THEORY PROBAB. APPL.Vol. 45, No. 4

ROUGH BOUNDARY TRACE FOR SOLUTIONS OF Lu = ψ(u)∗

E. B. DYNKIN† AND S. E. KUZNETSOV‡

Abstract. Let L be a second order elliptic differential operator in Rd and let E be a boundeddomain in Rd with smooth boundary ∂E. A pair (Γ, ν) is associated with every positive solutionof a semilinear differential equation Lu = ψ(u) in E, where Γ is a closed subset of ∂E and ν is aRadon measure on O = ∂E \ Γ. We call this pair the rough trace of u on ∂E. (In [E. B. Dynkinand S. E. Kuznetsov, Comm. Pure Appl. Math., 51 (1998), pp. 897–936], we introduced a fine traceallowing us to distinguish solutions with identical rough traces.)

The case of ψ(u) = uα with α > 1 was investigated using various methods by Le Gall, Dynkin,and Kuznetsov and by Marcus and Veron. In this paper we cover a wide class of functions ψ andsimplify substantially the proofs contained in our earlier papers.

Key words. boundary trace of a solution, moderate solutions, sweeping, removable and thinboundary sets, stochastic boundary value, diffusion, range of superdiffusion

PII. S0040585X97978579

1. Conditions on L and ψ. Suppose that E is a bounded domain of class C2,λ

in Rd and

Lu =∑i,j

aij∂2u

∂xi∂xj+∑i

bi∂u

∂xi

is a uniformly elliptic operator with bounded smooth coefficients. We assume that a func-tion ψ(x, u) on Rd × R+ has the following properties:

1.A. For every x, ψ(x, ·) is convex and ψ(x, 0) = 0, ψ(x, u) > 0 for u > 0.1.B. ψ(x, u) is continuously differentiable.

1.C. ψ is locally Lipschitz continuous in u uniformly in x; i.e., for every c > 0, thereexists a constant q(c) such that∣∣ψ(x; u1)− ψ(x; u2)

∣∣ � q(c)|u1 − u2| for all x ∈ E, u1, u2 ∈ [0, c].(1.1)

1.D. There is a constant a such that ψ(x, 2u) � aψ(x, u) for all x and u.We study the set U of all positive functions of class C2 on E such that

Lu = ψ(u) in E.(1.2)

In addition to 1.A–1.C, we suppose that all positive bounded solutions of (1.2) in any sub-domain D of E are uniformly bounded. Keller [8] and Osserman [9] have shown that thisproperty holds if a certain integral involving ψ is finite. All conditions listed in this sectionhold for ψ(x, u) = k(x)uα under mild conditions on k(x) if α > 1.

We define a boundary trace of u ∈ U by using two tools: (a) trace of moderate solutions;(b) sweeping of solutions.

∗Received by the editors July 26, 2000. This work was supported by National Science Foundationgrants DMS-9970942 and DMS-9971009.

http://www.siam.org/journals/tvp/45-4/97857.html†Department of Mathematics, Cornell University, Ithaca, NY 14853 (ebd1@cornell.edu).‡Department of Mathematics, University of Colorado, Boulder, CO 80309-0395

(Sergei.Kuznetsov@Colorado.edu).

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ROUGH BOUNDARY TRACE FOR SOLUTIONS OF Lu = ψ(u) 663

2. Trace of moderate solutions. Two linear operators play a key role in ourinvestigation: the Green operator G and the Poisson operator K. They can be defined byprobabilistic formulae

Gf(x) = Πx

∫ ζ

0

f(ξt) dt, Kf(x) = Πxf(ξζ),

where ξ = (ξt,Πx) is the diffusion in E with the generator L and ζ is the first exit time of ξfrom E.

Denote by H the set of all positive solutions of the linear equation Lu = 0 in E. Wecall elements of H harmonic functions.

