www.elsevier.com/locate/anihpc
Superharmonic functions are locally renormalized solutions
Tero Kilpeläinen
a,∗, Tuomo Kuusi
b, Anna Tuhola-Kujanpää
aaDepartment of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland bAalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland
Received 22 November 2010; accepted 9 March 2011 Available online 23 March 2011
Dedicated to Peter Lindqvist on the occasion of his 60th birthday
Abstract
We show that different notions of solutions to measure data problems involvingp-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.
©2011 Elsevier Masson SAS. All rights reserved.
MSC:35J92; 35A01
1. Introduction
Consider elliptic quasilinear type equations
−div
A(x, Du)
=μ, (1.1)
in an open setΩ⊂Rn, whereμis a nonnegative Radon measure and the operator div(A(x, Du))is a measurable perturbation of thep-Laplacian operator
pu=div
|Du|p−2Du
, 1< pn.
The natural domain of definition for the operator div(A(x, Du))isWloc1,p(Ω). Then, however,u→div(A(x, Du)) is locally in W−1,p(Ω). Consequently, Eq. (1.1) carries no solutionsu inWloc1,p(Ω)if the measure data μ is not in the dual. On the other hand, ifμ∈W−1,p(Ω), the existence of solutions is a straightforward consequence of duality methods in view of the weak continuity of the operator, see e.g. [23]. Moreover, the reader is asked to examine functions
u(x)= 1
|x|
rγ−1dr (1.2)
* Corresponding author.
E-mail addresses:tero.kilpelainen@jyu.fi (T. Kilpeläinen), tuomo.kuusi@tkk.fi (T. Kuusi), anna.tuhola-kujanpaa@jyu.fi (A. Tuhola-Kujanpää).
0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2011.03.004
that for γ =(p−n)/(p−1)yield a reasonable distributional solution to Eq. (1.1), where the operator is the p- Laplacian andμis a multiple of the Dirac measure – a measure outside the dual. From this example we also infer that the maximal regularity for a general solution cannot reachn-integrability of|Du|p−1.
In conclusion, in order to solve Eq. (1.1) with a general Radon measure one is forced to look outside the natural domain of the operator (see Section 2 for a more accurate description). A relevant existence theory for equations with general signed measure data was developed by Boccardo and Gallouët [6] for p >2−1/n (this restriction, dictated by the fact that the fundamental solution in (1.2) does not have a distributional derivative at the origin, can be dispensed with by using a weaker derivative, see [21]). Their method is based on a suitable approximation of the measureμ. The main task pursued there was showing necessary a priori summability estimates for the gradients of solutions that allow for viable compactness arguments. The solutions produced in [6] are often called SOLA (Solutions Obtained as Limits of Approximations), emphasizing the fact that these are limit functions of solutions to equations with regularized source measures from the dual ofW1,pconverging weakly to the original measure. Regularity theory for SOLA is a widely studied field, see for example [29–31] and the references therein.
As known e.g. by the example given by Serrin [35] the distributional solutions to (1.1) do not solve the Dirichlet problem in a unique manner. Thus there arose attempts to arrive at the unique solvability by imposing new require- ments foruto be a solution.
Whenμbelongs toL1, alternative solutions were called entropy or renormalized solutions, introduced indepen- dently by Bénilan et al. [4], Dall’Aglio [7], and by Lions and Murat [25], and in these works also the uniqueness of renormalized solutions was settled, but only when μ∈L1. Later, Dal Maso et al. [11] generalized the concept for general measures. Theserenormalized solutionsallow for testing the equation with Lipschitz functions of the solution itself provided that the derivative of the test function is compactly supported; see Section 2.3 for the precise definition.
Again, renormalized solutions are SOLA in the above sense.
In the case of nonnegative measures, Kilpeläinen and Malý [21] established a clear connection between exis- tence theory and nonlinear potential theory. In particular, it was shown that every nonnegative measure induces an A-superharmonic solution for allp >1 and that obtained solutions are SOLA as well. A class ofA-superharmonic functions consists of (pointwise defined) lower semicontinuous functions satisfying a comparison with respect to solutions to homogeneous equations. See Section 2.1 for definitions and [19] for the rich theory behind such func- tions. In the light of the fundamental convergence theorem, stating that under mild integrability conditions properly pointwise defined limits of A-superharmonic functions remainA-superharmonic, it is easy to see that SOLA have A-superharmonic representatives wheneverμcan be approximated with nonnegative smooth measures.
In this paper we study the connection betweenA-superharmonic functions and renormalized solutions. Our main result is thateveryA-superharmonic function is locally a renormalized solution. We also show the converse, i.e. that every renormalized solution has anA-superharmonic representative. In this respect,our result unifies the existence theory in the case of nonnegative measuresand allows for very sharp testing of superharmonic functions provided by the definition of renormalized solutions. More importantly, superharmonic functions form a class of pointwise defined solutions to (2.6) equivalent with SOLA and renormalized solutions whenever the source measure is nonnegative.
