Unidirectional evolution equations of diffusion type

0.37 MB·English

by  Goro Akagi

UNIDIRECTIONAL EVOLUTION EQUATIONS OF DIFFUSION TYPE
GOROAKAGI AND MASATOKIMURA
Abstract. Thispaperisconcerned withtheuniqueness,existence, comparisonprincipleand
5
long-timebehavior of solutions tothe initial-boundaryvalue problemfor aunidirectional dif-
1
fusion equation. The unidirectional evolution often appears inDamage Mechanics due to the
0
strong irreversibility of crack propagation or damage evolution. The existence of solutions is
2
provedinanL2-frameworkbyintroducingapeculiardiscretizationoftheunidirectionaldiffu-

n sionequationbymeansofvariationalinequitiesofobstacletypeandbydevelopingaregularity
a theoryforsuchvariationalinequalities. Thenoveldiscretizationargumentwillbealsoapplied
J toprovethecomparisonprincipleaswellastoinvestigatethelong-timebehaviorofsolutions.

6

]
P
A 1. Introduction
.
h InDamageMechanics,theunidirectional evolution isasignificantfeatureofcrackpropagation
t models. Indeed, crack propagation is an irreversible phenomena, and particularly, cracks in
a
m a specimen or the damage of a material (e.g., microcracks which break or weaken bonds of
microstructures) cannot disappear nor decrease. Hence if one introduces a phase parameter
[
(see [33, 29, 43, 27, 30]) or an internal variable (see [12, 13]) which describes the crack growth

1
or the damage accumulation, they are usually supposed to be unidirectional, i.e., nondecreasing
v
or nonincreasing. Such unidirectional evolution processes are often described by PDEs involving
2
the positive-part function, s 7→ (s) := s∨0 = max{s,0} for s ∈ R. Moreover, in a phase field
7 +
0 approach, the evolution of a phase parameter is governed by the gradient flow of an appropriate
1 energy functional subject to some unidirectional constraint.
0
In order to find out mathematical features of such unidirectional evolutions and to develop
.
1 mathematical devices to analyze them, in this paper, we shall treat, as a simplest case, the
0
evolution of u=u(x,t) governedby the (fully nonlinear) PDE,
5
1 ∂ u= ∆u+f , for x∈Ω, t>0, (1)
t +
:
v where Ω is a bounded Lipschitz do(cid:0)main of(cid:1)Rn with n ∈ N, ∂tu = ∂u/∂t, ∆ stands for the n-
Xi dimensionalLaplacian,f =f(x,t)isagivenfunctionand(s)+ :=s∨0fors∈R. Moreprecisely,
the main purpose of this paper is to prove the uniqueness, existence and comparison principle
r
a of strong solutions u = u(x,t) of the initial-boundary value problem for (1) and to reveal the
asymptotic behavior of u=u(x,t) as t→∞.
Solutions of (1) entail unidirectional nature, more precisely, the non-decrease of u=u(x,t) in
t,sincetheright-handsideof (1)isnon-negativeduetothepresenceofthepositivepartfunction.
There appear various kinds of unidirectionalevolutions in naturalsciences and engineering fields
(see,e.g.,[21]). Inparticular,aphasefieldmodelforcrackpropagationinanelasticmaterialwas
proposedwith the aidofa unidirectionalgradientflow by the secondauthorandhis collaborator
in [27, 42, 43] (see also [6, 20]). Equation (1) can be regarded as a simplified equation of their
model.
In mathematical points of view, (1) is classified as a fully nonlinear PDE, which is not fit for
energy methods in general; however, by taking a (multi-valued) inverse function of the positive
Date:January7,2015.
2010 Mathematics Subject Classification. Primary: 35K86;Secondary: 35K61,74A45.
Key words and phrases. Unidirectional diffusion equation, damage mechanics, discretization, variational in-
equalityofobstacletype,regularity,subdifferentialcalculus.
1
2 GOROAKAGI AND MASATOKIMURA
part function (·) , (1) can be formulated as a sort of doubly nonlinear evolution equations,
+
∂ u+α(∂ u)−∆u∋f a.e. in Ω×(0,∞), (2)
t t
where α is a (multi-valued) maximal monotone function in R given by α(0) = (−∞,0] and
α(s)={0}foranys>0withthedomainD(α)=[0,∞)(seeRemark2.3andSection4belowfor
more details). Equation (2) is fitter for energy methods and monotone techniques. On the other
hand, in view of the L2-theory of evolution equations, two operators v 7→α(v(·)) and v 7→−∆v
(defined for v ∈ L2(Ω)) are unbounded in L2(Ω), and hence, it is more delicate to establish a
prioriestimatesforprovingtheexistenceofstrongsolutions,ascomparedwithstandardequations
without unidirectional constraints, e.g., the classical and nonlinear diffusion equations.
