Corrigendum for the comparison theorems in : “A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations”

Ann. I. H. Poincaré – AN 24 (2007) 167–169

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Corrigendum

Corrigendum for the comparison theorems in: “A new definition of viscosity solutions for a class of second-order degenerate

elliptic integro-differential equations”

[Ann. I. H. Poincaré – AN 23 (5) (2006) 695–711]

Mariko Arisawa

GSIS, Tohoku University, Aramaki 09, Aoba-ku, Sendai 980-8579, Japan Available online 13 November 2006

In this note, we shall present the correction of the proofs of the comparison results in the paper [1]. In order to show clearly the correct way of the demonstration, we shall simplify the problem to the following.

(Problem (I)):

F

x, u,∇u,∇²u

−

R^N

u(x+z)−u(x)−1_|z|1

z,∇u(x)

q(dz)=0 inΩ, (1)

(Problem (II)):

F

x, u,∇u,∇²u

−

{z∈R^N|x+z∈Ω}

u(x+z)−u(x)−1_|z|1

z,∇u(x)

q(dz)=0 inΩ, (2)

whereΩ⊂R^Nis open, andq(dz)is a positive Radon measure such that

|z|1|z|²q(dz)+

|z|>11q(dz) <∞. Al- though in [1] only (II) was studied, in order to avoid the non-essential technical complexity, here, let us give the explanation mainly for (I). For (I), we consider the Dirichlet B.C.:

u(x)=g(x) ∀x∈Ω^c, (3)

wheregis a given continuous function inΩ^c. For (II), we assume thatΩ is a precompact convex open subset inR^N withC¹boundary satisfying the uniform exterior sphere condition, and consider either the Dirichlet B.C.:

u(x)=h(x) ∀x∈∂Ω, (4)

wherehis a given continuous function on∂Ω, or the Neumann B.C.:

∇u(x),n(x)

=0 ∀x∈∂Ω, (5)

wheren(x)∈R^Nthe outward unit normal vector field defined on∂Ω. The above problems are studied in the frame- work of the viscosity solutions introduced in [1]. Under all the assumptions in [1], for (I) the following comparison result holds, and for (II), although the proofs therein are incomplete, the comparison results stated in [1] hold, and we shall show in a future article.

DOI of original article: 10.1016/j.anihpc.2005.09.002.

E-mail address:arisawa@math.is.tohoku.ac.jp (M. Arisawa).

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.10.001

168 M. Arisawa / Ann. I. H. Poincaré – AN 24 (2007) 167–169

Theorem 1.1(Problem (I) with Dirichlet B.C.). Assume thatΩ is bounded, and the conditions forF in[1]hold. Let u∈USC(R^N)andv∈LSC(R^N)be respectively a viscosity subsolution and a supersolution of (1)inΩ, which satisfy uvonΩ^c. Then,uvinΩ.

To prove Theorem 1.1, we approximate the solutionsuandv by the supconvolution: u^r(x)=sup_y∈R^N{u(y)−

1

2r²|x−y|²}and the infconvolution:v_r(x)=inf_y∈RN{v(y)+_2r¹2|x−y|²}(x∈R^N), wherer >0.

Lemma 1.2(Approximation for Problem (I)). Letuandvbe respectively a viscosity subsolution and a supersolution of (1). For anyν >0there existsr >0such thatu^r andv_r are respectively a subsolution and a supersolution of the following problems.

F

x, u,∇u,∇²u

−

R^N

u(x+z)−u(x)−1_|z|1

z,∇u(x)

q(dz)ν, (6)

F

x, v,∇v,∇²v

−

R^N

v(x+z)−v(x)−1_|z|1

z,∇v(x)

q(dz)−ν, (7)

inΩ_r= {x∈Ω|dist(x, ∂Ω) >√

2Mr}, whereM=max{sup_Ω|u|,sup_Ω|v|}.

Remark thatu^r is semiconvex,v_r is semiconcave, and both are Lipschitz continuous inR^N. We then deduce from the Jensen’s maximum principle and the Alexandrov’s theorem (deep results in the convex analysis, see [2] and [3]), the following lemma, the last claim of which is quite important in the limit procedure in the nonlocal term.

Lemma 1.3.LetUbe semiconvex andV be semiconcave inΩ. Forφ(x, y)=α|x−y|²(α >0)considerΦ(x, y)= U (x)−V (y)−φ(x, y), and assume that(x,¯ y)¯ is an interior maximum ofΦinΩ×Ω. Assume also that there is an open precompact subsetOofΩ×Ω containing(x,¯ y), and that¯ μ=sup_OΦ(x, y)−sup_∂OΦ(x, y) >0. Then, the following holds.