As an application for our main result we characterize allW1,psolutions to Riccati type equation
−pu= |∇u|p, p >1. (1.3)
We show that the transformation
u→ep−1u (1.4)
gives an one-to-one correspondence between the solutions to (1.3) and thosep-superharmonic functions whose Riesz measures are singular with respect to thep-capacity. More precisely, for each nonnegative Radon measureμ, singular with respect to thep-capacity, any (SOLA) solution of−pv=μhas ap-superharmonic representative and it can be transformed to a solution uto (1.3) by the inverse of the transformation (1.4). Conversely, if uis a solution to the Riccati equation (1.3), then epu−1 is ap-superharmonic function whose Riesz measure is supported in a set of p-capacity zero.
A corresponding result was proved in the Laplacian case in [38] by using the linear potential theory. In the nonlinear case, results in the akin spirit were obtained independently by Abdel Hamid and Bidaut-Véron [1]; however our argument is fairly simple and our result completes the story.
The Riccati type equations, especially related existence and uniqueness questions, are widely studied, see for instance [1–3,8,12–17,20,26,27,33,34].
2. Tools from nonlinear potential theory
Throughout this paper we letΩstand for an open set inRn,n2, andμbe a nonnegative Radon measure inΩ. Moreover, we let 1< p <∞be a fixed number. Throughout,candC(andc(a, b, d)) will denote positive constants (depending on dataa, b, d) whose value is not necessarily the same at each occurrence.
LetA:Ω×Rn→Rn be a Carathéodory function, that is,(x, ξ )→A(x, ξ )is measurable for everyξ ∈Rn and ξ→A(x, ξ )is continuous for almost everyx∈Ω. We assume the growth conditions
A(x, ξ ), ξ
α0|ξ|p, and A(x, ξ )β0|ξ|p−1, (2.1)
for allξ∈Rnand for almost everyx∈Ω, and the monotonicity condition A(x, ξ )−A(x, ζ ), ξ−ζ
>0 (2.2)
for allξ=ζ inRnand for almost everyx∈Ω. Hereα0andβ0are positive constants.
2.1. A-superharmonic functions
A continuous functionh∈Wloc1,p(Ω)is said to beA-harmonicinΩif it is a weak solution to
−div
A(x,∇h)
=0, that is,
Ω
A(x,∇h),∇ϕ dx=0 for allϕ∈C0∞(Ω).
A lower semicontinuous functionu:Ω→R∪ {∞}is calledA-superharmonicifu≡ ∞in each component ofΩ, and for each openUΩ and eachh∈C(U )that isA-harmonic inU, the inequalityuh on∂U impliesuh inU.
The following characterization forA-superharmonicity is our starting point. For the proof, see for example [19].
2.3. Proposition.Suppose thatuis an a.e. finite function inΩ. Thenuhas anA-superharmonic representative if and only if the truncationsuk=min(u, k)are supersolutions to
−div
A(x,∇u)
0
for eachk >0, i.e.uk∈Wloc1,p(Ω)and
Ω
A(x,∇uk),∇ϕ dx0 for all nonnegativeϕ∈C0∞(Ω).
Recall that the pointwise values of anA-superharmonic function are uniquely determined by its values a.e. since u(x)=ess lim inf
y→x u(y), for eachx; see [19, Theorem 7.22].
We denote byTk(t )=min(k,max(t,−k))the usual truncation operator. Following the tradition of the potential theory we use the very weak gradient
Du= lim
k→∞∇Tk(u)
for suchuwhose truncations are Sobolev functions, see [19,21].
A frequently used property ofA-superharmonic functions is the local summability:
2.4. Theorem.(See [19, Theorem 7.46].) IfuisA-superharmonic inΩ, thenu∈Lsloc(Ω)and|Du|p−1∈Lqloc(Ω) whenever
0< s <n(p−1)
n−p and 0< q < n n−1;
forp=nany finitesis allowed;forp > n,u∈Wloc1,p(Ω).
A functionuis a solution to
−div
A(x,∇u)
=μ (2.5)
if
Ω
A(x, Du),∇ϕ dx=
Ω
ϕ dμ (2.6)
for all ϕ∈C∞0 (Ω). Here, of course, one must have thatA(x, Du) is locally integrable. For anA-superharmonic functionu this assumption is satisfied by Theorem 2.4 and, indeed, for any nonnegative measureμthere is anA- superharmonic function solving (2.5), see [21]. Conversely, for anyA-superharmonic function there exists a unique nonnegative Radon measureμsuch thatusolves Eq. (2.5). This measureμis called theRiesz measureofu, and it is often denoted byμ[u].