A function u ∈ U is called a moderate solution if u � h for some h ∈ H. Denote by U1

the set of all moderate solutions. We have proved in [4] that the formula

h = u+Gψ(u)(2.1)

defines a one-to-one mapping i from U1 onto a subset H1 of H. We denote the inversemapping by j. For every u ∈ U1, h = j(u) is the minimal harmonic majorant of u and, forevery h ∈ H1, u = i(h) is the maximal element of U dominated by h. Clearly, mappings iand j are monotone. Moreover, i(h1 + h2) � i(h1) + i(h2) (see section 4.3 in [6]).

Every h ∈ H has a representation

h(x) =

∫∂E

k(x, y) ν(dy),(2.2)

where k(x, y) is the Poisson kernel and ν is a finite measure on ∂E. Formula (2.2) establishesa one-to-one correspondence between the set of all finite measures on ∂E and H. If ν and hare related by formula (2.2), then we write h = Kν. Put ν ∈ N1 if Kν ∈ H1.

We say that ν is the trace of a moderate solution u if

u+Gψ(u) = Kν.(2.3)

Formula (2.3) defines a one-to-one correspondence between N1 and U1.

3. Sweeping. It is proved in [6] that U with the partial order � is a complete lattice;that is, for every subset C of U there exist SupC and Inf C in U (writing v = SupCmeans that (i) v ∈ U ; (ii) u � v for all u ∈ U ; (iii) if w ∈ U and u � w for all u ∈ U ,then v � w. Writing v = Inf C has a similar meaning). For every pair u, v ∈ U , we putu∨ v = Sup{u, v}, u∧ v = Inf{u, v}. We proved in [6] that, if C ⊂ U is closed under ∨, thenthere exists a sequence un ∈ C such that un(x) ↑ v(x) for all x ∈ E. We have v(x) = supC u(x)for all x ∈ E.

We use this result to introduce a sweeping QB(u) of u ∈ U to a closed subset B of ∂E.We put

QB(u) = Sup{v ∈ U : v � u, v = 0 on ∂E \B}.(3.1)

Operators QB have the following properties (see [7, section 3]):3.A. QB(u1) � QB(u2) for u1 � u2.3.B. QB(u) � u.3.C. If B1 ⊃ B2, then QB1(u) � QB2(u).3.D. For every B1, B2, QB1∪B2(u) � QB1(u) +QB2(u).3.E. Q∂E(u) = u.3.F. If u � u1 + u2, then QB(u) � QB(u1) +QB(u2).3.G. If u is a moderate solution with the trace ν, then v = QB(u) is a moderate solution

with the trace equal to the restriction νB of ν to B.Indeed, 3.A–3.C and 3.E follow easily from the definition; 3.F and 3.G could be estab-

lished by the arguments similar to those in [7]; and 3.D can be derived from [6, Lemma 5.2].

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664 E. B. DYNKIN AND S. E. KUZNETSOV

4. Rough trace. Now we are prepared to define a rough trace of an arbitrary u ∈ U .We say that a compact set B ⊂ ∂E is moderate for u if the solution uB = QB(u) ismoderate. Let νB stand for the trace of uB . By 3.D, the union of two moderate sets ismoderate. Suppose that B is moderate and let B ⊂ B. By 3.G, B is moderate and ν

Bis

the restriction of νB to B.A relatively open subset A of E′ is called moderate if all compact subsets of A are

moderate. The union O of all moderate open sets is moderate. Clearly, there exists a uniquemeasure ν on O such that its restriction to an arbitrary compact subset B coincides with νB .

The measure ν has the property that for every compact B ⊂ O the restriction of ν to Bbelongs to the class N1. We denote by N1(O) the class of measures with this property. Wecall closed set Γ = ∂E \O the special set of the solution u and we call pair (Γ, ν) the roughtrace of u on ∂E.

