Maximal generalized solution of eikonal equation

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ScienceDirect J. Differential Equations 257 (2014) 231–263 www.elsevier.com/locate/jde

Maximal generalized solution of eikonal equation Sandro Zagatti SISSA, Via Bonomea 265, 34136 Trieste, Italy Received 5 February 2012; revised 23 November 2013 Available online 18 April 2014

Abstract We study the Dirichlet problem for the eikonal equation: ⎧ 2 ⎨ 1  ∇u(x) − a(x) = 0 in Ω 2 ⎩ u(x) = ϕ(x) on ∂Ω, without continuity assumptions on the map a(·). We find a class of maps a(·) contained in the space L∞ (Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution. © 2014 Elsevier Inc. All rights reserved. MSC: 35F21; 35F20; 49L25; 46B50; 35F30 Keywords: Maximality; Viscosity solution; Eikonal equation; Strong compactness

1. Introduction We study the Dirichlet problem for the eikonal equation without continuity assumptions: P(a, ϕ, Ω):

  2 1 ∇u(x) − a(x) = 0 in Ω 2 u=ϕ on ∂Ω,

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jde.2014.04.001 0022-0396/© 2014 Elsevier Inc. All rights reserved.

(1.1)

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

where Ω is an open bounded subset of Rd , a ∈ L∞ (Ω) is a given nonnegative function and ϕ ∈ W 1,∞ (Ω) is an almost everywhere subsolution, i.e. satisfies the compatibility condition 2 1  ∇ϕ(x) ≤ a(x) 2

a.e. x ∈ Ω.

(1.2)

. Consider the nonempty set S(a, ϕ, Ω) = {v ∈ ϕ + W01,∞ (Ω) : |∇v|2 ≤ 2a a.e. in Ω} of a.e. subsolutions: it is immediate to see that S(a, ϕ, Ω) is compact in the strong L1 (Ω) topology and then there exists a unique element u of S(a, ϕ, Ω) which maximizes the integral on Ω, i.e. 

 u(x) dx ≥

Ω

u(x) dx

∀u ∈ S(a, ϕ, Ω).

Ω

By classical results the function u turns out to be pointwise maximal in S(a, ϕ, Ω) and, if the map a(·) is assumed to be continuous, u is the unique viscosity solution and a generalized solution of (1.1). However the existence of the maximal element u of S(a, ϕ, Ω) does not require the continuity of a(·) and then it is worth to investigate for which functions a ∈ L∞ (Ω) the maximal element u is a generalized solution, that is to say a map which satisfies the equation in (1.1) for almost every x ∈ Ω. More generally, it is interesting to ascertain for which functions a(·) problem P(a, ϕ, Ω) admits a generalized solution and, in addition, a selection criterium should be given in order to have uniqueness. In this direction maximality seems to be a good candidate, since it is a property enjoyed by viscosity solutions. While the continuous case has been widely studied in the realm of viscosity, few results have been devoted to the discontinuous one and we quote [9,6,11,5]. First of all we stress that, by a straightforward modification of the proof of Theorems 1 and 2 in [12] we may actually state the following Theorem 1. Assume that there exists a closed null set N ⊂ Ω such that the nonnegative map a ∈ L∞ (Ω) is continuous on Ω \ N . Then the maximal element of S(a, ϕ, Ω) is a generalized solution of P(a, ϕ, Ω). In this paper we try to give further answers to the above questions. The idea is to consider a sequence (an ) of regular functions pointwise converging almost everywhere to a(·) and the corresponding sequence (un ) of maximal (viscosity) solutions of P(an , ϕ, Ω). Making use of Lax formula for viscosity solutions, of some properties of semiconcave functions and imposing suitable conditions on the maps an (·), we prove that the sequence (un ) converges strongly in W 1,1 (Ω) and that its limit is actually a (maximal) generalized solution of P(a, ϕ, Ω). We exhibit also a class of functions a(·) for which our procedure actually works. The paper is organized as follows. In Section 2 we introduce notations and list a series of preliminary and well known results. In Section 3 we take a function a ∈ C 2 (Ω), the corresponding maximal (viscosity) solution of the Dirichlet problem P(a, ϕ, Ω) and derive a priori estimates which will be used in the following. In Section 4 we consider a sequence (an ) in C 2 (Ω), on which we impose the fundamental hypotheses used in the paper, and the corresponding sequence of maximal (viscosity) solutions of the Dirichlet problems P(an , ϕ, Ω). Using the estimates of Section 3, we prove the compactness criterion which is the core of the present work. In Section 5 we apply the compactness criterion to Dirichlet problem P(a, ϕ, Ω) with a discontinuous

S. Zagatti / J. Differential Equations 257 (2014) 231–263

233

function a(·), assuming that such function is the pointwise limit of a sequence (an ) satisfying the hypotheses of Section 4. By this way we prove in Theorem 4 our existence results. In Section 6 we exhibit a subset Z(Ω) of L∞ (Ω) whose elements satisfy hypotheses of Theorem 4, so that the above procedure can be actually performed and then the Dirichlet problem admits a solution for any a ∈ Z(Ω). With an additional hypothesis such solution turns out to be pointwise maximal and then, whenever a(·) is continuous, it coincides with the (unique) viscosity solution. This fact provide a selection criterion for generalized solutions obtained by our method and a generalization of the notion of viscosity solution. The last Section 7 is an appendix containing an alternative proof of our compactness result. 2. Notations and preliminaries In this paper Rd is the Euclidean d-dimensional space; we denote respectively by ·,· and . by | · | the inner product and the Euclidean norm in Rd , while E = {e1 , . . . , ed } is the canonical d d basis in R and a point x ∈ R is written as x = (x1 , . . . , xd ). The open ball in Rd of center x and radius r is written as B(x, r) = BRd (x, r). Given E ⊆ Rd , mk (E) is the k-dimensional Lebesgue measure, ∂E is the boundary, E c is the complement, and co(E) is the convex hull of E, by dist(x, E) we mean the distance of the point x from the set E. Given two points x and y in Rd we denote by [x, y] the closed line segment with endpoints x and y. Given an open subset U of Rd , by D(U ) and D (U ) we denote, respectively, the space of test functions and the space of distribution on U ; by D+ (U ) we mean the subset of nonnegative test functions. We use the spaces C k (U ), Lr (U ), W 1,r (U ), W01,r (U ), for k ∈ N0 = {0} ∪ N and 1 ≤ r ≤ ∞, with their usual (strong and weak) topologies. Dealing with a Sobolev function we assume to use the precise representative. By (ρα ) we denote a regularizing family and, given a . map u : U → R, we consider the convolution uα = ρα ∗ u assuming that it is defined on a set c Uδ = {x ∈ U : dist(x, U ) > δ} for α ∈ ]0, δ]. By the symbol Di2 we mean the second derivative in the direction ei , that is to say, given a function (or distribution) u = u(x1 , . . . , xd ) we have Di2 u =

∂ 2u . ∂xi2

If I is an open subset of R, by S(I ) we mean the space of real simple functions defined on I . If the set U ⊆ Rd is a Cartesian product of the form

×U , d

U=

j

j =1

where each Uj is an open subset of R, we consider the class of tensor products on U defined as f (x) = f (x1 , . . . , xd ) =

d 

fj (xj ),

(2.1)

j =1

where each function fj is defined on Uj . Given a function (distribution) space X(E) on the set E ⊆ R, by the symbol

234

S. Zagatti / J. Differential Equations 257 (2014) 231–263 d

X(Uj )

j =1

we denote the set of tensor products of the form (2.1), with fj ∈ X(Uj ) for every j , and by the symbol

L

d

 X(Uj )

j =1

we denote the linear span of the elements of the form: f=

d

j =1 S(Uj )

Kf

i.e. the linear space of functions f of

αk f k ;

k=0

with f k (x) = f k (x1 , . . . , xd ) =

d 

fjk (xj ),

x∈U

j =1

and αk ∈ R,

fjk ∈ X(Uj )

∀j.

In particular we will use the space L( dj =1 D(Uj )) and its subset L( dj =1 D+ (Uj )) of nonnegative elements f=

Kf

αk f k ,

k=0

i.e. f k (x) = f k (x1 , . . . , xd ) =

d 

fjk (xj ),

x ∈ U,

j =1

with αk ≥ 0,

fjk ∈ D+ (Uj )

∀j.

It is well known that L( dj =1 D(Uj )) and L( dj =1 D+ (Uj )) are dense, respectively, in D(U ) and in D+ (U ) and, as a consequence, we have the following

S. Zagatti / J. Differential Equations 257 (2014) 231–263

Lemma 1. Let U1 , . . . , Ud be open intervals of R and let U = is dense in L1 (U ).