The nonlinear PDE (2) may fall within the frame of abstract doubly nonlinear evolution
equations in a Hilbert space H of the form
∂Ψ(∂ u(t))+∂Φ(u(t))∋f(t) in H, 0<t<T, (3)
t
where ∂Ψ and ∂Φ denote the subdifferential operator of functionals Ψ : H → (−∞,∞] and
Φ : H →(−∞,∞], respectively. In a thermodynamic approach to continuum mechanics, Ψ and
Φ are often referred to as a dissipation functional and an energy functional, respectively. To
reduce (2) into the form (3), we set u(t):=u(·,t) and particularly choose
1 1
H =L2(Ω), Ψ(v)= |v|2dx+I (v), Φ(v)= |∇v|2dx for v ∈H,
2 [ ·≥0] 2
ZΩ ZΩ
where I is the indicator function over the set {v ∈ L2(Ω): v ≥ 0 a.e. in Ω}. Then we note
[ ·≥0]
that both subdifferentials ∂Ψ and ∂Φ are unbounded in H.
Let us briefly review the previous studies on abstract doubly nonlinear evolution equations
such as (3). Barbu [11] proved the existence of solutions for (3) with two unbounded operators
∂Ψ and ∂Φ by using the elliptic-in-time regularization and by imposing the differentiability (in
t) of f. This result was generalized by Arai [7], Senba [40] and so on. In these papers, the term
∂Ψ(∂ u(t))is estimatedbydifferentiating the equationandbytesting it with∂Ψ(∂ u(t)). There-
t t
forethedifferentiabilityoff (moreprecisely,f ∈W1,1(0,T;H)by[7])isessentiallyrequired,and
moreover, some strong monotonicity condition (i.e., the so-called ∂Ψ-monotonicity) is also im-
posedon∂Φ. Similarmethodsofestablishingaprioriestimatesarealsousedinindividualstudies
on irreversiblephase transitionmodels (see, e.g., [14]). On the other hand, Colli and Visintin es-
tablishedanalternativeapproachto(3)in[18],where∂Ψissupposedtobeboundedandcoercive
with linear growthinstead of assuming the regularityassumption on f and the ∂Ψ-monotonicity
of∂Φ(seealso[17]). Theirframeworkwouldbemoreflexibleinviewofapplicationstononlinear
PDEsandhasbeenextensivelyappliedtovarioustypesofdoublynonlinearproblems. Moreover,
their framework has been generalized in many directions, e.g., perturbation problems, long-time
behaviors (see, e.g., [1, 36, 35], [37, Sect. 11], [41, 39, 34, 38, 2, 3, 4, 5]). However, due to the
unboundedness of ∂Ψ, (2) seems to be beyond the scope of the latter approach. On the other
hand, the former approach due to Barbu and Arai is applicable to (2); however, the regularity
assumption f ∈ W1,1(0,T;H) has to be assumed for proving the existence of solutions. Aso et
al. [8, 9] also treated an irreversible phase transition system in a different fashion; however, the
regularity condition f ∈W1,2(0,T;H) is also assumed there.
Inthispaper,wepresentanovelapproachto(1)(orequivalently(2))byintroducingapeculiar
discretizationfor(1)bymeansofvariationalinequalitiesofobstacletype. Moreover,bydeveloping
a regularity theory of such variational inequalities, we shall establish new a priori estimates for
(1) (or (2)) without assuming the differentiability (in t) of f. Such a relaxation of the regularity
assumption on f may bring some advantage to consider perturbation problems, which will be
left for a forthcoming paper. Moreover, the novel discretization method will be also applied
to investigate the long-time behavior of solutions as well as to prove a comparison theorem.
In particular, we shall provide uniform (in t) estimates for solutions by employing the peculiar
discretization scheme. Furthermore, some variational inequality of obstacle type will play a
crucial role in asymptotic analysis; indeed, it will turn out that every solution will converge to
UNIDIRECTIONAL EVOLUTION EQUATIONS OF DIFFUSION TYPE 3
the unique solutionz =z(x) of a variationalinequality of obstacle type involvingthe initial data
as anobstaclefunction frombelow under suitable assumptions. Here it is worthmentioning that
the limit z of the solution u = u(x,t) depends on its initial data u ; indeed, one can construct
0
different limits of solutions for different initial data. On the other hand, the ω-limit set of each
solution must be singleton.
