In this note we study the limit as p(x) ! 1of solutions to − p(x)u = 0
in a domain 
, with Dirichlet boundary conditions. Our approach consists in considering
sequences of variable exponents converging uniformly to +1 and analyzing
how the corresponding solutions of the problem converge and what equation is satisfied
by the limit

Manfredi, Juan J.

Rossi, Julio D.

Urbano, José Miguel

University Marketing and Communications

Eastern Illinois University

Eastern Illinois UniversityThe Keep1997 Press Releases5-28-199705/28/1997 - Jerks Receives Special recognitionAt Commencement.pdfUniversity Marketing and CommunicationsFollow this and additional works at: http://thekeep.eiu.edu/press_releases_1997This Article is brought to you for free and open access by the Press Releases at The Keep. It has been accepted for inclusion in 1997 by an authorizedadministrator of The Keep. For more information, please contact tabruns@eiu.edu.Recommended CitationUniversity Marketing and Communications, "05/28/1997 - Jerks Receives Special recognition At Commencement.pdf " (1997). 1997.176.http://thekeep.eiu.edu/press_releases_1997/176f • 97-194 May 28, 1997 FOR IMMEDIATE RELEASE: JECKS RECEIVES SPECIAL REGOGNITION AT COMMENCEMENT CHARLESTON-- Eastern Illinois University junior Dan Jecks of East Moline, pictured above in the center, proudly poses with Nate Anderson of Rock Island, secretary of Eastern's Board ofTrustees, (left), and Eastern President David Jorns at the university's May 10 spring commencement. Anderson brought greetings from Eastern's Board of Trustees during the morning ceremony. Eastern held three commencement ceremonies with 1,200 graduates participating. Jecks, the son of Bob and Marcy Jecks of East Moline, was presented Eastern's Livingston C. Lord Scholarship at commencement. The scholarship, established by the Alumni Association in memory of Livingston C. Lord, Eastern's first president who served from 1899-1933, is the university's highest award for scholarship. Each Lord Scholar received a stipend and a medallion. -30-