5. Solutions uν and wB. The class U is closed under pointwise convergence. More-over, the class of solutions vanishing on a relatively open subset O of ∂E is also closed underpointwise convergence (see, for instance, [2, Theorem 1.2]). Suppose that Bn are closed andBn ↑O. If ν ∈ N1(O) and, if νn is the restriction of ν to Bn, then νn ∈ N1 and uνn is anincreasing sequence. We denote its limit by uν . It is easy to see that it does not depend onthe choice of Bn and that uµ � uν if and only if µ � ν.

For every closed B ⊂ ∂E, we put

wB = Sup{u ∈ U : u = 0 on ∂E \B}.(5.1)

Note that, for every x ∈ E, wB(x) is equal to the supremum of u(x) over all u ∈ U . We havethe following:

5.A. wB1 � wB2 if B1 ⊂ B2.5.B. QB(u) � wB for all u ∈ U .5.C. QB(wB) = wB .5.D. wB1∪B2 � wB1 + wB2 .5.E. If Bn ↓B, then wBn ↓wB .Items 5.A and 5.B follow from (3.1). By 5.B, QB(wB) � wB . On the other hand, v = wB

satisfies conditions v � wB , v = 0 on ∂E \B, and QB(wB) is a maximal solution with theseproperties. Hence, wB � QB(wB) which proves 5.C. To prove 5.D, we put B = B1 ∪B2 andwe note that, by 5.C, 3.D, and 5.B,

wB = QB(wB) � QB1(wB) +QB2(wB) � wB1 + wB2 .

Let us prove 5.E. Function wn = wBn is a maximal element of U vanishing on On = ∂E \Bn.By 5.A, wn ↓ v � wB . The limit v is a solution equal to 0 on O = ∂E \B. Hence v � wB .

6. Removable and thin boundary sets. We say that a compact set B ⊂ ∂E isremovable if 0 is the only element of U vanishing on ∂E \ B. (In the literature, such setsare called removable boundary singularities for solutions of (2.1).) Clearly, B is removableif and only if wB = 0. A set A is called thin if all its compact subsets are removable.1 Itfollows from 5.A and 5.D that

6.A. all closed subsets of a removable set are removable and all subsets of a thin set arethin.

6.B. the class of all removable sets and the class of all thin sets are closed under thefinite unions.

We also have that6.C. all thin Borel sets are not charged by any ν ∈ N1.Proof. Since N1 contains with every ν its restriction to any B, it is sufficient to show

that, if ν ∈ N1 is concentrated on a removable compact set B, then ν = 0. The Poisson kernel

1If X is an (L,ψ)-superdiffusion, then, by Theorem 1.2 in [5], a compact set B is removable ifand only if Px{R ∩B �= ∅} = 0, where R is the range of X.

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ROUGH BOUNDARY TRACE FOR SOLUTIONS OF Lu = ψ(u) 665

k(x, y) is bounded on every set {|x− y| > ε}, ε > 0, and it tends to 0 as y → y ∈ ∂E, y �= y.Therefore h = Kν = 0 on ∂E \ B. The solution i(h) satisfies the same condition becausei(h) � h. Therefore i(h) � wB . If wB = 0, then i(h) = 0. Thus h = 0 and ν = 0.

6.D. If compact B ⊂ Γ and if QB(wΓ) is moderate, then B is removable.Proof. By 5.A, wB � wΓ and, by 5.C and 3.A, wB = QB(wB) � QB(wΓ). If QB(wΓ)

is moderate, then so is wB . Hence, wB = i(h) for some h ∈ N1. By Theorem 4.3 in [6],2h ∈ N1.

2 Since i(2h) � 2i(h), we have v � 2wB . Hence, v = 0 on ∂E \ B which impliesthat v � wB . By the monotonicity of j, 2h = j(v) � j(wB) = h. Therefore h = 0 andwB = i(h) = 0.

6.E. Suppose Γ is removable and Bn = {x ∈ ∂E : d(x,Γ) � εn}. If εn ↓ 0, thenQBn(u) ↑u for every u ∈ U .