We shall later employ the fact that the truncationsuk=min(u, k)are alsoA-superharmonic and their Riesz mea- suresμ[uk]are locally in the dual of the Sobolev spaceW1,p(see Proposition 2.3); moreoverμ[uk] →μ[u]weakly inΩ.
Recall also the two-sided Wolff potential estimate [22,24,32,39,40]: ifuis a nonnegativeA-superharmonic solution to (2.5) inB(x,2r)⊂Ω, then there is a constantc=c(n, p, α0, β0)such that
1
cWμ,r(x)u(x)c
ess inf
B(x,r)u+Wμ,r(x) , (2.7)
where
Wμ,r(x)= r 0
μ(B(x, )) n−p
1/(p−1)
d .
Observe carefully that all A-superharmonic functions with the Riesz measure μsatisfy the estimate. This fact suggests a definition of a class of functions, namely
Sμ,r,L Ω
=
u: 1
cWμ,r(x)u(x)L+cWμ,r(x)∀x∈Ω
, for somer >0,L0, andΩΩ. We indeed have the following.
2.8. Proposition.Letube a nonnegativeA-superharmonic function with the Riesz measureμin a bounded domainΩ. LetΩΩ. For every0< r <dist(Ω, ∂Ω)/2, there is a constantL <∞for whichu∈Sμ,r,L(Ω).
Proof. The first inequality in the definition ofSμ,r,L(Ω)readily follows from the Wolff potential estimate (2.7). To deduce the second inequality from the same estimate we need to have an upper bound for infB(y,r)uwith an arbitrary y∈Ω. This easily follows from Theorem 2.4: there isγ=γ (n, p) >0 such that
inf
B(y,r)u
−
B(y,r)
uγdx 1/γ
c
r−n
Ω+Br
uγdx 1/γ
<∞
for ally∈Ω, as desired. 2
2.2. Decomposition of measures
The p-capacitycapp(B, Ω)of any setB⊂Ω is defined in the standard way: thep-capacity of a compact set K⊂Ωis
capp(K, Ω)=inf
Ω
|∇ϕ|pdx: ϕ∈C0∞(Ω), ϕ1 onK
.
Thep-capacity of an open setU⊂Ωis then capp(U, Ω)=sup
capp(K, Ω): Kcompact,K⊂U
; and for an arbitrary setE⊂Ω
capp(E, Ω)=inf
capp(U, Ω): Uopen,E⊂U . There is also a dual approach to the capacity. Indeed, define
capp(E, Ω):=sup
ν(E): ν∈
W01,p(Ω)
, suppν⊂E, ν0, −pw=νsuch that 0w1 forE⊂Ω. Then by [22, Theorem 3.5] we have
capp(E, Ω)=capp(E, Ω) wheneverE⊂Ωis a Borel set.
A setEis calledpolarif there is an open neighborhoodUof Eand anA-superharmonic functionuinU such thatu= ∞onE. We will later employ the fact (see e.g. [19]) that a setEis polar if and only if it is ofp-capacity zero, that is
capp(E∩U, U )=0 for all open setsU⊂Rn.
For every Radon measureμ we denote with μ0 the part which isabsolutely continuous with respect to thep- capacityand withμs thesingular part with respect to thep-capacity, i.e.
μ=μ0+μs,
whereμ0capp(meaning thatμ0(E)=0 for each setEofp-capacity zero), andμs⊥capp(meaning that there is a Borel setF ofp-capacity zero for whichμs(Rn\F )=0). The support of the singular part is contained in the polar set of correspondingA-superharmonic functions, as the next lemma shows.
2.9. Lemma.LetubeA-superharmonic with the Riesz measureμ. Then μs
{u <∞}
=0,
whereμs is the singular part ofμ(with respect to thep-capacity).
Proof. Our goal is to estimate the measureμs on the set{u <∞}by employing the dual definition of the capacity.
To this end, we first recall a general fact that if
Ω
Wν,r(x) dν <∞
for a measureνand for somer >0, thenνbelongs to(W01,p(Ω)), see [18] and also [28,41].
Let thenE⊂Ωbe a set such that capp(E)=0 andμs(Ω\E)=0. For everyk >0 denoteEk=E∩ {u < k}. Fix k >0 and take a compact subsetK⊂Ek. By the compactness, the distance ofKand∂Ω, sayr, is positive. Now the Wolff potential estimate (2.7) implies
WμK,r/8(x)Wμ,r/8(x)cu(x) < ck
for allx∈K. Thus
Ω
WμK,r/8(x) dμKckμ(K) <∞
and henceμK belongs to the dual ofW01,p(Ω).