English

05/28/1997 - Jerks Receives Special recognition At Commencement.pdf

The Keep

Estudo Geral

Pre´-Publicac¸o˜es do Departamento de Matema´tica
Universidade de Coimbra
Preprint Number 09–13
LIMITS AS p(x)→∞ OF p(x)-HARMONIC FUNCTIONS
JUAN J. MANFREDI, JULIO D. ROSSI AND JOSE´ MIGUEL URBANO
Abstract: In this note we study the limit as p(x)→∞ of solutions to −∆p(x)u = 0
in a domain Ω, with Dirichlet boundary conditions. Our approach consists in con-
sidering sequences of variable exponents converging uniformly to +∞ and analyzing
how the corresponding solutions of the problem converge and what equation is sat-
isfied by the limit.
Keywords: p(x)-Laplacian, infinity-Laplacian, variable exponents, viscosity solu-
tions.
AMS Subject Classification (2000): 35J20, 35J60, 35J70.
1. Introduction
Our goal in this note is to look for the limit, as the exponent p(x) → ∞,
of solutions to {
−∆p(x)u(x) = 0, x ∈ Ω ⊂ RN ,
u(x) = f(x), x ∈ ∂Ω, (1.1)
where ∆p(x)u(x) := div
(|∇u(x)|p(x)−2∇u(x)) is the p(x)-Laplacian operator
with a variable exponent p(x).
When p is constant in Ω, the limit of (1.1) as p→∞ has been extensively
studied in the literature (see [4] and the survey [2]) and leads naturally to
the infinity Laplacian
∆∞u :=
(
D2u∇u) · ∇u = N∑
i,j=1
∂u
∂xi
∂u
∂xj
∂2u
∂xixj
.
Infinity harmonic functions (solutions to −∆∞u = 0) solve the optimal Lip-
schitz extension problem (see [1] and the survey paper [2]) and find applica-
tions in optimal transportation, image processing and tug-of-war games (see,
e.g., [8], [9], [5], [13], [14] and the references therein). On the other hand,
problems related to PDEs involving variable exponents became popular a few
years ago due to applications in elasticity and the modeling of electrorheolog-
ical fluids. Meanwhile, the underlying functional analytical tools have been
Received March 11, 2009.
1
2 J. J. MANFREDI, J. D. ROSSI AND J.M. URBANO
extensively developed (cf. [10] and [7]) and new applications to image pro-
cessing have kept the subject as the focus of an intensive research activity.
Although a natural extension of the theory, the problem addressed here is a
follow-up from a recent paper of the authors [12], where the case of a variable
exponent that equals infinity in a subdomain of Ω is treated. Closely related
to this work is [11], where the authors prove existence and uniqueness (via
a comparison principle), as well as the validity of a Harnack inequality, for
solutions of our limit problem.
The approach in this paper is based on considering sequences pn(x) of
variable exponents converging uniformly to +∞ and analyzing how the cor-
responding solutions of the problem converge and what equation is satisfied
by the limit. Before introducing our main result, let us state the assumptions
on the data that will be assumed from now on.
• Ω ⊂ RN is a bounded smooth domain.
• f is a Lipschitz continuous function with Lipschitz constant less or
equal than one.
• pn(x) is a sequence of C1 functions in Ω such that
pn(x)→ +∞, uniformly in Ω; (1.2)
pn(x) ≥ α, for all x ∈ Ω; (1.3)
∇ ln pn(x) −→ ξ(x), uniformly in Ω, (1.4)
for a constant α > N and a function ξ ∈ C(Ω).
We now present some examples of possible sequences pn(x). In each case,
some smoothness assumptions have to be added, as well as conditions that
guarantee that (1.3) holds. We are primarily interested in understanding
(1.4) and hope the examples shed some light on the meaning of this assump-
tion.