Proof. Put Γn = {y ∈ ∂E : d(y,Γ) � 2εn}. Note that Γn∪Bn = ∂E and Γn ↓Γ. By 3.E,3.D, and 5.D, u = Q∂E(u) � QΓn(u) +QBn(u). By 5.B and 5.E, QΓn(u) � wΓn ↓wΓ = 0.

7. Principal results. We say that x is an explosion point of a measure ν and wewrite x ∈ Ex(ν) if ν(U) = ∞ for every neighborhood U of x. If B ∩ Ex(ν) = ∅ and if B iscompact, then ν(B) <∞. Note that O ∩ Ex(ν) = ∅ for every measure ν ∈ N1(O).

We say that (Γ, ν) is a normal pair if(a) Γ is a closed subset of ∂E;(b) ν ∈ N1(O), where O = ∂E \ Γ;(c) the following conditions hold:

Λ ⊂ Γ is thin and contains no explosion points of ν,

Γ \ Λ is closed, implying that Λ = ∅.(7.1)

We demonstrate that these conditions hold for the trace of an arbitrary solution u. First,we prove a few auxiliary propositions.

7.A. If u ∈ U vanishes on a compact set B ∈ ∂E, then QB(u) = 0.Indeed, by (3.1), v = QB(u) = 0 on ∂E \B and v � u. Hence, v = 0 on ∂E, and v = 0

by the comparison principle (see, e.g., [1, p. 113]).7.B. Suppose that tr (u) = (Γ, ν). If u = 0 on an open subset O1, then O1 ∩ Γ = ∅ and

ν(O1) = 0.Indeed, by 7.A, QB(u) = 0 for all compact subsets of O1.7.C. Let tr (u) = (Γ, ν). If Γ is removable and ν is finite, then u is moderate.To prove this, we apply 6.E. Note that h = Kν ∈ H and un = QBn(u) � Kνn � h.

By 6.E, un ↑u. Hence, u � h.Theorem 7.1. The trace of an arbitrary solution u is a normal pair.Proof. Properties (a) and (b) follow immediately from the definition of the trace. Let us

prove (c). Suppose that Λ satisfies the conditions (7.1) and let Γ0 = Γ \Λ. The theorem willbe proved if we show that v = QB1(u) is moderate for every closed subset B1 of O0 = ∂E\Γ0.Indeed, this implies O0 ⊂ O and therefore Γ0 ⊃ Γ, Λ = ∅.

Let (Γ1, ν1) be the trace of v. By 7.C, it is sufficient to prove that Γ1 is removable and ν1is finite.

By 3.B, v � u and, by 3.A, all moderate sets for u are also moderate for v. HenceΓ1 ⊂ Γ. By 7.B, ∂E \B1 ⊂ ∂E \ Γ1 because v = 0 on ∂E \B1. Hence, ∂E \B1 is moderatefor v and is contained in O1 = ∂E \ Γ1. We conclude that Γ1 ⊂ B1 ∩ Γ ⊂ Λ. Hence, Γ1 isremovable.

Note that B1 ⊂ O ∪Λ does not contain explosion points of ν and therefore ν(B1) <∞.Since QB(v) = 0 for B ∩ B1 = ∅, measure ν1 vanishes on ∂E \ B1. Since v � u, ν1 � ν onO ⊂ O1. We have

ν1(O1) = ν1(O1 ∩B1) = ν1[(O1 \O) ∩B1

]+ ν1(O ∩B1) � ν(O ∩B1) <∞

because Γ ∩B1 is removable and therefore ν1(Γ ∩B1) = 0 by 6.C. Theorem 7.1 is proved.

2Here is the only place we use the property 1.D of ψ.