Next, letvbe a nonnegativeA-superharmonic function solving
−pv=μK
inΩwithv∈W01,p(Ω). By the Wolff potential estimate (2.7) we have that v(x)L+ck
for allx∈K(see Proposition 2.8). SincevisA-harmonic inΩ\K, the maximum principle yields 0vL+ck
inΩ. Hence, forM=L+ck,w=v/M solves
−pw=Mp−1μK∈
W01,p(Ω)
, 0w1,
and therefore it is an admissible function to test the dual capacity ofK. It follows that μ(K)Mp−1capp(K, Ω)=Mp−1capp(K, Ω)Mp−1capp(E, Ω)=0, where we used the equivalence of capacities. Thusμ(Ek)=0, and hence
μs
{u <∞}
μs Ω\E
+ ∞ k=1
μs(Ek)=0. 2
2.3. Locally renormalized solutions
Ifμis a nonnegative Radon measure in an open setΩ, we say that a functionuis alocal renormalized solutionto (2.5) inΩ if
Tk(u)∈Wloc1,p(Ω) for allk >0,
|u|p−1∈Lsloc(Ω) for all 1s < n n−p,
|Du|p−1∈Lqloc(Ω) for all 1q < n
n−1, (2.10)
and
Ω
A(x, Du), Du
h(u)φ dx+
Ω
A(x, Du),∇φ h(u) dx
=
Ω
h(u)φ dμ0+h(+∞)
Ω
φ dμs (2.11)
is satisfied for allφ∈C0∞(Ω)andh∈W1,∞(R)such thathhas a compact support; here h(∞)= lim
t→∞h(t ).
This definition is a local version for a nonnegative measureμof a renormalized solution used by Dal Maso, Murat, Orsina, and Prignet in [11] for general signed measures. The localization was then made by Bidaut-Véron in [5]. The most important feature in the localization is that the test functionφis required to be compactly supported in (2.11).
We would like to write the condition (2.11) for short as
Ω
A(x, Du), D h(u)φ
dx=
Ω
h(u)φ dμ, (2.12)
wherehandφare as above. This, however, requires some care: the left-hand sides of both (2.11) and (2.12) clearly agree for all a.e. representatives ofu. The same is not true for the right-hand sides. Indeed,
Ω
h(u)φ dμ=
Ω
h(u)φ dμ0+
Ω
h(u)φ dμs.
The first integral on the right is easily settled: the integration against μ0 is independent of the chosen p- quasicontinuous representative ofuas these agree q.e. and henceμ0-a.e. That
Ω
h(u)φ dμs=h(∞)
Ω
φ dμs
for allhandφis more tricky: it requiresuto be Borel measurable (orμs-measurable) and, more importantly, that u= ∞μs-a.e. By Lemma 2.9A-superharmonic representatives (if exist) have these properties, since they are lower semicontinuous.
We will proceed in showing that locally renormalized supersolutions haveA-superharmonic representatives when- ever μ is nonnegative. For such functions the condition (2.12) is a legitimate way to write Eq. (2.11). The first task is to show that renormalized solutions are locally bounded below. This will readily imply by the assumption Tk(u)∈Wloc1,p(Ω)that also min(u, k)∈Wloc1,p(Ω)for allk >0.
2.13. Lemma. Let μ be nonnegative and letu be a local renormalized solution to (2.5)in Ω. Then u is locally essentially bounded below.
Proof. Choose first the test function hε(u)=1
εmin{ε, u+} −1, ε >0,
and leth∈W1,∞(R)be nonnegative withhhaving a compact support. Letφ∈C∞0 (Ω)be nonnegative. On the one hand, we have
hε(+∞)h(+∞)
Ω
φ dμs=0,
Ω
hε(u)h(u)φ dμ00,
and
Ω
A(x, Du),∇hε(u)
h(u)φ dx0.
On the other hand, the dominated convergence theorem gives
Ω
A(x, Du),∇ h(u)φ
hε(u) dx→
Ω
−A(x,−Du−),∇ h(u)φ
dx asε→0. Thusv:=u−satisfies
min(v, k)∈Wloc1,p(Ω), k >0,
and
Ω
A(x, Dv),∇ h(v)φ
dx0 (2.14)
for allφandhas above. HereA(x, z):= −A(x,−z). This means thatvis a nonnegative distributional subsolution for which a priori integrability requirements are not necessarily fulfilled. We now proceed to show thatvis actually locally bounded and thus a usual weak subsolution. We establish this using the method in [22].