(1) pn(x) = n; we have ξ = 0.
(2) pn(x) = p(x) + n; we get again ξ = 0.
LIMITS AS p(x)→∞ OF −∆p(x)u = 0 3
(3) pn(x) = np(x); we get a nontrivial vector field
ξ(x) = ∇(ln(p(x))).
This example is what motivates the study of the limit equation in [11].
(4) pn(x) = n
ap(x/n) [scaling in x]; in this case, we have
∇(ln pn(x)) = ∇p
p
(x/n)
1
n
−→ 0
and so ξ = 0. This also happens for pn(x) = n+ p(x/n).
(5) pn(x) = n
ap(nx); we get
∇(ln pn(x)) = n∇p
p
(nx),
which does not have a limit as n → ∞. The same happens with
pn(x) = n+ p(nx), for which
∇(ln pn(x)) = n∇p(nx)
n+ p(nx)
that does not have a uniform limit (although it is bounded).
(6) We can modify the previous example to get a nontrivial limit. Assume
that q(x) is a function of the angular variable and that 0 6∈ Ω; then
consider pn(x) = n+ q(nx) to obtain
∇(ln pn(x)) = n ∇q(θ)
n+ q(nx)
−→ ∇q(θ).
(7) Finally, we can combine examples (3) and (6). Let pn(x) = np(x) +
q(nx), with q and Ω as in (6). We get
∇(ln pn(x)) = n∇p(x) + n∇q(θ)
np(x) + q(nx)
−→ ∇p(x) +∇q(θ)
p(x)
.
The following is the main result of this paper. We prove, under the above
assumptions, that the limit of solutions of (1.1) with p(x) = pn(x) exists
and can be characterized as the unique viscosity solution of a PDE that
involves the∞−Laplacian and an extra term in which the vector field ξ(x) =
limn∇ ln pn(x) appears.
4 J. J. MANFREDI, J. D. ROSSI AND J.M. URBANO
Theorem 1.1. Let un be the solution of (1.1) with p(x) = pn(x). Then
un −→ u, uniformly in Ω, (1.5)
where u is the unique viscosity solution of the problem{ −∆∞u− |∇u|2 ln |∇u| 〈ξ,∇u〉 = 0, in Ω,
u = f, on ∂Ω.
(1.6)
Remark 1.2. Uniqueness of solutions to the limit problem (1.6) is a conse-
quence of the results of [11]. See also [3].
Remark 1.3. Notice that we are taking F (0) = 0 for F (s) = s2 ln(s), hence
(1.6) makes sense when evaluated at a test function with vanishing gradient.
Remark 1.4. In dimension one, we get as the limit problem{
u′′(x) + ln |u′(x)| 〈ξ(x), u′(x)〉 = 0, x ∈ (0, 1),
u(0) = f(0), u(1) = f(1),
(1.7)
which is uniquely and explicitly solvable. We just have to observe that we can
assume, without loss of generality, that f(0) = 0 (just consider v = u−f(0))
and f(1) > 0. Then we solve the equation as follows:
u′′(x) + ln |u′(x)| 〈ξ(x), u′(x)〉 = 0, x ∈ (0, 1)
is equivalent to ∫ t
0
u′′(x)
ln |u′(x)|u′(x)dx = −
∫ t
0
ξ(x)dx,
that is, ∫ u′(t)
C
1
ln(z)z
dz = −
∫ t
0
ξ(x)dx = −H(t).
This gives
ln(ln(u′(t))) = (C −H(t))
and thus
u(x) =
∫ x
0
exp(exp(C −H(t)))dt.
Finally, we only have to choose C such that
f(1) =
∫ 1
0
exp(exp(C −H(t)))dt.
LIMITS AS p(x)→∞ OF −∆p(x)u = 0 5
The rest of the paper is organized as follows: in Section 2 we collect some
properties of the approximate problems and in Section 3 we prove our main
result, Theorem 1.1.
2. The approximate problem
We first consider the problem corresponding to (1.1) when p(x) is replaced
by pn(x). For convenience, we refer to this problem as (1.1)n.
Lemma 2.1. There exists a unique weak solution un to (1.1)n, which is the
unique minimizer of the functional
Fn(u) =
∫
Ω
|∇u|pn(x)
pn(x)
dx (2.1)
in the set
Sn =
{
u ∈ W 1,pn(·)(Ω) : u|∂Ω = f
}
. (2.2)
Proof : Functions in the variable exponent Sobolev space W 1,pn(·)(Ω) are nec-
essarily continuous thanks to the assumption pn(x) ≥ α > N . Indeed, the
continuous embedding in
W 1,pn(·)(Ω) ↪→ W 1,α(Ω) ⊂ C (Ω) (2.3)
follows from [10, Theorem 2.8 and (3.2)].
We can then take the boundary condition u|∂Ω = f in the classical sense