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666 E. B. DYNKIN AND S. E. KUZNETSOV

For every u, v ∈ U we put u⊕ v = Sup{u ∈ U : u � u+ v}. Note that uµ ⊕ uν = uµ+ν .Theorem 7.2. If (Γ, ν) is a normal pair, then u = wΓ ⊕ uν is a solution with the trace

(Γ, ν). Moreover, u is maximal among such solutions.Proof. 1◦. Let B ⊂ O = ∂E\Γ. We claim that QB(u) = uνB , where νB is the restriction

of ν to B. Indeed, B is contained in an open subset O1 of ∂E such that O1 ⊂ O. Let ν1 andν2 be the restrictions of ν to O1 and to O \O1. Note that uν = uν1 ⊕ uν2 . By 3.D,

QB(u) � QB(uν) +QB(wΓ), QB(uν) � QB(uν1) +QB(uν2).

Since ν2 does not charge a neighborhood of B, uν2(B) = 0. The same is true for wB .By 7.A, QB(wΓ) = QB(uν2) = 0. Hence, QB(u) � QB(uν1) and, by 3.A and 3.G, QB(u) =QB(uν1) = uνB .

2◦. Denote the trace of u by (Γ0, ν0). It follows from 1◦ that O ⊂ O0 and ν = ν0 on O0.Since ν is concentrated on O, we have ν � ν0 and therefore Ex(ν) ⊂ Ex(ν0) ⊂ Γ0 ⊂ Γ. Everycompact B ⊂ Λ = Γ \ Γ0 is moderate for u (because B ⊂ O0) and it is removable by 6.D.Thus, Λ is thin. Since Λ ∩ Ex(ν) = ∅ and Γ \Λ = Γ0 is closed, Λ = ∅ by the definition of anormal pair.

3◦. Let us show that an arbitrary solution v with the trace (Γ, ν) is dominated byu = wΓ ⊕ uν . Since u is the maximal element of U dominated by wΓ + uν , it is sufficientto show that v � wΓ + uν . Consider compact sets Bn and Γn such that Bn ↑O, Γn ↓Γ, andBn ∪ Γn = ∂E for all n. By 3.E and 3.D,

v = Q∂E(v) � QBn(v) +QΓn(v).(7.2)

Let νn be the restriction of ν to Bn. Note that QBn(v) = uνn � uν . By 5.B, QΓn(v) � wΓn

and, by 5.E, wΓn ↓wΓ. Therefore (7.2) implies that v � uν + wΓ. Theorem 7.2 is proved.Remark. If X is a (L,ψ)-superdiffusion, then

(wΓ ⊕ uν)(x) = − logPx{R ∩ Γ = ∅, e−Zν},(7.3)

where R is the range of X and Zν is the stochastic boundary value of h = Kν (see [3]).

REFERENCES

[1] E. B. Dynkin, A probabilistic approach to one class of nonlinear differential equations, Probab.Theory Related Fields, 81 (1991), pp. 89–115.

[2] E. B. Dynkin, Superdiffusions and parabolic nonlinear differential equations, Ann. Probab., 20(1992), pp. 942–962.

[3] E. B. Dynkin, Stochastic boundary values and boundary singularities for solutions of the equa-tion Lu = uα, J. Funct. Anal., 153 (1998), pp. 147–186.

[4] E. B. Dynkin and S. E. Kuznetsov, Solutions of Lu = uα dominated by L-harmonic functions,J. Anal. Math., 68 (1996), pp. 15–37.

[5] E. B. Dynkin and S. E. Kuznetsov, Superdiffusions and removable singularities for quasi-linear partial differential equations, Comm. Pure Appl. Math., 49 (1996), pp. 125–176.

[6] E. B. Dynkin and S. E. Kuznetsov, Fine topology and fine trace on the boundary associatedwith a class of quasilinear differential equations, Comm. Pure Appl. Math., 51 (1998),pp. 897–936.

[7] E. B. Dynkin and S. E. Kuznetsov, Trace on the boundary for solutions of nonlinear differ-ential equations, Trans. Amer. Math. Soc., 350 (1998), pp. 4499–4519.

[8] J. B. Keller, On the solutions of ∆u = f(u), Comm. Pure Appl. Math., 10 (1957), pp. 503–510.

[9] R. Osserman, On the inequality ∆u � f(u), Pacific J. Math., 7 (1957), pp. 1641–1647.

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