Define
hk,d,ε(v)=1−
1+min
(v−k)+ d ,1
ε 1−τ
, k, d, ε >0, τ >1,
which we can substitute into (2.14). Note thathk,d,ε0. We then have by the monotone convergence together with the assumed summability of|Dv|p−1that
0
Ω
A(x, Dv),∇
hk,d,ε(v)φp dx
1
C
Ω
|Dv|phk,d,ε(v)φpdx−C
Ω
|Dv|p−1hk,d,ε(v)φp−1|Dφ|dx
→ 1 C
Ω
|Dv|phk,d(v)φpdx−C
Ω
|Dv|p−1hk,d(v)φp−1|Dφ|dx
asε→0, where hk,d(v):=1−
1+(v−k)+ d
1−τ
, k, d >0, τ >1.
This energy estimate is enough for showing [22, Lemma 4.1]. Indeed, now one may mimic the proof starting from [22, (4.5)] with obvious changesv≡uandμ≡0. We then continue as in the proof of [22, Theorem 4.8], with only slight differences: letx0∈Ωbe a Lebesgue point ofvγ and letr <dist(x0, ∂Ω)/2. Assume that
p−1< γ < n(p−1) n−p+1.
DenoteBj=B(x0, rj), whererj=21−jr. Leta0=0 and forj1 let aj+1=aj+δ−1
−
Bj+1
(v−aj)γ+dx 1/γ
,
whereδ >0 is a suitable small constant. Note thataj<∞sincev=u−∈Lγloc(Ω)by the assumptions. Now applying [22, Lemma 4.1] one can deduce as in the proof of [22, Theorem 4.8] thataj+1−aj (aj−aj−1)/2 implying by telescoping argument that
a:= lim
j→∞aj2a1=C
−
B1
vγdx 1/γ
.
Hence the sequence(aj)is bounded and increasing. Therefore, we have v(x0)−aγ
+= lim
j→∞−
Bj
(v−a)γ+dx lim
j→∞−
Bj
(v−aj)γ+dx= lim
j→∞Cδ(aj−aj−1)=0.
Thus
u−(x0)=v(x0)aC
−
B0
|u|γdx 1/γ
and henceuis locally essentially bounded below by the assumed summability ofu. 2
We are ready to prove that for nonnegative measuresμeach local renormalized solution has anA-superharmonic representative.
2.15. Theorem.Suppose thatuis a local renormalized solution to(2.5)inΩ with a nonnegativeμ. Then there is an A-superharmonic functionu˜such thatu˜=ua.e. and, moreover,u˜satisfies(2.12), i.e.
Ω
A(x, Du),˜ ∇ h(u)φ˜
dx=
Ω
h(u)φ dμ˜
for allφ∈C0∞(Ω)andh∈W1,∞(R)such thathhas a compact support.
Proof. In the light of the discussion after (2.12) it suffices to find anA-superharmonic representative foru. To this end, letφ∈C0∞(Ω)be nonnegative. Forε >0 andk >0 write
hk,ε(t )=1 εmin
(k+ε−t )+, ε . Sincehk,ε(t )0, we have
Ω
A(x, Du), Du
hk,ε(u)φ dx0.
Moreover, the nonnegativity ofμandφimplies
Ω
hk,ε(u)φ dμ0+hk,ε(+∞)
Ω
φ dμ+s 0.
Thus, (2.11) yields
Ω
A(x,∇uk),∇φ dx0
once we letε→0 and refer to the dominated convergence theorem; hereuk=min(u, k).
Sinceuis locally bounded from below by Lemma 2.13,uk∈Wloc1,p(Ω)is an ordinary supersolution. Therefore each uk has anA-superharmonic representativeu˜k. We conclude the proof by observing that the desired representative of uis then given by
˜ u= lim
k→∞u˜k
as it isA-superharmonic, being an increasing limit ofA-superharmonic functions. 2 3. Superharmonic functions are locally renormalized
Before proving our main theorem, we establish the existence of an auxiliary comparison function. The result relies on the existence of renormalized solutions.
3.1. Lemma.Letμbe a nonnegative Radon measure supported inB(0, R). Then there is anA-superharmonic func- tionwsolving
−div(A(x, Dw))=μ inB(0,4R),
w=0 on∂B(0,4R),
such that for all0< r < Rthere is a positive constantL <∞such that w∈Sμ,r,L
B(0,4R)
and
B(0,4R)
∇
min(w,2λ)−λ
+pdx λ α0
μ
{Wμ,r> λ/L} ∩B(0, R)
for allλ > L.
Proof. We first obtain by [11, Definition 2.25] the existence of a renormalized solutionvto the equation in the formu- lation vanishing on∂B(0,4R)in theW01,p-sense. Theorem 2.15 implies thatvhas anA-superharmonic representative wsatisfying (2.12).
Observe next that sincewis anA-superharmonic function, it is nonnegative by the minimum principle. Proposi- tion 2.8 gives a constantL0<∞such that
w∈Sμ,r,L
B(0,2R)
forLL0. The Wolff potential estimate (2.7) implies thatwis locally bounded outside the support ofμ, hencew is locally inW1,p there; in particular wis A-harmonic inB(0,4R)\B(0, R)(cf. [28, Corollary 3.19]). Thus the maximum principle gives
sup
B(0,4R)\B(0,2R)
wL0.