(recall that f is assumed to be Lipschitz) and the unique solvability is stan-
dard in view of the regularity of the variable exponent.
It is also standard that the minimizer of Fn in Sn is the unique weak solution
of (1.1)n, i.e., un = f on ∂Ω and∫
Ω
|∇un|pn(x)−2∇un · ∇ϕdx = 0, ∀ ϕ ∈ C∞0 (Ω). (2.4)
Let us now recall the definition of viscosity solution (cf. [6]) for a problem
like (1.1) or (1.6). Assume we are given a continuous functions
F : Ω× RN × SN×N → R.
Definition 2.2. Consider the problem
F (x,∇u,D2u) = 0 in Ω (2.5)
6 J. J. MANFREDI, J. D. ROSSI AND J.M. URBANO
with a boundary condition
u = f on ∂Ω. (2.6)
A lower semi-continuous function u is a viscosity supersolution of (2.5)–(2.6)
if u ≥ f on ∂Ω and for every φ ∈ C2(Ω) such that u−φ has a strict minimum
at the point x0 ∈ Ω, with u(x0) = φ(x0), we have
F (x0,∇φ(x0), D2φ(x0)) ≥ 0.
An upper semi-continuous function u is a viscosity subsolution of (2.5)–(2.6)
if u ≤ f on ∂Ω and for every ψ ∈ C2(Ω) such that u−ψ has a strict maximum
at the point x0 ∈ Ω, with u(x0) = ψ(x0), we have
F (x0,∇ψ(x0), D2ψ(x0)) ≤ 0.
Finally, u is a viscosity solution if it is both a viscosity supersolution and a
viscosity subsolution.
In the sequel, we will use the notation as in the definition: φ will always
stand for a test function touching the graph of u from below and ψ for a test
function touching the graph of u from above.
Proposition 2.3. Let un be a continuous weak solution of (1.1)n. Then un
is a viscosity solution of (1.1)n in the sense of Definition 2.2.
Proof : The proof is contained in the proof of Proposition 2.4 in [12]. We
reproduce it here for the sake of completeness and readability.
We omit the subscript n in this proof. Let x0 ∈ Ω and a let φ be a test
function such that u(x0) = φ(x0) and u− φ has a strict minimum at x0. We
want to show that
−∆p(x0)φ(x0) = −|∇φ(x0)|p(x0)−2∆φ(x0)
−(p(x0)− 2)|∇φ(x0)|p(x0)−4∆∞φ(x0)
−|∇φ(x0)|p(x0)−2 ln(|∇φ|)(x0) 〈∇φ(x0),∇p(x0)〉
≥ 0.
LIMITS AS p(x)→∞ OF −∆p(x)u = 0 7
Assume, ad contrarium, that this is not the case; then there exists a radius
r > 0 such that B(x0, r) ⊂ Ω and
−∆p(x)φ(x) = −|∇φ(x)|p(x)−2∆φ(x)
−(p(x)− 2)|∇φ(x)|p(x)−4∆∞φ(x)
−|∇φ(x)|p(x)−2 ln(|∇φ|)(x)〈∇φ(x),∇p(x)〉
< 0,
for every x ∈ B(x0, r). Set
m = inf
|x−x0|=r
(u− φ)(x)
and let Φ(x) = φ(x) +m/2. This function Φ verifies Φ(x0) > u(x0) and
−∆p(x)Φ = −div(|∇Φ|p(x)−2∇Φ) < 0 in B(x0, r). (2.7)
Multiplying (2.7) by (Φ− u)+, which vanishes on the boundary of B(x0, r),
we get ∫
B(x0,r)∩{Φ>u}
|∇Φ|p(x)−2∇Φ · ∇(Φ− u) dx < 0.
On the other hand, taking (Φ − u)+, extended by zero outside B(x0, r), as
test function in the weak formulation of (1.1)n, we obtain∫
B(x0,r)∩{Φ>u}
|∇u|p(x)−2∇u · ∇(Φ− u) dx = 0.
Upon subtraction and using a well know inequality, we conclude
0 >
∫
B(x0,r)∩{Φ>u}
(
|∇Φ|p(x)−2∇Φ− |∇u|p(x)−2∇u
)
· ∇(Φ− u) dx
≥ c
∫
B(x0,r)∩{Φ>u}
|∇Φ−∇u|p(x) dx,
a contradiction.
This proves that u is a viscosity supersolution. The proof that u is a
viscosity subsolution runs as above and we omit the details.
We next obtain uniform estimates (independent of n) for the sequence
(un)n.
8 J. J. MANFREDI, J. D. ROSSI AND J.M. URBANO
Proposition 2.4. The minimizer of Fn in Sn, un, satisfies
Fn(un) =
∫
Ω
|∇un|pn(x)
pn(x)
dx ≤ |Ω|.
Hence, the sequence (Fn(un))n is uniformly bounded and the sequence (un)n
is uniformly bounded in W 1,α(Ω) and equicontinuous.
Proof : Note that, since f has a Lipschitz constant less or equal than one, the
set
S =
{
