid points x_i =ih, i = 0, . . . , N_h+1, the max-min problem is

λ^h₁= − min

U∈R^Nh



i=1,...,Nmax_h

U_i+1+U_i−1− 2U_i h²U_i



(with U₀=U_Nh+1=0).

h Err (λ₁) Order (λ₁) Err∞(w₁) Err₂(w₁) 1.00 · 10⁻¹ 8.0908 · 10⁻² 3.3662 · 10⁻¹¹ 5.7732 · 10⁻¹¹ 5.00 · 10⁻² 2.0277 · 10⁻² 1.9964 1.4786 · 10⁻¹⁰ 3.8119 · 10⁻¹⁰ 2.50 · 10⁻² 5.0723 · 10⁻³ 1.9991 6.6613 · 10⁻¹⁶ 1.8731 · 10⁻¹⁵ 1.25 · 10⁻² 1.2683 · 10⁻³ 1.9998 1.5543 · 10⁻¹⁵ 6.2524 · 10⁻¹⁵ 6.25 · 10⁻³ 3.1708 · 10⁻⁴ 1.9999 1.2212 · 10⁻¹⁵ 7.1576 · 10⁻¹⁵

Table: Space step (first column), eigenvalue error (second column), eigenfunction error in L^∞(fourth column), eigenfunction error in L²(last column)

Example: A linear equation with a discontinuous coefficient

 a(x )ϕ⁰⁰+ λϕ =0 x ∈ (0, π)

ϕ(x ) = 0 x = 0, π (2)

where

a(x ) =

 1 for x ∈ [0,_2k^π ) 2 for x ∈ [_2k^π , π]

and k := ²⁺

√ 2 2√

2 >1. The principal eigenvalue λ₁is given by k². Note that the principal eigenfunction

ϕ₁(x ) =

( sin(kx ) for x ∈ [0,_2k^π ), b sin(^√kx

2+c) for x ∈ [_2k^π , π].

is not C²

The scheme is

λ1,h = − min

U∈R^Nh

 max

i=1,...,Nh

a(ih)U_i+1+U_i−1− 2Ui

h²U_i



(with U₀=U_Nh+1=0).

h Err (λ₁) Err∞(w₁) Err₂(w₁) 0.1571 0.1197 0.0213 0.0563 0.0785 0.0476 0.0090 0.0383 0.0393 0.0347 0.0065 0.0391 0.0196 0.0157 0.0030 0.0264 0.0098 0.0061 0.0012 0.0149

Table: Space steps (first column), Error eigenvalue (second column), Error eigenfunction in L^∞(fourth column), Error eigenfunction in L²(last column)

Figure:Exact and approximate eigenfunctions for h = 10⁻¹

Example: The p-Laplacian

Consider the p-Laplace operator (Birindelli-Demengel, CPAA,2006) D(|Dϕ|^p−2Dϕ) + λp|ϕ|^p−2ϕ =0.

Even if the operator is not uniformly elliptic, the formula λ^h_p := − inf

ϕ>0sup

x ∈Ωh

 F_h,p[ϕ](x ) ϕ(x )^p−1



where F_h,p is a finite-difference approximations of F_p gives an approximation of the principal eigenvalue λ_p= ^2π

√p

p−1 (b−a)p sin(^π_p)

1/p

. h Err (λ₄) Order (λ₄)

1.00 · 10⁻¹ 2.6770

5.00 · 10⁻² 0.6210 2.1079 2.50 · 10⁻² 0.1457 2.0912 1.25 · 10⁻² 0.0347 2.0724 6.25 · 10⁻³ 0.0083 2.0581

Table: Space step(1^st column), eigenvalue error (2^nd column), convergence order (3^rd column) for the 4-Laplace operator

If Ω is a ball, ϕpconverges for p → ∞ to d (x , ∂Ω). We draw approximations of ϕ_p for various values of p and we observe the convergence to d (x , {0, 1}) for p increasing

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure:Approximate eigenfunction ϕ^h_pfor p = 2, 4, 6, 8, 10 and h = 10⁻³