235

d

×dj=1 Uj . Then L(

j =1 S(Uj ))

Proof. Let f ∈ L1 (Ω) and  > 0. By classical result we may find ϕ ∈ D(Ω) such that f − φL1 (Ω) ≤ . Then we take ψ ∈ L( dj =1 D(Uj )) such that ψ − ϕL1 (Ω) ≤ . Given an open interval I ⊆ R and a map φ ∈ D(I ), we may find a simple function arbitrarily close to φ 1 (I ) norm. Hence, by an elementary application of Fubini–Tonelli theorem, we may find in the L g ∈ L( dj =1 S(Uj )) such that ψ − gL1 (Ω) ≤ . 2 Special notations. Let Ω ⊆ Rd be an open convex set and v : Ω → R. Given j ∈ {1, . . . , d} and x ∈ Rd we write xj = (x1 , . . . , xj −1 , xj +1 , . . . , xd ) ∈ Rd−1 and x = (x1 , . . . , xd ) = (xj , xj ). Given xj ∈ Rd−1 consider the line rxj touching the point xj and parallel to the direction ej ; then we call Jxj the line segment . Jxj = rxj ∩ Ω.

(2.2)

vxj (t) = v(x1 , . . . , xj −1 , t, xj +1 , . . . , xd ),

(2.3)

In addition we set

with t ∈ Jxj and xj ranging in the set  .  Ωjd−1 = xj ∈ Rd−1 : Jxj = ∅ .

(2.4)

Definition 1. Let Ω ⊆ Rd be open, i ∈ {1, . . . , d}, r ≥ 0, h ≥ 0 and let x = (xi , xi ) ∈ Rd . We introduce the closed cylinder      .  R(x, r, h, i) = y = yi , yi ∈ Rd : yi ∈ B Rd−1 xi , r , yi ∈ [xi − h, xi + h] .

(2.5)

Let a ∈ C 2 (Ω, R). We set    . A(a, x, r, h, i) = max Di2 a(y), y ∈ R(x, r, h, i) ∩ Ω .

(2.6)

Remark 1. The continuity of the map Di2 a(·) implies that there exists a positive constant C such that for any index i A(a, x, r, h, i) ≤ C

∀(x, r, h) ∈ Ω × [0, +∞[ × [0, +∞[

and that the application Ω × [0, +∞[ × [0, +∞[  (x, r, h) → A(a, x, r, h, i) is continuous. We recall some properties of semiconcave functions with linear modulus (see [3]).

(2.7)

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Definition 2. Let U ⊆ Rd and u : U → R. We say that u is semiconcave with linear modulus (SCLM) with constant C ≥ 0 if for every x, y ∈ A such that the line segment [x, y] is contained in U and for every λ ∈ [0, 1] the following inequality holds:   1 λu(x) + (1 − λ)u(y) − u λx + (1 − λ)y ≤ λ(1 − λ)C|x − y|. 2

(2.8)

The following properties are well known (see for example [3]). Proposition 1. Let U ⊆ Rd be convex and u : U → R be continuous. The following properties are equivalent. (i) u is SCLM with constant C ≥ 0. (ii) For every x ∈ U and for every h ∈ Rd such that x − h, x + h ∈ U we have u(x + h) + u(x − h) − 2u(x) ≤ C. |h|2 (iii) The map U  x → u(x) − 12 C|x|2 is concave. (iv) For every unit vector ν ∈ Rd we have ∂ 2u ≤C ∂ν 2

in D (U ),

that is to say 

∂ 2φ u(x) 2 (x) dx ≤ C ∂ν

U

 φ(x) dx,

∀φ ∈ D+ (U ).

U

(v) There exist a concave function u1 : U → R and a C 2 -function u2 : U → R such that u = u1 + u2 and D 2 u2 L∞ (U ) ≤ C. We shall use the following consequence of Proposition 1 Corollary 1. Let U ⊆ Rd be open and w ∈ C 2 (U ). Then for every Λ  U open, bounded and convex, i ∈ {1, . . . , d}, λ ∈ [0, 1], x ∈ Λ, h ∈ R such that x + hei ∈ Λ, we have:   λw(x) + (1 − λ)w(x + hei ) − w λx + (1 − λ)(x + hei )   1 ≤ λ(1 − λ) maxDi2 w(y)|h|. 2 y∈Λ Proof. It is a trivial consequence of points (iv)–(v) of Proposition 1. We now prove some preliminary results.

(2.9) 2

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237

˜ → R be a Lemma 2. Let I ⊆ R be an open interval, v ∈ C(I ) and h˜ > 0. Let F : I × [0, h] function satisfying the following properties: ˜ (i) I  t → F (t, h) is nonnegative and belongs to L1 (I ) for every h ∈ [0, h]; ˜ (ii) there exists a map F˜ ∈ L1 (I ) such that F (·, h) ≤ F˜ a.e. in I for every h ∈ [0, h]; . (iii) F (t, h) → F 0 (t) = F (t, 0) as h → 0+ for almost every t ∈ I . Suppose that we have:   1 λv(t) + (1 − λ)v(t + h) − v λt + (1 − λ)(t + h) ≤ λ(1 − λ)F (t, h), 2

(2.10)

˜ such that [t − h, t + h] ⊆ I . for every t ∈ I , for every h ∈ R with |h| ≤ h and for every h ∈ ]0, h] Then we have v ≤ F 0

in D (I ).

(2.11)

Proof. By Proposition 1, inequality (2.10) is equivalent to v(t − h) + v(t + h) − 2v(t) ≤ F (t, h)|h|2

(2.12)

˜ such that [t − h, t + h] ⊆ I . for every t ∈ I , for every h ∈ R with |h| ≤ h and for every h ∈ ]0, h] + Let φ ∈ D (I ). Multiplying both sides of (2.12) by φ and integrating on the interval I , we have   I

  v(t − h) + v(t + h) − 2v(t) F (t, h)φ(t) dt. φ(t) dt ≤ |h|2

(2.13)

I

By a change of variable in (2.13):  I

   φ(t − h) + φ(t + h) − 2φ(t) v(t) dt ≤ F (t, h)φ(t) dt. |h|2

(2.14)

I

Since the square parenthesis in the l.h.s. of (2.14) converges uniformly to φ as |h| goes to zero, taking the limit we obtain: 

v(t)φ (t) dt ≤

I

 F (t, h)φ(t) dt.

(2.15)

I

Passing to the limit h → 0+ in (2.15) we obtain (2.11) by assumptions (ii)–(iii) and by dominated convergence. 2 Lemma 3. Let Ω be an open bounded convex subset of Rd and let f, g ∈ L1loc (Ω). Given i ∈ {1, . . . , d} and s ∈ N assume that Dis fxi ≤ gxi

in D (Ixi ) for a.e. xi ∈ Ωid−1 .

(2.16)

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

Then we have in D (Ω).

Dis f ≤ g

(2.17)

×d

Proof. Assume that Ω is a polyinterval, i.e. the Cartesian product j =1 Ij of intervals Ij of R. The general case will be obtained considering a polyinterval containing Ω and extending f and g by zero out of Ω. We shall use this lemma for s = 2, hence we perform the proof for such value of s, being identical the general case. In addition, for the sake of simplicity, we fix i = 1. For a.e. x1 ∈ Ω1d−1 we have that gx1 ∈ L1loc (I1 ) and, by assumption (2.16), we have D12 fx1 ≤ gx1

in D (I1 ) for a.e. x1 ∈

×I , d

j

j =2

that is to say 

fx1 (t)φ (t) dt ≤

I1

 gx1 (t)φ(t) dt

(2.18)

I1

×d

for every φ ∈ D+ (I1 ) and for almost every x1 ∈ j =2 Ij . Let ψ ∈ D+ (Ω) be a tensor product of the form ψ(x1 , . . . , xd ) =

d 

φj (xj ),

j =1

where φj ∈ D+ (Ij ) for j = 1, . . . , d. Using (2.18), we have: 



D12 f, ψ D (Ω),D(Ω)

 =

f (x1 , . . . , xd )φ1 (x1 )

 ...  ... I2



=

  d Id j =2

I2

≤

φj (xj ) dx1 . . . dxd

j =2

Ω

=

d 

  d

  φj (xj ) fx1 (x1 )φ1 (x1 ) dx1 dx2 . . . dxd I1

  φj (xj ) gx (x1 )φ1 (x1 ) dx1 dx2 . . . dxd

Id j =2

1

I1

g(x)ψ(x) dx. Ω

This inequality is clearly preserved for any test function ψ ∈ L( dj =1 D+ (Ij )) and, since this last space is dense in D+ (Ω), the proof of (2.17) is achieved. 2

S. Zagatti / J. Differential Equations 257 (2014) 231–263

239

The following result is well known (see Lemma 3.1 in [10], p. 89) and we recall it for convenience of the reader. Lemma 4. Let Ω be an open subset of Rd and (un ) a bounded sequence in W 1,∞ (Ω). Let (Gn ) be a sequence of nonnegative functions in L1loc (Ω) such that for every Λ  Ω there exists CΛ > 0 such that  Gn (x) dx ≤ CΛ ∀n ∈ N. (2.19) Λ

Assume un ≤ Gn

in D (Ω) ∀n ∈ N.