Consequently, we may take anyLL0to obtain w∈Sμ,r,L
B(0,4R) .
The potential estimate also leads to the inclusion {w > L0+ck} ⊂ {Wμ,r> k}
for allk∈R. Fixk=λ/c−L0,λ >0. We have for allλ >2cL0thatk > λ/(2c). Thus, {w > λ} ⊂
Wμ,r> λ/(2c) holds for allλ >2cL0.
Finally, we test the renormalized equation ofwwith h(w)=
min(w,2λ)−λ
+,
λ >2cL0>0, which is clearly admissible since h is Lipschitz continuous andh has a compact support. More- over, since wvanishes continuously on the boundary ofB(0,4R),h(w)has a compact support inB(0,4R)and, in particular,h(w)∈W01,p(B(0,4R)). We have
λμ
Wμ,r> λ/(2c)
∩B(0, R)
B(0,R)
h(w) dμ
=
B(0,4R)
A(x,∇w),∇h(w)
dxα0
{λ<w<2λ}∩B(0,4R)
|∇w|pdx
=α0
B(0,4R)
∇
min(w,2λ)−λ
+pdx and the result follows forL:=2cmax{L0,1}. 2
The heart of this paper is the following.
3.2. Theorem.Suppose thatuis anA-superharmonic solution to(2.5). Suppose further thatvisA-superharmonic and that for allΩΩand for all smallr >0there isL <∞such that
u, v∈Sμ,r,L
Ω .
Leth:R×R→Rbe Lipschitz and let∇hhave a compact support. Then
Ω
h(u, v)φ dμ=
Ω
A(x, Du),∇
h(u, v)φ dx
for allφ∈C0∞(Ω).
Proof. Denoteuj=min(u, j ),j >0. Letkbe so large thath(u, v)=h(uk, vk). Letφ∈C0∞(Ω)and letΩΩ be a smooth domain such that the support ofφbelongs toΩ.
Letε >0 and
Kε⊂ {Wμ,1= +∞} ∩Ω
be a compact set such that μs
Ω\Kε
< ε.
Set
r=1 2min
dist
Ω, ∂Ω ,dist
Kε,
max(u, v)k
>0 and denote
Sε:=
x∈Rn: dist(x, Kε)r . Take
θε∈C0∞
min(u, v) > k
, 0θε1, such that
θε=1 onSε.
In particular, in the support ofθεφ,h(u, v)=h(k, k)is a constant. Define further με=μΩ\Kε,
i.e., the restriction ofμto the setΩ\Kε. Note thatμε(E)μ0(E)+εwheneverEis a Borel set. Observe that we have
Wμ,r=Wμε,r inΩ\Sε. This yields the inclusion
Sμε,r,L
Ω
⊂Sμ,r,L
Ω\Sε for allL0.
Next, letRbe large enough so thatΩ⊂B(0, R)and letwεbe anA-superharmonic renormalized solution to −div(A(x, Dwε))=με inB(0,4R),
wε=0 on∂B(0,4R).
By Lemma 3.1, there is a constantL <∞such that wε∈Sμε,r,L
Ω
⊂Sμ,r,L
Ω\Sε , and for
ψλ=(min{wε,2λ} −λ)+
λ ,
the estimate
Ω
|∇ψλ|pdxCλ1−pμε
{Wμ,r> λ/C} ∩Ω Cλ1−p
μ0
{Wμ,r> λ/C} ∩Ω +ε
(3.3) holds for allλ >L.
Furthermore, the assumption of the theorem provides usLsuch that u, v∈Sμ,r,L
Ω\Sε
and thus
wε, u, v∈Sμ,r,max{L,L}
Ω\Sε .
Consequently,wε,u, andvare comparable. In particular, there is a constantC <∞such that max(u, v) > Cλ
∩
Ω\Sε
⊂ {wε> λ} ∩
Ω\Sε
⊂
min(u, v) > C−1λ
∩ Ω\Sε holds for allλ > C.
Next we observe that by the choice ofθεwe have
Ω
h(u, v)φθεdμ=h(k, k)
Ω
φθεdμ
=h(k, k)
Ω
A(x, Du),∇(φθε) dx
=
Ω
A(x, Du),∇
h(u, v)φθε
dx. (3.4)
Indeed,θεhas been chosen so that its support does not intersect the support of∇h. Our goal is hence to show that
Ω
h(u, v)φ(1−θε) dμ−
Ω
A(x, Du),∇
h(u, v)φ(1−θε) dx
is small by means ofε, eventually leading to the result of the theorem. To prove this, we use the truncated equation ofu, i.e.