Example: A bi-dimensional example

Consider the eigenvalue problem for the Ornstein-Uhlenbeck operator

∆ϕ −x · Dϕ + λϕ = 0, x ∈ (0, 1)²

with homogeneous boundary conditions. The eigenvalue and the corresponding eigenfunction are given by

λ1=4, ϕ1(x₁,x₂) = (1 − x₁²)(1 − x₂²) The Laplacian is discretized by a five-point formula.

h Err (λ₁) Order (λ₁) 4.00 · 10⁻¹ 0.1524

2.00 · 10⁻¹ 0.0392 1.9592 1.00 · 10⁻¹ 0.0103 1.9250 5.00 · 10⁻² 0.0027 1.9580

Ongoing work (with Simone Cacace)

It is known that

Among allrectanglesof same area, the one that minimizes the first Laplace-Dirichlet eigenvalue is thesquare.

The Faber-Krahn’s inequality affirms that in any dimension, among all domains of same volume, the euclidean ball has the smallestfirst Laplace-Dirichlet eigenvalue.

The idea seems to be that the λ₁(Ω)is decreasing with respect to the symmetry of the domain Ω. Is Faber-Krahn inequality true for the Pucci operator?

The idea is to investigate numerically this conjecture using the previous approximation scheme and, for the computation of the eigenvalue, the nonlinear least square method developed in

S.CACACE, F. CAMILLI, Ergodic problems for Hamilton-Jacobi equations: yet another but efficient numerical method , 2016

Ongoing work (with Simone Cacace)

It is known that

Among allrectanglesof same area, the one that minimizes the first Laplace-Dirichlet eigenvalue is thesquare.

The Faber-Krahn’s inequality affirms that in any dimension, among all domains of same volume, the euclidean ball has the smallestfirst Laplace-Dirichlet eigenvalue.

The idea seems to be that the λ₁(Ω)is decreasing with respect to the symmetry of the domain Ω. Is Faber-Krahn inequality true for the Pucci operator?

The idea is to investigate numerically this conjecture using the previous approximation scheme and, for the computation of the eigenvalue, the nonlinear least square method developed in

S.CACACE, F. CAMILLI, Ergodic problems for Hamilton-Jacobi equations: yet another but efficient numerical method , 2016

Ongoing work (with Simone Cacace)

It is known that

Among allrectanglesof same area, the one that minimizes the first Laplace-Dirichlet eigenvalue is thesquare.

The Faber-Krahn’s inequality affirms that in any dimension, among all domains of same volume, the euclidean ball has the smallestfirst Laplace-Dirichlet eigenvalue.

The idea seems to be that the λ₁(Ω)is decreasing with respect to the symmetry of the domain Ω. Is Faber-Krahn inequality true for the Pucci operator?

The idea is to investigate numerically this conjecture using the previous approximation scheme and, for the computation of the eigenvalue, the nonlinear least square method developed in

S.CACACE, F. CAMILLI, Ergodic problems for Hamilton-Jacobi equations: yet another but efficient numerical method , 2016

Ongoing work (with Simone Cacace)

It is known that

Among allrectanglesof same area, the one that minimizes the first Laplace-Dirichlet eigenvalue is thesquare.

The Faber-Krahn’s inequality affirms that in any dimension, among all domains of same volume, the euclidean ball has the smallestfirst Laplace-Dirichlet eigenvalue.

The idea seems to be that the λ₁(Ω)is decreasing with respect to the symmetry of the domain Ω. Is Faber-Krahn inequality true for the Pucci operator?

The idea is to investigate numerically this conjecture using the previous approximation scheme and, for the computation of the eigenvalue, the nonlinear least square method developed in

S.CACACE, F. CAMILLI, Ergodic problems for Hamilton-Jacobi equations: yet another but efficient numerical method , 2016

In the picture, approximation of the Laplace-Dirichlet eigenvalue λ₁of the rectangle [0, a] × [0, 1/a] as a function of a. The minimum is attained for a = 1, i.e. by the square

The Laplace-Dirichlet eigenfunction in a flower domain (courtesy S.Cacace)

Thank You!