(2.20)

Then the sequence (un ) is (strongly) relatively compact in W 1,r (Ω) for every r ∈ [1, ∞[. Proof. Let Λ  Ω and assume un ∈ D(Ω) with supp un ⊆ Λ for ever n ∈ N. We have 

  un (x) dx =

Λ



  un (x) − Gn (x) + Gn (x) dx

Λ



≤

  un (x) − Gn (x) dx +

Λ



= Λ

 Gn (x) dx Λ





 Gn (x) − un (x) dx + 

 un (x) dx + 2

=− Λ

Gn (x) dx Λ

Gn (x) dx ≤ 2CΛ , Λ

where we have performed an integration by parts and used (2.19) and (2.20). Then the sequence (|un |) is bounded in L1 (Λ) and, by classical Sobolev embedding, it turns out to be bounded in 1 3 W − 2 ,q (Λ) for q ∈ [1, 2d/(d − 1)[. Hence the sequence (un ) is bounded in W 2 ,q (Λ) and then, in particular, it is relatively compact in W 1,1 (Λ). The claim follows easily by the arbitrariness of Λ and by interpolation. Assume now un ∈ W 1,∞ (Ω). Let Λ  Ω, φ ∈ D(Ω) with 0 ≤ φ ≤ 1, φ ≡ 1 on Λ, and let (ρ )>0 be a regularizing family. Define v = ρ ∗ (φ · un ). If  is sufficiently small, (vn ) is a sequence in D(Ω) and there exists Λ˜  Ω such that supp vn ⊆ Λ˜ for ever n ∈ N. In addition we have (φ · un ) = (φ) · un + 2∇φ · ∇un + φun and

in D (Ω)

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

   ρ ∗ (φ · un ) = ρ ∗ (φ · un )

in D(Ω).

It follows that vn ≤ C + Gn

in D (Ω)

for a suitable positive constant C depending on the fixed cut off φ. By previous step (vn ) turns out to be relatively compact in W 1,r (Ω) and then, by the arbitrariness of Λ, the same holds true for the sequence (un ) itself. 2 We shall use arguments taken from the theory of viscosity solutions for Hamilton–Jacobi equations and of semiconcave functions, for which we refer to the wide existing literature, mentioning for example the monographs [1–4,10] and paper [7]. In addition we shall need the results contained in [12] (see also [13]) and the representation formula for viscosity solutions due to P.D. Lax and illustrated, for example, by P.L. Lions in [10]. To this aim we introduce suitable notations and recall the following results. Definition 3. Let Λ ⊆ Rd be open and bounded. Given T > 0 and x, y ∈ Λ we set    .  Ξ (x, y, T , Λ) = ξ ∈ W 1,∞ [0, T ], Rd : ξ(t) ∈ Λ a.e. t ∈ [0, T ], ξ(0) = x, ξ(T ) = y . Theorem 2. Let Ω ⊆ Rd be an open, bounded domain and let H ∈ C(Ω × Rd ) be a function satisfying the following conditions: (i) Rd  p → H (x, p) is convex for every x ∈ Λ; (ii) there exist constants α > 0, β ≥ 0 such that H (x, p) ≥ α|p| − β

∀x ∈ Ω, ∀p ∈ Rd ;

(iii) infp∈Rd H (x, p) ≤ 0 for every x ∈ Ω. Let ϕ ∈ W 1,∞ (Ω) satisfy the compatibility condition   H x, ∇ϕ(x) ≤ 0

a.e. x ∈ Ω

and introduce the (nonempty) set    .  SH,ϕ = v ∈ ϕ + W01,∞ (Ω) : H x, ∇v(x) ≤ 0 a.e. x ∈ Ω . Then there exists a unique element u ∈ SH,ϕ such that u(x) ≥ v(x) for every x ∈ Ω and for every v ∈ SH,ϕ which turns out to be a viscosity solution of the Dirichlet problem 

H (x, ∇u) = 0 u=ϕ

in Ω, on ∂Ω.

In addition, calling H ∗ the Fenchel transform of H , i.e.

S. Zagatti / J. Differential Equations 257 (2014) 231–263

241

  . H ∗ (x, q) = sup − p, q − H (x, p) , p∈Rd

we have the following (Lax) formula:  T u(x) = inf

  e H ξ(t), ξ (t) dt + u(w) : T > 0, w ∈ ∂Λ, ξ ∈ Ξ (x, w, T , Λ) , −t

∗





(2.21)

0

for every x ∈ Λ and for every Λ  Ω open, bounded, simply connected with smooth boundary ∂Λ. The proof of the first part is performed in [12] (Theorems 1 and 2), while the second one is treated in Section 5 (p. 115 and ff.) of [10]. 3. C 2 a priori estimates In this section we consider problem P(a, ϕ, Ω) as in (1.1), assuming that the given nonnegative function a(·) is sufficiently regular, and derive estimates that will be used in the proof of our main result. We take a ∈ C 2 (Ω), ϕ ∈ W 1,∞ (Ω), assuming that the compatibility condition (1.2) is satisfied, and a positive constant α such that aL∞ (Ω) ≤ α.

(3.1)

We introduce the nonempty set S(a, ϕ, Ω) of a.e. subsolutions of P(a, ϕ, Ω): Definition 4.   2 1  . 1,∞  S(a, ϕ, Ω) = v ∈ ϕ + W0 (Ω) : ∇v(x) ≤ a(x) a.e. x ∈ Ω . 2

(3.2)

Applying Theorem 2 we infer the existence of a unique map u ∈ S(a, ϕ, Ω) such that u(x) ≥ v(x) for every x ∈ Ω and for every v ∈ S(a, ϕ, Ω) which turns out to be a viscosity and a generalized solution of problem P(a, ϕ, Ω). It is immediate to see there exist positive constants 1 . K0 = K0 (α, ϕ, Ω) and K1 = (2α) 2 such that uL∞ (Ω) ≤ K0 = K0 (α, ϕ, Ω),

1

∇uL∞ (Ω) ≤ K1 = (2α) 2 .

(3.3)

Let Λ  Ω be an open, convex subset of Ω with smooth boundary ∂Λ. For γ > 0 we set    .  Λγ = x ∈ Λ : dist x, Λc > γ .

(3.4)

We recall from Section 2 that the solution u may be represented according to formula (2.21). In our case we have 1 H (x, p) = |p|2 − a(x), 2

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

and by easy computations (see for example chapter 5 in [8]) we find that the Fenchel transform H ∗ of H is 1 H ∗ (x, q) = |q|2 + a(x). 2 Hence we have:   u(x) = inf J (ξ, T , w); T > 0, w ∈ ∂Λ, ξ ∈ Ξ (x, w, Λ, T )

∀x ∈ Λ,

(3.5)

where . J (ξ, T , w) =

T

  2   1 e−t a ξ(t) + ξ (t) dt + u(w). 2

(3.6)

0

The section is devoted to the proof of some preliminary estimates provided by the following lemmas. Lemma 5. Let Ω, a, α, ϕ, u as above and let Λ  Ω be an open, convex subset of Ω with smooth boundary ∂Λ. Let γ > 0 and Λγ as in (3.4). Then (i) there exists M = M(α, ϕ, Ω) > 0 such that the infimum in formulas (3.5)–(3.6) can be taken on the elements ξ ∈ Ξ (x, w, T , Λ) such that   ξ 

L2 ([0,T ∗ ])

≤ M,

(3.7)

where . T ∗ = min{1, T };

(3.8)

(ii) there exist T0 = T0 (γ , α, ϕ, Ω) > 0 such that the infimum in formulas (3.5)–(3.6) can be taken on the values T such that T ≥ T0 (and, correspondingly, on paths ξ ∈ Ξ (x, w, T , Λ)) for every x ∈ Λγ . Proof. Step 1. Proof of (i). It is clear that in (3.5), we may limit ourselves to paths ξ ∈ Ξ (x, w, T , Λ) such that J (ξ, T , w) ≤ u(x) + 1 ≤ K0 + 1. We have T

1 −t  2 e ξ (t) dt ≤ J (ξ, T , w) − u(w) ≤ u(x) + 1 − u(w), 2

0

and it follows that T 0

1 −t  2 e ξ (t) dt ≤ 2K0 + 1. 2

S. Zagatti / J. Differential Equations 257 (2014) 231–263

243

∗

Since e−T ≤ e−t for every t ∈ [0, T ∗ ], we have

e

−T ∗

T ∗

1  2 ξ (t) dt ≤ 2

T ∗

0

1 −t  2 e ξ (t) dt ≤ 2

0

T

1 −t  2 e ξ (t) ≤ 1 + 2K0 , 2

0

and finally T ∗  2 ξ (t) dt ≤ 2eT ∗ (1 + 2K0 ) ≤ 2e(1 + 2K0 ). 0 1 . Hence we set M = (2e(1 + 2K0 )) 2 and, recalling (3.1) and (3.3), the claim is proved. Step 2. Proof of (ii). Let x ∈ Λγ , T > 0, w ∈ ∂Λ and ξ ∈ Ξ (x, w, T , Λ) be arbitrary; write I = [0, T ] and, given ρ ≥ 0, set

   .  Iρ = t ∈ I : ξ (t) ≥ ρ . We have  T  T        γ ≤ |x − w| ≤ ξ(0) − ξ(T ) =  ξ (t) dt  ≤ ξ (t) dt   0 0       ξ (t) dt ≤ ξ (t) dt + I \Iρ

Iρ



≤

  ξ (t) dt + ρT .