−div
A(x,∇um)
=μ[um], m∈N.
First, since bothuk andvk arep-quasicontinuous and inW1,p(Ω), there are sequencesuk,j andvk,j of smooth functions converging in W1,p(Ω)andp-quasieverywhere to uk andvk, respectively. In particular,uk,j →uk and vk,j→vk μ0-almost everywhere. This readily implies thath(uj,k, vj,k)converges weakly toh(uk, vk)inW1,p(Ω).
Recall thath(uk, vk)=h(u, v)by the choice ofk. We have by the weak convergence ofμ[um]toμthat
Ω
h(uk,j, vk,j)(1−θε)φ dμ[um] →
Ω
h(uk,j, vk,j)(1−θε)φ dμ.
Furthermore, by thep-quasieverywhere convergence and the dominated convergence theorem, we obtain
Ω
h(uk,j, vk,j)(1−θε)φ dμ0→
Ω
h(u, v)(1−θε)φ dμ0
asj→ ∞and the estimate
Ω
h(uk,j, vk,j)(1−θε)φ dμsεh∞φ∞
holds by Lemma 2.9 and the choice ofθε. Hence we obtain lim sup
j→∞
Ω
h(uk,j, vk,j)(1−θε)φ dμ−
Ω
h(u, v)(1−θε)φ dμ
Cε (3.5)
withCindependent ofε.
Next, rewrite
Ω
h(uk,j, vk,j)(1−θε)φ dμ[um]
=
Ω
ψλh(uk,j, vk,j)(1−θε)φ dμ[um] +
Ω
(1−ψλ)h(uk,j, vk,j)(1−θε)φ dμ[um]. (3.6) We estimate the first integral on the right as
Ω
ψλh(uk,j, vk,j)(1−θε)φ dμ[um]h∞
Ω
ψλ(1−θε)φ dμ[um] (3.7)
and then use the structure ofAto obtain
Ω
ψλ(1−θε)φ dμ[um]
=
Ω
A(x,∇um),∇
ψλ(1−θε)φ dx
β0φ∞
Ω\Sε
|∇um|p−1|∇ψλ|dx+β0∇
φ(1−θε)∞
Ω∩supp(ψλ)
|Du|p−1dx. (3.8)
Sincewε, u∈Sμ,r,C(Ω\Sε),
uC+cWμ,rC+c2wε< C+2c2λ in{wε<2λ} ∩ Ω\Sε
(3.9)
and henceuCλin the intersection of the support of∇ψλandΩ\Sεfor all sufficiently largeλ. In this setum=uCλ for allm > Cλ. It follows by Hölder’s inequality and (3.3) that
Ω\Sε
|∇um|p−1|∇ψλ|dx =
Ω\Sε
|∇uCλ|p−1|∇ψλ|dx
Ω
|∇uCλ|pdx
(p−1)/p
B(0,4R)
|∇ψλ|pdx 1/p
Cλ(p−1)/pCλ−(p−1)/p μ0
{Wμ,r> λ/C} ∩Ω +ε1/p
=C μ0
{Wμ,r> λ/C} ∩Ω +ε1/p
→Cε1/p (3.10)
asλ→ ∞since capp({Wμ,r> λ/C} ∩Ω)→0; here we have also employed the estimate
Ω∩{uλ}
|∇u|pdxCλ, (3.11)
with some constantCindependent ofλ, from the proof of [21, Theorem 1.13]. Note that the upper bound in (3.10) is independent ofj andm. Moreover, the local summability of|Du|p−1, see Theorem 2.4, implies that
Ω∩supp(ψλ)
|Du|p−1dx→0 (3.12)
asλ→ ∞since capp({ψλ>0} ∩Ω)→0. Inserting estimates (3.10) and (3.12) into (3.8) and then using (3.7) leads to
lim sup
λ,j,m→∞
Ω
h(uk,j, vk,j)ψλ(1−θε)φ dμ[um]
Cε1/p. (3.13)
Hence, by (3.5) and (3.6), lim sup
λ,j,m→∞
Ω
h(uk,j, vk,j)(1−ψλ)(1−θε)φ dμ[um] −
Ω
h(u, v)(1−θε)φ dμ C
ε+ε1/p
, (3.14)
withC independent ofε.
Next, we consider the first term on the left in (3.14). By (3.9) we have that(1−ψλ)(1−θε)φvanishes outside {uCλ} ∩(Ω\Sε)for all sufficiently largeλ. Hence, for allm > Cλ,
Ω
h(uk,j, vk,j)(1−ψλ)(1−θε)φ dμ[um]
=
Ω\Sε
A(x,∇uCλ),∇φ
h(uk,j, vk,j)(1−θε)(1−ψλ) dx
+
Ω\Sε
A(x,∇uCλ),∇(1−θε)
h(uk,j, vk,j)(1−ψλ)φ dx
+
Ω\Sε
A(x,∇uCλ),∇h(uk,j, vk,j)
(1−θε)(1−ψλ)φ dx
−
Ω\Sε
A(x,∇uCλ),∇ψλ
h(uk,j, vk,j)(1−θε)φ dx.