The organization of this paper is as follows. The initial-boundary value problem for (1) and
basic notions (e.g., strong solution) and assumptions are formulated, and main results of this
paper will be stated in the next section. Section 3 is devoted to establishing a regularity theory
as well as to verifying a comparison theorem for variational inequalities of obstacle type. In
Section 4, we discuss a reduction of (1) to an evolution equation of doubly nonlinear type in
L2(Ω)andprovetheuniquenessofsolutionsfortheinitial-boundaryvalueproblem. InSection5,
we carry out the backward-Euler time-discretization of (1) and construct a strong solution of
(1) by proposing a new a priori estimate based on the regularity theory developed in Section 3.
A comparison theorem for (1) is also proved here. The long-time behavior of solutions will be
investigated in Section 6. In the last section, we shall discuss other equivalent formulations of
solutions for (1).
Notation. For each normed space N, we denote by N′ the dual space of N with the duality
pairinghg,viN :=N′hg,viN =g(v)forv ∈N andg ∈N′. ForBanachspacesU andW,thesetof
all bounded linear operatorsfrom U to W is denoted by B(U,W). Moreover,the set of all linear
topologicalisomorphismsfromU to W isdenotedbyIsom(U,W),thatis, A∈Isom(U,V)means
that A is bijective from U to W, A∈B(U,W) and A−1 ∈B(W,U). Furthermore, Hk stands for
the k-dimensional Hausdorff measure in Rn for k = 1,2,...,n. We also write a∨b = max{a,b}
anda∧b=min{a,b}for a,b∈R. Finally, C denotesa non-negativeconstantindependentofthe
elements of the corresponding space and set and may vary from line to line.
2. Main results
Let Ω be a bounded Lipschitz domain in Rn with n ∈ N. Let Γ be the boundary of Ω such
that Γ is the disjoint union of two subsets Γ and Γ , that is,
D N
Γ ∪Γ =Γ, Γ ∩Γ =∅.
D N D N
Moreover, assure that Γ is (relatively) open in Γ. Let ν denote the outward-pointing unit
D
normal vector on Γ. One of these two subsets may be empty. In such a case, the other set
coincideswiththe wholeofΓ. Mainresultsofthe presentpaperareconcernedwiththe following
initial-boundary value problem for a unidirectional evolution equation of diffusion type,
∂ u= ∆u+f in Q:=Ω×(0,∞), (4)
t +
u=(cid:0)0 (cid:1) on ΓD×(0,∞), (5)
∂ u=0 on Γ ×(0,∞), (6)
ν N
u| =u in Ω, (7)
t=0 0
where ∂ =∂/∂t, f =f(x,t) and u =u (x) are given functions of class L2 ([0,∞);L2(Ω)) and
t 0 0 loc
L2(Ω), respectively, and ∂ u := γ (∇u)·ν denotes the normal derivative of u. Moreover, (·)
ν 0 +
stands for the positive part function, i.e., (s) :=s∨0 for s∈R. If Γ (resp., Γ ) is empty, the
+ D N
corresponding boundary condition (5) (resp., (6)) is ignored.
Remark 2.1. By change of variable, one can reduce another unidirectional diffusion equation,
∂ u= ∆u+f in Q, (8)
t −
where (s)− :=s∧0 for s∈R, to (4). Inde(cid:0)ed, set v(cid:1):=−u and g :=−f. Then (8) is transformed
to
−∂ v = −∆v−g =− ∆v+g ,
t − +
whence v solves (4) with f replaced by g.
(cid:0) (cid:1) (cid:0) (cid:1)
4 GOROAKAGI AND MASATOKIMURA
Let us start with defining strong solutions of (4)–(7). To this end, we set up notation. Let
γ ∈B(H1(Ω),H1/2(Γ)) denote the trace operator defined on H1(Ω) (throughout the paper, we
0
may omit γ if no confusion can arise). Moreover,define
0
V :={v ∈H1(Ω): γ v =0 Hn−1-a.e. on Γ },
0 D
X :={v ∈H2(Ω): γ (∇v)·ν =0 Hn−1-a.e. on Γ }
0 N
equippedwiththeinducednormsandinnerproducts,i.e.,k·kV =k·kH1(Ω)and(·,·)V =(·,·)H1(Ω)
for V; k·kX = k·kH2(Ω) and (·,·)X = (·,·)H2(Ω) for X. Then V and X are closed subspaces of
H1(Ω) and H2(Ω), respectively; hence, they are Hilbert spaces. If either Γ or Γ is empty, the
D N
corresponding boundary condition specified in the definition of V or X above is ignored.