u ∈ W 1,∞(Ω) : ‖∇u‖L∞(Ω) ≤ 1 and u|∂Ω = f
}
(2.8)
is nonempty. Recalling (2.2), the definition of Sn, observe that S ⊂ Sn, for
every n. Since un is a minimizer, we have
Fn(un) ≤ Fn(v), ∀ v ∈ S.
Hence, picking an element v ∈ S 6= ∅,
Fn(un) =
∫
Ω
|∇un|pn(x)
pn(x)
dx ≤
∫
Ω
|∇v|pn(x)
pn(x)
dx ≤ |Ω|.
In order to estimate the Sobolev norm, we first use Poincare´ inequality and
the boundary data, to obtain
‖un‖W 1,α(Ω) ≤ ‖un − f‖W 1,α0 (Ω) + ‖f‖W 1,α(Ω)
≤ C ‖∇(un − f)‖Lα(Ω) + ‖f‖W 1,∞(Ω)
≤ C ‖∇un‖Lα(Ω) + (C + 1)‖f‖W 1,∞(Ω).
We proceed, using Ho¨lder inequality and elementary computations, to obtain
‖∇un‖Lα(Ω) =
(∫
Ω
|∇un|α dx
)1/α
≤ |Ω|
1
α− 1p−n
(∫
Ω
|∇un|p−n dx
)1/p−n
≤ (1 + |Ω|)
{
|Ω|+
(
p+n
∫
Ω
|∇un|pn(x)
pn(x)
dx
)1/p−n}
≤ (1 + |Ω|)
{
|Ω|+ (p+nFn(un))1/p−n}
≤ (1 + |Ω|) |Ω|
{
1 +
(
p+n
)1/p−n} ≤ C,
LIMITS AS p(x)→∞ OF −∆p(x)u = 0 9
since we have the bound (
p+n
)1/p−n ≤ C (2.9)
due to assumption (1.4).
Indeed, one can prove a Harnack inequality for a nonnegative function ψ
such that∇ lnψ is bounded. For completeness, we include here the argument.
Suppose ψ ≥ 0 and |∇ lnψ| ≤ K in Ω, and take arbitrary points x, y ∈ Ω.
By the mean value theorem,
ln
ψ(x)
ψ(y)
≤ |lnψ(x)− lnψ(y)| ≤ K|x− y|
and thus
ψ(x)
ψ(y)
≤ eK|x−y|.
Since Ω is bounded,
ψ(x) ≤ eK|x−y| ψ(y) = C ψ(y).
Applying this reasoning to each pn, (2.9) follows from the uniform bound-
edness of |∇ ln pn|.
We conclude that (un)n is uniformly bounded inW
1,α(Ω) and, recalling the
embedding in (2.3), that it is equicontinuous.
Remark 2.5. If f has Lipschitz constant greater than one then Fn(un) is
unbounded, see [12].
3. Passing to the limit: proof of theorem 1.1
Owing to Proposition 2.4, it follows from Ascoli’s theorem, extracting a
subsequence if necessary, that
un −→ u, uniformly in Ω,
for a certain continuous function u. Since un = f on ∂Ω we have that u = f
on ∂Ω.
To prove that u is a viscosity supersolution of (1.6), let φ be such that
u−φ has a strict local minimum at x0 ∈ Ω, with φ(x0) = u(x0). We want to
prove that
−∆∞φ(x0)− |∇φ(x0)|2 ln |∇φ(x0)|〈ξ(x0),∇φ(x0)〉 ≥ 0. (3.1)
Since un → u uniformly, there is a sequence (xn)n such that xn → x0 and
un− φ has a local minimum at xn. As un is a viscosity solution of (1.1)n (cf.
10 J. J. MANFREDI, J. D. ROSSI AND J.M. URBANO
Proposition 2.3), we have
− |∇φ(xn)|
2∆φ(xn)
pn(xn)− 2 −∆∞φ(xn)
−|∇φ(xn)|2 ln |∇φ(xn)|
〈
∇φ(xn), ∇pn(x)
pn(xn)− 2
〉
≥ 0.
Using the fact that xn → x0 and the assumptions (1.2) and (1.4), we obtain
|∇φ(xn)|2∆φ(xn)
pn(xn)− 2 −→ 0,
∆∞φ(xn) −→ ∆∞φ(x0),
|∇φ(xn)|2 ln(|∇φ(xn)|) −→ |∇φ(x0)|2 ln(|∇φ(x0)|),〈
∇φ(xn), ∇pn(x)
pn(xn)− 2
〉
−→ 〈∇φ(x0), ξ(x0)〉
and (3.1) follows.
This proves that u is a viscosity supersolution; the fact that it is also a
viscosity subsolution follows analogously. 
Acknowledgements. Parts of this work were done during visits of JDR
and JMU to the University of Pittsburgh. They both acknowledge the
stimulating atmosphere found there. JDR partially supported by project
MTM2004-02223, MEC, Spain, by UBA X066 and by CONICET, Argentina.
JMU partially supported by CMUC/FCT.
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Juan J. Manfredi
Department of Mathematics, University of Pittsburgh, Pittsburgh PA 15260, USA
E-mail address: manfredi@pitt.edu
Julio D. Rossi
Departamento de Matema´tica, FCEyN UBA (1428), Buenos Aires, Argentina
E-mail address: jrossi@dm.uba.ar
Jose´ Miguel Urbano
CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal
E-mail address: jmurb@mat.uc.pt