Iρ

Hence 

  ξ (t) dt ≥ γ − ρT .

Iρ

Recalling (3.3) and that a(·) ≥ 0, we may minorize J (ξ, T , w): T J (ξ, T , w) =

  2   1 e−t a ξ(t) + ξ (t) dt + u(w) 2

0

T ≥ −K0 + 0

1 −t  2 e ξ (t) dt 2

(3.9)

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T

e−T ≥ −K0 + 2

 2 ξ (t) dt

0



e−T ≥ −K0 + 2

 2 ξ (t) dt

Iρ

≥ −K0 +



e−T 2

Iρ

  ρ ξ (t) dt 

  ξ (t) dt

= −K0 + e

−T

ρ 2

≥ −K0 + e

−T

ρ (γ − ρT ), 2

Iρ

(3.10)

where we have used (3.9). Recall again that we may limit ourselves to elements ξ , T and w such that J (ξ, T , w) ≤ K0 + 1;

(3.11)

then collect (3.10) and (3.11) and obtain the following chain −K0 −

ρ ρ 2 −T T e + γ e−T ≤ J (ξ, T , w) ≤ K0 + 1. 2 2

Setting 1 . ρ = T −2

it follows that 1

e−T T − 2 ≤

4K0 + 4 . γ

(3.12)

1

Since e−T T − 2 → +∞ as T → 0+, (3.12) and (3.3) imply the existence of the required positive T0 = T0 (γ , A, ϕ, Ω). 2 Lemma 6. Assume hypotheses and notations of Lemma 5 and let γ > 0. Then ˜ x ∈ Λγ , i ∈ {1, . . . , d}, (i) there exist δ0 > 0 and h˜ > 0 such that for every δ ∈ ]0, δ0 ], h ∈ ]0, h], λ ∈ ]0, 1[ and h ∈ R such that |h| ≤ h, we have   λu(x) + (1 − λ)u(x + hei ) − u λx + (1 − λ)(x + hei )     1 1 1 ≤ λ(1 − λ) + δA a, x, Mδ 2 , h, i |h|2 ; 2 δ (ii) the positive constant δ0 depends only on γ , α, ϕ and Ω.

(3.13)

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245

Proof. Step 1. Recall the definition of M and T0 from Lemma 5 and choose δ0 > 0 such that δ0 ≤ min{1, T0 }.

(3.14)

Recall (2.5), (3.4) and choose h˜ > 0 sufficiently small, in such a way that for every x ∈ Λγ we have   1 ˜ i ⊆ Ω. R x, Mδ 2 , h, (3.15) ˜ For i ∈ {1, . . . , d} and h ∈ R such that |h| ≤ h, Let now λ ∈ ]0, 1[, x ∈ Λγ and h ∈ ]0, h]. . write y = x + hei ; then, recalling Definition 3, choose arbitrary w ∈ ∂Λ, T ≥ T0 and ξ ∈ Ξ (z, w, T , Λ),

(3.16)

z = λx + (1 − λ)y = x + (1 − λ)hei .

(3.17)

where

Define the following paths:  ξ1 (t) =  ξ2 (t) =

ξ(t) − (1 − λ)( δ−t δ )hei

t ∈ [0, δ]

ξ(t)

t ∈ ]δ, T ],

ξ(t) + λ( δ−t δ )hei

t ∈ [0, δ] t ∈ ]δ, T ];

ξ(t)

(3.18) (3.19)

remarking that ξ1 ∈ Ξ (x, w, T , Λ),

ξ2 ∈ Ξ (y, w, T , Λ)

(3.20)

and that ξ(t) = λξ1 (t) + (1 − λ)ξ2 (t)

t ∈ [0, T ].

(3.21)

By definitions (3.18) and (3.19), the three considered paths ξ1 , ξ2 and ξ coincide for t ≥ δ. By (3.7), (3.8), (3.14) and by Hölder inequality, we have  t  t       ξ(t) − ξ(0) =  ξ (s) ds  ≤ ξ (s) ds   0

0

t

1

≤

 2 ξ (s) ds

2

1

1

t 2 ≤ Mδ 2

∀t ∈ [0, δ].

(3.22)

0

˜ we It follows from the above construction, from (2.5), (3.15) and (3.22) that for every h ∈ ]0, h] have     1 1 ˜ i ∀t ∈ [0, δ]. ξ(t), ξ1 (t), ξ2 (t) ∈ R x, Mδ 2 , h, i ⊆ R x, Mδ 2 , h, (3.23)

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Now we estimate the quantity . A = λJ (ξ1 , T , w) + (1 − λ)J (ξ2 , T , w) − J (ξ, T , w).

(3.24)

By (3.6) we have: δ A=λ

e

−t



  1  2    a ξ1 (t) + ξ1 (t) dt 2

0

δ + (1 − λ)

   1  2    e a ξ2 (t) + ξ2 (t) dt 2 −t

0

δ −

e

−t



  1  2   a ξ(t) + ξ (t) dt 2 

0

δ =

       e−t λa ξ1 (t) + (1 − λ)a ξ2 (t) − a ξ(t) dt

0

δ +

 2  2  1 −t   2 e λ ξ1 (t) + (1 − λ)ξ2 (t) − ξ (t) dt 2

0

= A1 + A2 .

(3.25)

Step 2: estimate of A1 . The C 2 -regularity of the map a(·), definition (2.6), inclusions (3.15), (3.23), formula (3.21) and inequality (2.9) in Corollary 1 imply that       λa ξ1 (t) + (1 − λ)a ξ2 (t) − a ξ(t)   1 1 ≤ λ(1 − λ)A a, x, Mδ 2 , h, i |h|2 2

∀t ∈ [0, δ].

Hence we have   1 1 A1 ≤ λ(1 − λ) · δ · A a, x, Mδ 2 , h, i |h|2 2

˜ ∀|h| ≤ h ≤ h.

(3.26)

Step 3: estimate of A2 . Observe that for every q1 , q2 ∈ Rd and for every λ ∈ [0, 1], we have  2 λ|q1 |2 + (1 − λ)|q2 |2 − λq1 + (1 − λ)q2  = λ(1 − λ)|q1 − q2 |2 and that, by direct computations, ξ1 (t) − ξ2 (t) =

h , δ

a.e. t ∈ [0, δ].

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247

Deriving (3.21) with respect to the variable t ∈ [0, δ], it follows that 1 A2 ≤ λ(1 − λ) 2

δ

e−t

|h|2 1 |h|2 dt ≤ λ(1 − λ) 2 δ δ2

˜ ∀|h| ≤ h ≤ h.

(3.27)

0

Step 4. Collecting (3.25), (3.26) and (3.27) we have   1   1 1 A ≤ λ(1 − λ) + δA a, x, Mδi2 , h, i |h|2 2 δ

˜ ∀|h| ≤ h ≤ h.

(3.28)

By the arbitrariness of w ∈ ∂Λ, T ≥ T0 and ξ ∈ Ξ (z, w, T , Λ), recalling the representation (3.5)–(3.6), formulas (3.16), (3.17), (3.20), (3.21) and (3.24), inequality (3.28) implies that the claimed inequality (3.13) holds for every λ ∈ ]0, 1[, h ∈ R such that |h| ≤ h ≤ h˜ and for every i ∈ {1, . . . , d}, as claimed. Step 5. It is clear from the above construction that the positive constant δ0 depends only on γ , T0 and M, that is to say, recalling Lemma 5, only on γ , α, ϕ and Ω. 2 Lemma 7. Assume hypotheses and notations of Lemmas 5 and 6. Let γ > 0 and take δ0 > 0 and h˜ > 0 as in Lemma 6. For every i ∈ {1, . . . , d} let δi ∈ L1 (Λγ ) be functions such that 0 < δi (x) ≤ δ0

for a.e. x ∈ Λγ

(3.29)

and 1 ∈ L1 (Λγ ). δi (·)

(3.30)

˜ define For almost every x ∈ Λγ and for every h ∈ [0, h] . Gi (x, h) =

  1 1 + δi (x)A a, x, Mδi (x) 2 , h, i . δi (x)

(3.31)

Then, for every i ∈ {1, . . . , d}, the map Λγ  x → Gi (·, h) belongs to L1 (Λγ ) and is bounded ˜ In addition, for every i ∈ {1, . . . , d}, λ ∈ ]0, 1[, by an integrable function uniformly in h ∈ [0, h]. ˜ h ∈ R such that |h| ≤ h ≤ h, and for almost every x ∈ Λγ , we have   λu(x) + (1 − λ)u(x + hei ) − u λx + (1 − λ)(x + hei ) 1 ≤ λ(1 − λ)Gi (x, h)|h|2 . 2

(3.32)

Proof. The functions Λγ  x → δi (x),

Λγ  x →

1 δi (x)

are integrable and independent on h. By Remark 1 the function

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  1 Λγ  x → A a, x, Mδi (x) 2 , h, i is well defined and measurable. In addition, recalling (2.7), it is bounded by a constant uniformly with respect to h. For a.e. x ∈ Λγ and for i ∈ {1, . . . , d}, we may set δ = δi (x) in the construction of Lemma 6 obtaining formula (3.32) by comparing (3.13) and (3.31). 2 Lemma 8. Assume hypotheses and notations of Lemmas 5, 6 and 7. Let γ > 0 and take δ0 > 0 and h˜ > 0 as in Lemma 6. For any i ∈ {1, . . . , d} let δi ∈ L1 (Λγ ) satisfy (3.29)–(3.30). Define . Gi (x) = Gi (x, 0) =

  1 1 + δi (x)A a, x, Mδi (x) 2 , 0, i , δi (x)

x ∈ Λγ ,

(3.33)

and . G(x) = Gi (x) d

x ∈ Λγ .