(i) There exists a sequence of points(x_m, y_m)∈O (m∈N)such thatlimm→∞(x_m, y_m)=(x,¯ y), and¯ (p_m, X_m)∈ J_Ω^2,+U (xm),(p_m, Ym)∈J_Ω^2,−V (ym)such that

mlim→∞pm= lim

m→∞p_m=2α(xm−ym)=p, andX_mY_m∀m.

(ii) ForP_m=(p_m−p,−(p_m−p)),Φ_m(x, y)=Φ(x, y)− P_m, (x, y)takes a maximum at(x_m, y_m)in O.

(iii) The following holds for anyz∈R^Nsuch that(x_m+z, y_m+z)∈O.

U (x_m+z)−U (x_m)− p_m, zV (y_m+z)−V (y_m)− p_m, z. (8) By admitting these lemmas here, let us show how Theorem 1.1 is proved.

Proof of Theorem 1.1. We use the argument by contradiction, and assume that max_Ω(u−v)=(u−v)(x₀)=M₀>0 forx₀∈Ω. Then, we approximateubyu^r (supconvolution) andv byv_r (infconvolution), which are a subsolution and a supersolution of (6) and (7), respectively. Clearly, max_Ω(u^r −v_r)M₀>0. Letx¯∈Ω be the maximizer of u^r −v_r. In the following, we abbreviate the index and writeu=u^r,v=v_r without any confusion. As in the PDE theory, considerΦ(x, y)=u(x)−v(y)−α|x−y|², and let(x,ˆ y)ˆ be the maximizer ofΦ. Then, from Lemma 1.3 there exists(x_m, y_m)∈Ω (m∈N) such that limm→∞(x_m, y_m)=(x,ˆ y), and we can takeˆ (ε_m, δ_m)a pair of positive numbers such that

u(x_m+z)u(x_m)+ p_m, z +1

2X_mz, z +δ_m|z|², v(y_m+z)v(y_m)+ p_m, z +1

2Y_mz, z −δ_m|z|², for∀|z|ε_m. From the definition of the viscosity solutions, we have

M. Arisawa / Ann. I. H. Poincaré – AN 24 (2007) 167–169 169

F

x_m, u(x_m), p_m, X_m

−

|z|εm

1 2

(X_m+2δmI )z, z dq(z)

−

|z|εm

u(xm+z)−u(xm)−1_|z|1z, pmq(dz)ν,

F

y_m, v(y_m), p_m, Y_m

−

|z|ε_m

1 2

(Y_m−2δmI )z, z dq(z)

−

|z|εm

v(y_m+z)−v(y_m)−1_|z|1z, p_mq(dz)−ν.

By taking the difference of the above two inequalities, by using (8), and by passingm→ ∞(thanking to (8), it is now available), we can obtain the desired contradiction. The claimuvis proved. 2

Remark 1.1.To prove the comparison results for (II) (in [1]), we do the approximation by the supconvolution:u^r(x)= sup_y∈Ω{u(y)−_2r¹2|x−y|²}, and the infconvolution:v_r(x)=inf_y∈Ω{v(y)+_2r¹2|x−y|²}as in Lemma 1.2. Because of the restriction of the domain of the integral of the nonlocal term and the Neumann B.C., a slight technical complexity is added. The approximating problem for (2)–(5) inΩ is as follows.

min

F

x, u(x),∇u(x),∇²u(x)

+ min

y∈Ω,|x−y|√

2Mr −

{z∈R^N|y+z∈Ω}

u(x+z)−u(x)

−1_|z|1

z,∇u(x)

q(dz), min

y∈∂Ω,|x−y|√ 2Mr

n(y),∇u(x) +ρ

ν,

max

F

x, v(x),∇v(x),∇²v(x)

+ max

y∈Ω,|x−y|√ 2Mr

−

{z∈R^N|y+z∈Ω}

v(x+z)−v(x)

−1_|z|1

z,∇v(x)

q(dz), max

y∈∂Ω,|x−y|√ 2Mr

n(y),∇v(x)

−ρ −ν.

We deduce the comparison result from this approximation and Lemma 1.3, by using the similar argument as in the proof of Theorem 1.1.

References

[1] M. Arisawa, A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations, Ann. I. H.

Poincaré – AN 23 (5) (2006) 695–711.

[2] M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math.

Soc. 27 (1) (1992).

[3] W.H. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, 1992.