We are concerned with strong solutions of (4)–(7) defined by
Definition 2.2 (Strong solution). For T >0, a function u∈C([0,T];L2(Ω)) is called a strong
solution of (4)–(7) on [0,T], if the following three conditions are satisfied:
(i) u∈W1,2(0,T;L2(Ω))∩L2(0,T;X∩V),
(ii) the equation ∂ u=(∆u+f) holds a.e. in Q :=Ω×(0,T),
t + T
(iii) the initial condition u| =u holds a.e. in Ω.
t=0 0
A function u∈C([0,∞);L2(Ω)) is called a strongsolutionof (4)–(7)on [0,∞), if for anyT >0,
the restriction of u onto [0,T] is a strong solution of (4)–(7) on [0,T].
Remark 2.3. One can further derive that u ∈ C([0,T];V) from (i) by employing a chain-rule
for convex functionals. See Lemma 4.4 in §4 below for more details.
We are now in position to state main results, whose proofs will be given in later sections. We
begin with the uniqueness of solutions.
Theorem 2.4 (Uniqueness). Let T > 0, u ∈ V and f ∈ L2(Q ). Then the strong solution of
0 T
(4)–(7) on [0,T] is unique.
To state our existence result, we shall introduce some assumptions for the domain Ω and the
boundary Γ , Γ . For λ∈R, we define a mapping A ∈B(V,V′) by
D N λ
hA u,vi = (∇u·∇v+λuv) dx for u,v ∈V. (9)
λ V
ZΩ
It is well known that A ∈ Isom(V,V′) holds if λ > 0. Hence one can define u = A−1g for
λ λ
g ∈L2(Ω) as the unique solution u of the elliptic problem in a weak form,
(∇u·∇v+λuv) dx= gvdx for all v ∈V,
ZΩ ZΩ
(i.e., A u=g in V′). Then we assume that
λ
A−1g ∈H2(Ω) for all g ∈L2(Ω). (10)
1
Condition (10) is often called an elliptic regularity condition and deeply related to the geometry
of the domain and boundary conditions. Indeed, it holds true for smooth domains with a single
boundary condition (i.e., Γ = ∅ or Γ = ∅). However, it is more delicate to consider the
N D
validity of (10) for situations with nonsmooth domains or mixed boundary conditions. On the
other hand, in order to take account of physical backgrounds of crack growth models and their
numerical simulations, the regularity of the boundary may be at most Lipschitz continuous, and
mixed boundary conditions seem to be natural as well. We shall give conditions equivalent to
(10) in Proposition 3.6 below.
Remark 2.5. Let us exhibit a couple of examples of Ω, Γ and Γ for which the condition (10)
D N
is satisfied.
(i) If Γ ∩ Γ = ∅ and Γ is of class C1,1, then (10) is satisfied (see Theorem 2.2.2.3 and
D N
Theorem 2.2.2.5 of [25]).
UNIDIRECTIONAL EVOLUTION EQUATIONS OF DIFFUSION TYPE 5
(ii) Let Ω be convex. If Γ = Γ or Γ = ∅, then (10) is satisfied (see Theorem 3.2.1.2 and
D D
Theorem 3.2.1.3 of [25]).
(iii) If n = 1 or if Ω is a rectangle in R2 and Γ is a union of some of four edges of Γ, then
D
(10) is satisfied. Indeed, since the weak solution u = A−1g can be extended to an open
1
neighborhood of Ω by reflection, it follows that u=A−1g ∈H2(Ω).
1
Our existence result reads,
Theorem2.6(Existence). Wesupposethatthecondition (10)holds true. LetT >0,u ∈X∩V
0
and f ∈L2(Q ) be given. If there exists f∗ ∈L2(Ω) satisfying
T
f(x,t)≤f∗(x) a.e. in Q , (11)
T
then there exists a strong solution u=u(x,t) to the problem (4)–(7) on [0,T].
Remark2.7(Assumptionsonf). Condition(11)isweakerthanf ∈L∞(Q )orf ∈W1,1(0,T;L2(Ω))
T
(cf.[7]). Indeed,iff ∈W1,1(0,T;L2(Ω)),thenf∗(x):=f(x,0)+ T |∂ f(x,t)|dtbelongstoL2(Ω)
0 t
and satisfies (11). On the other hand, (11) is stronger than f := f ∨0 ∈ L∞(0,T;L2(Ω)). In
+R
fact, (11) yields f ∈ L∞(0,T;L2(Ω)). However, even if f ∈ L∞(0,T;L2(Ω)), (11) might not
+ +
hold true. One may easily find a counterexample, e.g., f(x,t)=|x−t|−α, Ω=(0,1), T =1 and
0<α<1/2.