(3.34)

i=1

Then: (i) the functions Gi and G belong to L1 (Λγ ) for every i ∈ {1, . . . , d}; (ii) we have ∂ 2u ≤ Gi ∂xi2

in D (Λγ ) ∀i ∈ {1, . . . , d}

(3.35)

and u ≤ G in D (Λγ ).

(3.36)

Proof. By Lemma 7 the functions Gi (·) are well defined and integrable on Λγ . In addition, by Remark 1, for every i ∈ {1, . . . , d}, we have that lim Gi (x, h) = Gi (x),

h→0+

for a.e. x ∈ Λγ .

(3.37)

˜ and for every |h| ≤ h ≤ h˜ Lemma 6 implies that for almost every x ∈ Λγ , for every h ∈ ]0, h] formula (3.32) holds true. Now, recalling notations (2.2), (2.3) and (2.4), we set . F (t, h) = Gix (t, h)

a.e. t ∈ Jxi , xi ∈ (Λγ )d−1 , i

. F 0 (t) = (Gi )xi (t)

a.e. t ∈ Jxi , xi ∈ (Λγ )d−1 , i

i

and . v(t) = uxi (t)

a.e. t ∈ Jxi , xi ∈ (Λγ )d−1 , i

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249

where in the definition of F , F 0 and v we omit for convenience the index i. The properties of Gi (·,·) and Gi (·) given by Lemma 7 and formula (3.37) imply that F , F 0 and v satisfy hypotheses (i)–(iii) of Lemma 2. Furthermore, formula (3.32) implies formula (2.10). Hence, applying the result of Lemma 2, we have v ≤ F 0

in D (Ixi ), a.e. xi ∈ (Λγ )d−1 , i = 1, . . . , d. i

Then inequality (3.35) follows by an application of Lemma 3, while (3.36) is an immediate consequence of (3.35). 2 4. Approximate solutions In this section we consider a sequence of regular function (an ) and the corresponding Dirichlet problems for the associated eikonal equation as in (1.1). For every n ∈ N we take the maximal (viscosity) solution un of the problem and apply to it the results of previous Section 3, defining the functions Gn as in (3.34). The goal is to provide a uniform integral estimate for such sequence (Gn ) in order to guarantee the strong relative compactness of the sequence (un ) in the Sobolev space W 1,1 (Ω). This program requires a suitable assumption on the sequence (an ) (Hypothesis 1). Let Ω ⊆ Rd be open and bounded and ϕ ∈ W 1,∞ (Ω). Hypothesis 1. Let (an ) be a sequence of nonnegative functions in C 2 (Ω) and assume that there exists α > 0 such that an ∈L∞ (Ω) ≤ α

∀n ∈ N.

(4.1)

Recalling formulas (2.5) and (2.6), set    . A(an , x, r, h, i) = max Di2 an (y), y ∈ R(x, r, h, i) ∩ Ω

(4.2)

and recall formulas (3.33)–(3.34) too. Let Λ  Ω open, bounded, convex with smooth boundary and γ > 0; define Λγ as in (3.4). We say that the sequence (an ) satisfies Hypothesis 1 if, given γ > 0 and M > 0, for every i ∈ {1, . . . , d} and n ∈ N there exist functions δin ∈ L1 (Λγ ) and a positive constant C > 0 such that the following conditions hold: (i) for every i ∈ {1, . . . , d}, for every n ∈ N and for a.e. x ∈ Λγ 0 < δin (x) ≤ 1;

(4.3)

(ii) for every i ∈ {1, . . . , d}, for every n ∈ N 1 ∈ L1 (Λγ ); δin (·)

(4.4)

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(iii) setting . Gni (x) =

1 δin (x)

  1 + δin (x)A an , x, Mδin (x) 2 , 0, i

a.e. x ∈ Λγ

(4.5)

and . n Gn (x) = Gi (x), d

a.e. x ∈ Λγ ,

(4.6)

∀n ∈ N.

(4.7)

i=1

we have  Gn (x) dx ≤ C Λγ

Remark 2. It is clear that if a family δin (·) satisfies the requirement in Hypothesis 1, then, given δˆ0 ∈ ]0, 1], conditions (4.4)–(4.7) are still satisfied replacing δin (·) by a family δˆin (·) such that, instead of (4.3), we require 0 < δˆin (x) ≤ δˆ0 . Indeed, setting   . δˆin (x) = min δin (x), δˆ0 , it is immediate to see that all conditions (4.4)–(4.7) remains valid replacing δin with δˆin . Example 1. In order to focalize the meaning of Hypothesis 1, we observe that if the sequence (Di2 an ) is bounded in C(Ω) for every index i, then Hypothesis 1 is trivially satisfied by taking as maps δi any constant function δi (x) ≡ δ0 , where δ0 ∈ ]0, 1] is a positive constant depending only on γ . Hence we exhibit a simple case of a sequence (an ) satisfying Hypothesis 1 while the sequence of second derivatives (Di2 an ) is unbounded. Let B the unit ball in Rd , with d ≥ 3, and (an ) a sequence in C 2 (B). Assume that there exists a non increasing map f : ]0, 1] → R such that  B  x → |x|2 f

|x| 2



is integrable (for example we may take f (t) = t −4 ) and   2   D an (y) ≤ f |y| i

y ∈ B, i ∈ {1, . . . , d}, n ∈ N.

Recall the definition of the constant M from Lemma 5 and set

S. Zagatti / J. Differential Equations 257 (2014) 231–263

. |x|2 δin (x) = 4M 2

251

x ∈ B, i ∈ {1, . . . , d}, n ∈ N.

By elementary computations, recalling definitions (4.2)–(4.5), one can see that Gni (x) ≤

  4M 2 |x|2 |x| + f 2 |x|2 4M 2

x ∈ B, i = 1, . . . , d, n ∈ N.

Hence, recalling (4.5)–(4.7), Hypothesis 1 is satisfied. We are ready to prove our strong compactness property in Sobolev spaces. Let (an ) be a sequence of nonnegative functions in C 2 (Ω) satisfying the compatibility condition: 2 1  ∇ϕ(x) − an (x) ≤ 0 a.e. x ∈ Ω ∀n ∈ N. 2

(4.8)

For every n ∈ N we consider the problem P(an , ϕ, Ω):

⎧ 2 ⎨ 1  ∇u(x) − an (x) = 0 in Ω ⎩2 u(x) = ϕ(x) on ∂Ω.

(4.9)

By Theorem 2, for every n ∈ N there exists un ∈ ϕ + W01,∞ (Ω) maximal (viscosity) solution of P(an , ϕ, Ω) for which the representation (3.5)–(3.6) holds true. We have the following Theorem 3. Let (an ) be a sequence satisfying Hypothesis 1 and assume that compatibility conditions (4.8) hold true for every n ∈ N. Let (un ) be the sequence of maximal viscosity solutions of P(an , ϕ, Ω). Then there exists a subsequence, still denoted by (un ), and a map u ∈ ϕ +W01,∞ (Ω) n→∞

such that un −→ u strongly in W 1,r (Ω) for every r ∈ [1, ∞[ and, in particular, n→∞

∇un (x) −→ ∇u(x) for a.e. x ∈ Ω.

(4.10) 1

Proof. Step 1. Recall from (3.3) that there exist K0 = K0 (α, ϕ, Ω) and K1 = (2α) 2 such that un L∞ (Ω) ≤ K0 ,

∇un L∞ (Ω) ≤ K1

∀n ∈ N.