Now we fix the “marching order” for the limiting processes by sending firstm, thenj, and finallyλto infinity; observe that the estimates in previous limiting processes were independent of the particular order.
First, it follows by the dominated convergence theorem that
λlim→∞ lim
j→∞ lim
m→∞
Ω\Sε
A(x,∇uCλ),∇φ
h(uk,j, vk,j)(1−θε)(1−ψλ) dx
=
Ω
A(x, Du),∇φ
h(u, v)(1−θε) dx
and
λlim→∞ lim
j→∞ lim
m→∞
Ω\Sε
A(x,∇uCλ),∇(1−θε)
h(uk,j, vk,j)(1−ψλ)φ dx
=
Ω
A(x, Du),∇(1−θε)
h(u, v)φ dx.
Second, the weak convergence of∇h(uk,j, vk,j)to∇h(uk, vk)= ∇h(u, v)together with the dominated convergence gives
λlim→∞ lim
j→∞ lim
m→∞
Ω\Sε
A(x,∇uCλ),∇h(uk,j, vk,j)
(1−θε)(1−ψλ)φ dx
=
Ω
A(x, Du),∇h(u, v)
(1−θε)φ dx.
Third, estimating as in (3.8) and (3.10), we have
Ω\Sε
A(x,∇uCλ),∇ψλ
h(uk,j, vk,j)(1−θε)φ dx
Ch∞φ∞
Ω
|∇uCλ|p−1|∇ψλ|dx
C
μ0
{Wμ,r> λ/C} ∩Ω +ε1/p
, which readily implies
lim sup
λ,j,m→∞
Ω\Sε
A(x,∇uCλ),∇ψλ
h(uk,j, vk,j)(1−θε)φ dx
Cε1/p.
Inserting above estimates into (3.14) we infer that
Ω
h(u, v)(1−θε)φ dμ−
Ω
A(x, Du),∇
h(u, v)(1−θε)φ dx
C
ε+ε1/p . This together with (3.4) yields
Ω
h(u, v)φ dμ−
Ω
A(x, Du),∇
h(u, v)φ dx
C
ε+ε1/p , concluding the proof after lettingε→0. 2
Now we arrive at our main theorem by choosingu=vin Theorem 3.2.
3.15. Theorem.LetubeA-superharmonic with Riesz measure μ= −divA(x,∇u).
Thenuis a local renormalized solution, i.e.,
Ω
A(x, Du),∇ h(u)φ
dx=
Ω
h(u)φ dμ
for allφ∈C0∞(Ω)and for all Lipschitz functionsh:R→Rwhose derivativeshare compactly supported.
3.16. Remark.Dal Maso and Malusa [9] defined a concept of areachable solution. They showed that such a solution satisfies the formula (forhandϕas in Theorem 3.15)
Ω
A(x, Du),∇ h(u)φ
dx=
Ω
h(u)φ dμ1+h(+∞)
Ω
φ dμ2−h(−∞)
Ω
φ dμ3
forsomedecompositionμ=μ1+μ2−μ3for the measureμ. The novelty in our result is that we may now specify the decomposition by takingμ1capp,μ2⊥capp such that spt(μ2)⊂
k>0{u > k}, andμ3=0. Our theorem seems not to be easily deduced from results in [9], since the weak convergence of measures, used by Dal Maso and Malusa to obtain the measuresμ2andμ3, seems to be as such inadequate to conquer the concentration phenomenon.
4. Nonlinear Riccati type equations
Theorem 3.15 enables us to employ all the properties of the renormalized solution when studying equations of type (2.5), regardless of the nature of the solutions. As an example we consider the following two problems:
−pu= |∇u|p inΩ,
u∈W01,p(Ω) (4.1)
and ⎧
⎨
⎩
−pv=μ,
μ∈M+(Ω) and μ⊥capp,
0min(v, k)∈W01,p(Ω) for allk >0,
(4.2) whereΩ is bounded. Recall that, as emphasized in (2.5), the equations are understood in the sense of distributions.
In this section we show that these two problems are essentially equivalent:
4.3. Theorem.There is a one-to-one correspondence between problems(4.1)and(4.2)via the transformation v=ep−u1 −1.
That is, if u solves(4.1), then v=eu/(p−1)−1 solves(4.2); and conversely, ifv is a solution to(4.2), thenu= (p−1)log(v+1)is a solution to(4.1).