(4.11)

By (4.11) the sequence (un ) is bounded in W 1,∞ (Ω), hence there exists u ∈ ϕ + W01,∞ (Ω) and ∗

a subsequence, still denoted by (un ), such that un  u in W 1,∞ (Ω) and uniformly on Ω. Step 2. Let now any Λ  Ω open, bounded, convex with smooth boundary, a positive γ and consider the set Λγ . Since an ∈ C 2 (Ω), we may reproduce the construction of Section 3 for every n ∈ N. By (4.1) the sequence (an ) is bounded in L∞ (Ω) by the positive constant α and then, as a consequence of (4.11) and point (ii) in Lemma 6, the positive constant δ0 provided by point (i) in Lemma 6 depends on ϕ, α, Ω and γ , but is independent on n ∈ N. Hence the whole constructions of Section 3 can be performed for any n ∈ N with the same δ0 . Let δin (·) be the family provided by Hypothesis 1. By Remark 2 we may assume that for every i ∈ {1, . . . , d},

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n ∈ N and for a.e. x ∈ Λγ , we have 0 < δin (x) ≤ δ0 . Hence Hypothesis 1 is satisfied with the constant δ0 in place of the number 1 in (4.3). This implies that the functions Gn constructed as in Section 3 satisfy all conditions in Hypothesis 1, and in particular (4.7). Inequality (3.36) in Lemma 8 takes the following form: un ≤ Gn

in D (Λγ ) ∀n ∈ N.

Then inequality (4.7) in Hypothesis 1 and Lemma 4 imply that the sequence (un ) is relatively compact in W 1,r (Λγ ) for every r ∈ [1, ∞[. Hence the thesis follows by the arbitrariness of Λ and γ . 2 Remark 3. The crucial point in the application of the above result is the following: given the sequence (an ), construct sequences (δni ) (i ∈ {1, . . . , d}) such that Hypothesis 1 is satisfied. 5. Eikonal equation We now apply the strong compactness criterion proved in previous section to the solution of the Dirichlet problem for eikonal equation P(a, ϕ, Ω) given in (1.1). The idea is to take a sequence (an ) in C 2 (Ω) and to find a solution of P(a, ϕ, Ω) as strong limit in W 1,1 (Ω) and pointwise almost everywhere of the sequence (un ) of maximal solutions of problems P(an , ϕ, Ω) given in (4.9) by the use of Theorem 3. We start by imposing suitable assumptions on the limit function a(·). Hypothesis 2. Let Ω ⊆ Rd be open and bounded and let a ∈ L∞ (Ω) be a nonnegative function. We say that a(·) satisfies Hypothesis 2 if there exists a sequence (an ) in C 2 (Ω) satisfying Hypothesis 1 such that lim an (x) = a(x) a.e. x ∈ Ω.

n→∞

(5.1)

Hypothesis 3. Assume Hypothesis 2 and require, in addition, that a(x) ≤ an (x) a.e. x ∈ Ω, ∀n ∈ N.

(5.2)

Theorem 4. Let Ω ⊆ Rd be open and bounded and ϕ ∈ W 1,∞ (Ω). 1. Let a ∈ L∞ (Ω) satisfy Hypothesis 2 and assume that the compatibility conditions (1.2) and (4.8) hold true for every n ∈ N. Then problem P(a, ϕ, Ω) admits at least one generalized solution. 2. Let a ∈ L∞ (Ω) satisfy Hypothesis 3 and assume that the compatibility conditions (1.2) and (4.8) hold true for every n ∈ N. Then the unique maximal element u(·) of S(a, ϕ, Ω) is a generalized solution of P(a, ϕ, Ω). If in addition the map a is assumed to be continuous, then u(·) is the unique viscosity solution of P(a, ϕ, Ω). Proof. Part 1. Take the sequence (an ), the corresponding Dirichlet problems P(an , ϕ, Ω) introduced in (4.9) and the sequence (un ) in ϕ + W01,∞ (Ω) of maximal (viscosity) solutions considered in Section 4. By Theorem 3 there exists u ∈ ϕ + W01,∞ (Ω) such that ∇un converges

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253

strongly and pointwise almost everywhere to ∇u in L1 (Ω) and almost everywhere. For every n ∈ N and for a.e. x ∈ Ω we have 2 1  ∇un (x) = an (x). 2

(5.3)

By convergence (4.10) and (5.1) we may pass to the limit n → ∞ in (5.3) obtaining 2 1  ∇u(x) = a(x) 2

for a.e. x ∈ Ω.

(5.4)

Hence u is a generalized solution. Part 2. Let u ∈ ϕ + W01,∞ (Ω) be the generalized solution of P(a, ϕ, Ω) provided by point 1. Recall (3.2) in Definition 4 and observe that condition (5.2) implies that S(a, ϕ, Ω) ⊆ S(an , ϕ, Ω) ∀n ∈ N. Hence, calling u the maximal element of S(a, ϕ, Ω), we have necessarily u ≤ un on Ω for every n ∈ N. On the other hand (5.4) implies that u ∈ S(a, ϕ, Ω), so that u(x) ≤ u(x) ≤ un (x) a.e. x ∈ Ω, ∀n ∈ N. Since un converges uniformly to u on Ω, we have necessarily u = u and the thesis is achieved. The last assertion is a trivial consequence of Theorems 1 and 2 in [12]. 2 6. A class of functions satisfying Hypotheses 2 and 3 In this section we exhibit a class of functions satisfying Hypotheses 2 and 3. Step 1. Take β ∈ R+ and define a sequence of continuous functions hn : R → R as follows:

t ∈ ]−∞, 0]:

. hn (t) =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ β n5 (t − 2

⎪ − β n5 (t ⎪ ⎪ ⎪ 2 ⎪ ⎩

0

t ∈ ]0, +∞]:

t ∈ ]−∞, − n1 − 1 n

−

−

1 ) n2 1 1 n + n2 )

t t t

1 ], n2 ∈ ]− n1 − n12 , − n1 ], ∈ ]− n1 , − n1 + n12 ], ∈ ]− n1 + n12 , 0],

hn (t) = −hn (−t).

Then, for any n ∈ N, construct the function fn : R → R of class C 2 such that fn (t) = 0 for t < 2 and fn = hn . By elementary computations we see that: ⎧ ⎪ ⎪ fn (t) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 ≤ fn (t) ≤ β ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ fn (t) = β

  1 1 t ∈ −∞, − − 2 , n n   1 1 1 1 t ∈ − − 2, + 2 , n n n n   1 1 t∈ + 2 , +∞ . n n

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It is immediate to see that n→∞

fn (t) −→ f (t)

a.e. t ∈ R,

where  . 0 f (t) = β

t ∈ ]−∞, 0], t ∈ [0, +∞[.

Now consider a bounded open interval I ⊆ R containing the point t = 0 and the sequence fn : I → R for n sufficiently large. Define:  δn (t) =

t ∈ I : |fn (t)| ≤ 1,

1 n

− 32

t ∈ I : |fn (t)| > 1.

We have   ⎧ ⎨ δn (t)fn (t) ≤ 1   ⎩ δn (t)fn (t) ≤ β n 32 2

  t ∈ I : fn (t) ≤ 1,   t ∈ I : fn (t) > 1.

Observe that         1 1 1 1 1 1 1 1 − 2, + 2 . t ∈ I : fn (t) > 1 ⊆ − − 2 , − + 2 ∪ n n n n n n n n Hence, by elementary computations:  

1 1 dt ≤ m1 (I ) + 4n− 2 , δn (t)

I

  1 δn (t)fn (t) dt ≤ m1 (I ) + 2βn− 2 .

I

Remark 4. We may define sequences (fˇn ) and (fˆn ) with the same properties of (fn ) and satisfying the following additional inequalities: fˇn (t) ≤ fn (t)

∀t ∈ R, ∀n ∈ N,

fˆn (t) ≥ fn (t)

∀t ∈ R, ∀n ∈ N.

Indeed it is sufficient to set   1 1 . ˇ fn (t) = fn t − − 2 , n n

  1 1 . ˆ fn (t) = fn t + + 2 ∀t ∈ R, ∀n ∈ N. n n

2

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255

Step 2. Consider a bounded open interval of R and a nonnegative simple function f with k jumps of the form f=

k

βi χIi ,

i=0

where βi ≥ 0 and {Ii , i = 0, . . . , k} is a finite family of semiopen disjoint subinterval of I covering I itself. With a simple generalization of the construction of previous step, and taking into account Remark 4, it is possible to construct a sequence (fn ) of functions belonging to C 2 (I ) and a sequence (δn ) of positive simple functions defined on I such that: fn (t) ≤ f (t) ≤ max{βi , i = 1, . . . , k} n→∞

fn (t) −→ f (t)

for a.e. t ∈ I, ∀n ∈ N,

for a.e. t ∈ I

and such that the following inequalities hold:  1 1 dt ≤ m1 (I ) + 4k · n− 2 , δn (t) 

I

  1 δn (t)fn (t) dt ≤ m1 (I ) + 2 max {βi } · k · n− 2 . i=1,...,k

I

Choosing n sufficiently large we have:  

1 dt ≤ m1 (I ) + 1, δn (t)

I

  δn (t)fn (t) dt ≤ m1 (I ) + 1.

I

We have so proved the following Lemma 9. Let I be an open bounded interval of R and let s(·) be a nonnegative simple function defined on I . Then, given  > 0, there exist a simple function δ(·) defined on I and a function s˜ ∈ C 2 (I ) such that: 0 < δ(t) ≤ 1 for a.e. t ∈ I ; s˜ (t) ≤ s(t) ≤ s∞   I

for a.e. t ∈ I ;

s − s˜ L1 (I ) ≤ ; 1 dt ≤ m1 (I ) + 1, δ(t)

I

  δ(t)s˜ (t) dt ≤ m1 (I ) + 1.

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Step 3. Let now I1 , . . . , Id be open bounded intervals of R and let Ω ⊆ Rd be the polyinterval

×I . d

Ω=

j

j =1

For any j take a nonnegative simple function sj ∈ S(Ij ), the element g of g(x) = g(x1 , . . . , xd ) =

d 

d

j =1 S(Ij )

given by

sj (xj )

j =1

and take a positive constant b > 0 such that 0 ≤ sj (t) ≤ b

∀t ∈ Ij , ∀j = 1, . . . , d.

(6.1)

Fix  > 0. Applying Lemma 9 we may find simple functions δj and functions s˜j ∈ C 2 (I j ) such that, ∀j = 1, . . . , d: 0 < δj (t) ≤ 1 for a.e. t ∈ Ij ; s˜j (t) ≤ sj (t) ≤ b

for a.e. t ∈ Ij ;

(6.3)

˜sj − sj L1 (Ij ) ≤ ;

(6.4)

1 dt ≤ 1 + m1 (Ij ); δj (t)

(6.5)

  δj (t)s˜j (t) dt ≤ 1 + m1 (Ij ).

(6.6)





(6.2)

Ij

Ij

Now define .  g(x) ˜ = g(x ˜ 1 , . . . , xd ) = s˜ (xj ), d

x ∈ Ω.

(6.7)

j =1

We have g˜ ∈

d

j =1 C

2 (I

j)

and, as a consequence of (6.1) and (6.3), g(x) ˜ ≤ g(x) ≤ bd

for a.e. x ∈ Ω,

(6.8)

while (6.4) implies that g − g ˜ L1 (Ω) ≤ bd−1 P ,

(6.9)

  . P = max m1 (Ij ), i = 1, . . . , d .

(6.10)

where we have set

j =i

S. Zagatti / J. Differential Equations 257 (2014) 231–263

257

For every i ∈ {1, . . . , d} we may compute the second derivatives of g, ˜ obtaining Di2 g(x) ˜ = Di2 g(x ˜ 1 , . . . , x i , . . . , xd ) =



 s˜ (xj ) s˜ (xi ),

x ∈ Ω.

j =i

Recalling Definition 2.6, we have then   A(g, ˜ x, r, 0, i) ≤ bd−1 s˜i (xi ),

∀x ∈ Ω, ∀r > 0.

(6.11)

By abuse of notations, and recalling (6.2), we define the positive simple functions  .  δi (x) = δ xi , xi = δi (xi ),

∀x ∈ Ω,

(6.12)

and, recalling (6.5) and (6.10), we have  Ω

  1 dx ≤ 1 + m1 (Ii ) P . δi (x)

(6.13)

In addition, as a consequence of (6.6) and (6.11), we have 

      δi (t)A g, ˜ xi , t , r, 0, i dt ≤ bd−1 1 + m1 (Ii )

∀r > 0

Ii

and then 

  δi (x)A(g, ˜ x, r, 0, i) dt ≤ bd−1 1 + m1 (Ii ) P

∀r > 0.

(6.14)

Ω

Step 4. Let now g be a nonnegative element of L( dj =1 Sj (Ij )) written in the following form g=

N

λk gk =

k=1

N

λk

k=1

d 

sk,j ,

(6.15)

j =1

where λk ≥ 0, sk,j ∈ S(Ij ) for every j = 1, . . . , d and for every k = 1, . . . , N . Let b > 0 and L > 0 such that 0 ≤ sk,j ≤ b

∀j = 1, . . . , d, ∀k = 1 . . . , N;

(6.16)

and N

k=1

λk ≤ L.

(6.17)

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

Fix  > 0. Applying previous step, for every k = 1, . . . , N we may construct a map g˜ k ∈ C 2 (Ω) satisfying all properties (6.2)–(6.14). In particular, by (6.8), we have g˜ k (x) ≤ gk (x) ≤ bd

for a.e. x ∈ Ω ∀k = 1, . . . , N

(6.18)

and, by (6.9), we may impose g˜ k − gk L1 (Ω) ≤ bd−1 P

 2k

∀k = 1, . . . , N.

(6.19)

Define . g˜ = λk g˜ k . N

(6.20)

k=1

Clearly g˜ belongs to C 2 (Ω) and, by (6.8), (6.15), (6.18), (6.20) and (6.17), we have g(x) ˜ ≤ g(x) ≤ L · bd

for a.e. x ∈ Ω,

(6.21)

while, combining (6.9), (6.17), (6.19) and (6.20), we have g˜ − gL1 (Ω) ≤ bd−1 P L.

(6.22)

For every i ∈ {1, . . . , d} we compute second derivatives: Di2 g(x ˜ 1 , . . . , xi , . . . , xd ) =

N

λk



k=1

 s˜k,j (xj ) s˜k,i (xi ),

j =i

so that, recalling (6.11), we have A(g, ˜ x, r, 0, i) ≤ bd−1

N

  λk s˜k,i (xi ),

∀x ∈ Ω, ∀r > 0.

(6.23)

k=1

For every i = 1, . . . , d and k = 1, . . . , N we may construct positive simple functions δk,i as in previous step and then define the strictly positive simple functions δi by setting   . δi (xi ) = min δk,i (xi ), k = 1, . . . , N ,

xi ∈ Ii .

(6.24)

By the same abuse of notation of (6.12) we define the positive simple functions  .  δi (x) = δ xi , xi = δi (xi ),

∀x ∈ Ω,

(6.25)

obtaining the same result of formula (6.13):  Ω

  1 dx ≤ 1 + m1 (Ii ) P δi (x)

∀i = 1, . . . , d.

(6.26)

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259

In addition, recalling (6.14), (6.17) and (6.23), for every i = 1, . . . , d and for every r > 0, we have: 

N

      δi (t)A g, ˜ xi , t , r, 0, i dt ≤ λk · bd−1 1 + m1 (Ii ) k=1

Ii

  ≤ L · bd−1 1 + m1 (Ii ) .

Hence, for every i = 1, . . . , d and for every r > 0: 

  δi (x)A(g, ˜ x, r, 0, i) dt ≤ L · bd−1 1 + m1 (Ii ) P .

(6.27)

Ω

Step 5. Let  > 0, g(·) ˜ and δi (·) (i = 1, . . . , d) as in Step 4. Define the functions . ˜ i (x) = G

  1 1 ˜ x, Mδin (x) 2 , 0, i , + δi (x)A g, δi (x)

x ∈ Ω, i = 1, . . . , d

and . ˜ ˜ G(x) = Gi (x), d

x ∈ Ω.

i=1

By formulas (6.17), (6.24), (6.26), (6.25), (6.20) and (6.27) we have 

˜ G(x) dx ≤

d



  1 + m1 (Ii ) 1 + Lbd−1 P .

(6.28)

i=1

Ω

Inequality (6.21) says that g˜ ≤ g and is bounded by a constant depending only on g, while (6.22) means that g˜ can be constructed arbitrarily close to g in L1 (Ω) norm and almost everywhere. Remarking that the r.h.s. in (6.28) does not depend on , we formulate the following hypothesis which constitutes the assumption of the subsequent results and provides a class of nonnegative functions in L∞ satisfying Hypotheses 2–3.

×d

Definition 5. Let Rd ⊇ Ω = j =1 Ij , where I1 , . . . , Id are open bounded intervals of R. Let ∞ f be in L1 (Ω)-norm of the space da nonnegative element of L (Ω) belonging to the closure ∞ L( j =1 S(Ij )). We say that f belongs to the cone Z(Ω) ⊆ L (Ω) if the following conditions hold true. There exist two positive constants M and L and a sequence (fn ) in L( dj =1 S(Ij )) of the form fn =

n

k=1

λnk fkn =

n

k=1

λnk

d  j =1

n fk,j

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

such that n→∞

fn (x) −→ f (x) n 0 ≤ fk,j

λnk ≥ 0

≤M

for a.e. x ∈ Ω;

∀j = 1, . . . , d, ∀k = 1, . . . , n, ∀n ∈ N;

∀k = 1, . . . n, ∀n ∈ N

and

n

λnk ≤ L ∀n ∈ N.

k=1

Then we have

×d

Theorem 5. Let Rd ⊇ Ω = j =1 Ij , where I1 , . . . , Id are open bounded intervals of R and f ∈ Z(Ω). Then f satisfies Hypothesis 2. Proof. As we have seen in Steps 1–5 we may construct a sequence (fˇn ) in C 2 (Ω) such that fˇn ≤ fn ≤ f for every n. All properties required in Hypothesis 1 turn out to be satisfied. 2 Remark 5. Since f is defined as a series of nonnegative elements, the sequences (fˇn ) and (fn ) approximate f necessarily from below. Hence our construction seems to exclude maps f satisfying Hypothesis 3 which requires the existence of a regular sequence approximating f from above. Observe however that given a function g ∈ C 2 (Ω), the sequence given by gn ≡ g for every n ∈ N satisfy Hypothesis 1 for any choice of the maps δin . In addition, if f belongs to Z(Ω), f ≤ g almost everywhere and (fˇn ) is the C 2 -approximating sequence of f provided by Theorem 5, we have C 2 (Ω)  g(x) − fˇn (x) ≥ g(x) − f (x) for a.e. x ∈ Ω, ∀n ∈ N and g(x) − fˇn (x) −→ g(x) − f (x) n→∞

for a.e. x ∈ Ω.

Then we obtain immediately the following Theorem 6. Let Rd ⊇ Ω =

×dj=1 Ij , where I1, . . . , Id are open bounded intervals of R. Let

f ∈ Z(Ω) and g ∈ C 2 (Ω) be such that f ≤ g almost everywhere in Ω. Then g − f satisfies Hypothesis 3. 7. An alternative proof of the strong compactness property In Section 4 we have proved a strong compactness property in W 1,r (1 ≤ r < ∞) using a distributional inequality involving Laplace operator and Sobolev embeddings. In this appendix we show that the same property can be obtained by elementary computations. We start with a Lemma 10. Let E ⊆ Rd be open, bounded and convex and let (vn ) be a sequence in W 1,∞ (E) such that

S. Zagatti / J. Differential Equations 257 (2014) 231–263 ∗

vn  v

261

in W 1,∞ (E).

Suppose that for every j ∈ {1, . . . , d} and for almost every xj ∈ Ejd−1 , the sequence (vn,x ) is j

relatively compact in L1 (Jxj ). Then n→∞

vn −→ v

strongly in W 1,r (E) ∀r ∈ [1, ∞[.

. Proof. Assume E = Qq = ]0, 1[d and, by the substitution vn ←→ vn − v, v = 0. First of all remark that the sequence (vn ) converge uniformly to zero on E and, consequently, for every j ∈ {1, . . . , d} and for every xj ∈ Ejd−1 , the sequence vn,xj converges uniformly to zero 1 on Jxj . Hence, by relative compactness of the sequence of derivatives (vn,x ) in L (Jx ), the j j

sequence vn,xj converges strongly to zero in W 1,1 (Jxj ) for every j ∈ {1, . . . , d} and for almost every xj ∈ Ejd−1 . Suppose, by contradiction, that there exists a positive ρ such that 

  ∇vn (x) dx ≥ ρ.

Qd

Necessarily there exist a positive η, an index j ∈ {1, . . . , d} and a subsequence, still denoted by (vn ), such that 

  Dj vn (x) dx ≥ η,

Qd

that is to say  Qd−1

1

  Dj vn (x1 , . . . , xj −1 , t, xj +1 , . . . , xd ) dt dx ≥ η > 0. j

(7.1)

0

It follows that there exists a subset G of Qd−1 of positive measure md−1 (G) > 0 and, for almost every xj ∈ G, positive numbers α(xj ) > 0 such that 1

  Dj vn (x1 , . . . , xj −1 , t, xj +1 , . . . , xd ) dt =

0

1

 v

    dt ≥ α x > 0.

n,xj (t)

0

Indeed, if not, we would have, for almost every xj ∈ Qd−1 , 1 lim

n→0 0

  Dj vn (x1 , . . . , xj −1 , t, xj +1 , . . . , xd ) dt = 0,

j

(7.2)

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S. Zagatti / J. Differential Equations 257 (2014) 231–263

and then, remembering the boundedness of the integrands and by dominated convergence,  lim

1

n→0 Qd−1 0

  Dj vn (x1 , . . . , xj −1 , t, xj +1 , . . . , xd ) dt dx = 0, j

contradicting (7.1). By assumption the sequences (vn,xj ) converges strongly to zero in W 1,1 (0, 1) for almost every

xj ∈ Ejd−1 and then inequalities (7.2) give the required contradiction. The strong convergence for r > 1 follows trivially by interpolation. The general case of an arbitrary open, bounded, convex subset can be easily obtained by extension or invoking Vitali covering lemma. 2 Remark 6. It is obvious that Lemma 10, by the aid of covering theorems, can be generalized to arbitrary open subsets of Rd . Consider now the following slightly different version of Hypothesis 1: Hypothesis 4. In the notations of Hypothesis 1 we say that the sequence (an ) satisfies Hypothesis 4 if it satisfies Hypothesis 1 with the only change that condition (iii) is replaced by the following one: (iii)’ setting . Gni (x) =

1 δin (x)

  1 + δin (x)A an , x, Mδin (x) 2 , 0, i

a.e. x ∈ Λγ ,

(7.3)

we have  Gni,x (t) dt ≤ C i

for a.e. xi ∈ (Λγ )d−1 ∀n ∈ N. i

(7.4)

Jx i

Remark 7. In (7.4) we have used special notations (2.2)–(2.4). In simple words the only difference between Hypothesis 1 and 4 is that in Case 1 the uniform bound on the integral of the functions Gni is on the whole set Λγ , while in Case 4 it is imposed line by line almost everywhere. It follows from Fubini Tonelli theorem that Hypothesis 4 implies Hypothesis 1. We may now state the equivalent of Theorem 3. Theorem 7. Let (an ) be a sequence satisfying Hypothesis 4 and assume that compatibility conditions (4.8) hold true for every n ∈ N. Let (un ) be the sequence of maximal viscosity solutions of P(an , ϕ, Ω). Then there exists a subsequence, still denoted by (un ), and a map u ∈ ϕ +W01,∞ (Ω) n→∞

such that un −→ u strongly in W 1,r (Ω) for every r ∈ [1, ∞[ and n→∞

∇un (x) −→ ∇u(x) for a.e. x ∈ Ω.

S. Zagatti / J. Differential Equations 257 (2014) 231–263

263

Proof. As in the proof of Theorem 3 there exists u ∈ ϕ + W01,∞ (Ω) and a subsequence, still ∗

denoted by (un ), such that un  u in W 1,∞ (Ω) and uniformly on Ω. Let now Λ be any open, bounded, convex subset of Ω with smooth boundary; let γ > 0 and consider the set Λγ defined as in (3.4). From Lemma 8 we take inequality (3.35), according to which, for every i ∈ {1, . . . , d}, we have ∂ 2 un ≤ Gni ∂xi2

in D (Λγ ).

(7.5)

Given an index i ∈ {1, . . . , d} and recalling definition (2.4), consider the set (Λγ )d−1 and look i to inequality (7.5) line by line. Recall definitions (2.2) and (2.3) and, for every i ∈ {1, . . . , d}, for , define the following functions: almost every xi ∈ (Λγ )d−1 i .   H n (t) = Gni x (t) t ∈ Jxi ; i . vn (t) = un,xi (t) t ∈ Jxi , where we omit for convenience the dependence on i. Setting J = Jxi , formula (7.5) takes the form vn ≤ H n

in D (J ) ∀n ∈ N.

(7.6)

We may apply the one-dimensional version of Lemma 4 and, by virtue of hypotheses (7.3)–(7.4), we obtain from (7.6) that the sequence (vn ) is relatively compact in W 1,r (J ) for every r ∈ [1, ∞[. Since this holds for almost every xi ∈ (Λγ )d−1 and for every i ∈ {1, . . . , d}, we achieve the proof i of the theorem by Lemma 10 and by the arbitrariness of Λ and γ . 2 References [1] G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Math. Appl., vol. 17, Springer-Verlag, Berlin, 1994. [2] M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Birkhäuser, Boston, 1997. [3] P. Cannarsa, C. Sinestrari, Semiconcave Functions, Hamilton–Jacobi Equations and Optimal Control, Birkhäuser, Basel, 2004. [4] B. Dacorogna, P. Marcellini, Implicit Partial Differential Equations, Birkhäuser, Basel, 1999. [5] M. Garavello, P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running costs, NoDEA Nonlinear Differential Equations Appl. 11 (2004) 271–298. [6] Y. Giga, P. Gorka, P. Rybka, A comparison principle for Hamilton–Jacobi equations with discontinuous Hamiltonians, Proc. Amer. Math. Soc. 139 (2011) 1777–1785. [7] H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55 (1987) 369–384. [8] R. Lucchetti, Convexity and Well Posed Problems, Springer-Verlag, New York, 2006. [9] D.N. Ostrov, Viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape-fromshading equations with discontinuous intensities, SIAM J. Appl. Math. 59 (1999) 2060–2085. [10] P.L. Lions, Generalized Solutions of Hamilton–Jacobi Equations, Res. Notes Math., vol. 69, Pitman, London, 1982. [11] P. Soravia, Boundary value problems for Hamilton–Jacobi equations with discontinuous Lagrangian, Indiana Univ. Math. J. 51 (2002) 451–476. [12] S. Zagatti, On viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 361 (1) (2009) 41–59. [13] S. Zagatti, An integro-extremization approach for non coercive and evolution Hamilton–Jacobi equations, J. Convex Anal. 18 (4) (2011).