Theorem 2.6 will be proved in Section 5. Our method of proof relies on the backward-Euler
time-discretizationof (4). Thediscretizedequationwillbe rewrittenassomevariationalinequal-
ities of obstacle type. Moreover, developing a regularity theory for such variational inequalities,
we shall obtain a new a priori estimate for discretized solutions. As a result, we shall construct
a strong solution without assuming the differentiability of f in t, which is usually assumed in
standard frameworks of doubly nonlinear evolution equations such as (32) (see, e.g., [11, 7, 40]).
Moreover, such a basic strategy of proving the existence result will be also applied to prove
the following comparison theorem for strong solutions of (4)–(7):
Theorem 2.8 (Comparison principle). Let T > 0 and suppose that (10) is satisfied. For each
i=1,2, let ui ∈X ∩V and fi ∈L2(Q ) be such that there exists f∗ ∈L2(Ω) satisfying
0 T
fi(x,t)≤f∗(x) a.e. in Q .
T
For i = 1,2, let ui = ui(x,t) be the unique strong solution of (4)–(7) with u = ui and f = fi
0 0
on [0,T]. If u1 ≤u2 a.e. in Ω and f1 ≤f2 a.e. in Q , then u1 ≤u2 a.e. in Q .
0 0 T T
Thepeculiardiscretizationargumentwillbealsoemployedtoinvestigatethelong-timebehav-
iorofsolutionsfor(4)–(7). Furthermore,thecomparisontheoremstatedabovewillplayacrucial
role to identify the limit of each solution u=u(x,t) as t→∞.
Theorem 2.9 (Convergenceofsolutions ast→∞). Let u ∈X∩V and assume that (10) holds
0
and that
(H1) Hn−1(Γ )>0;
D
(H2) there exists a function f ∈L2(Ω) such that f −f belongs to L2(0,∞;L2(Ω));
∞ ∞
(H3) f ∈L∞(0,∞;L2(Ω)), and (11) is satisfied.
Thentheuniquesolutionu=u(x,t)of (4)–(7)on[0,∞)convergestoafunctionz =z(x)∈X∩V
strongly in V as t→∞. Moreover, the limit z satisfies
z ≥u and −∆z ≥f a.e. in Ω.
0 ∞
In addition, if f(x,t) ≤ f (x) for a.e. (x,t) ∈ Q, then the limit z coincides with the unique
∞
solution z¯∈X ∩V of the following variational inequality:
z¯∈K (u ):={v ∈V : v ≥u a.e. in Ω},
0 0 0
(VI)(u ,f )
0 ∞ 
∇z¯·∇(v−z¯)dx≥ f (v−z¯)dx for all v ∈K (u ).
 ∞ 0 0
ZΩ ZΩ

6 GOROAKAGI AND MASATOKIMURA
Remark 2.10. Assumption(H1)isessentiallyrequiredtoensuretheconvergenceofthesolution
u = u(x,t) as t → ∞. Indeed, suppose that Γ = ∅ (i.e., Γ = Γ) and set u (x) ≡ 1 and
D N 0
f(x,t)≡1. The unique strong solution of (4)–(7) is given by
u(x,t)=1+t for (x,t)∈Q,
and then, u(x,t) is divergent to ∞ at each x∈Ω as t→∞.
3. Regularity theory for variational inequalities of obstacle type
In this section, based on the approach of Gustafsson [26], we revisit a regularity theory for
variationalinequalitiesofobstacletype. Inclassicalliteratureonvariationalinequalitiesofobsta-
cle type, the W2,p(Ω)-regularity of solutions is often obtained by using a penalization technique
(see,e.g.,[28,Chap.4],[22]). Ontheotherhand,Gustafsson[26]gaveasimpleralternativeproof
by introducing an auxiliary variationalinequality andby proving the coincidence of solutions for
both problems. Let us also remark that, in previous results, it is assumed that W2,p(Ω)⊂C(Ω)
(namely, p>n/2)inordertoutilize the classicalmaximumprincipleforlinearelliptic equations.
We shall establish a W2,p(Ω)-regularity result for variational inequalities of obstacle type
equipped with a mixed boundary condition by properly modifying the argument of Gustafs-
son [26]. It is noteworthy that we do not assume that W2,p(Ω) ⊂ C(Ω), i.e., p > n/2, as we
employ Stampacchia’s truncation technique instead of the classical maximum principle.
First, we shall set up notation. For σ ≥0, let A:=A ∈B(V,V′) be defined as in (9). Define
σ
a symmetric bilinear form a(·,·):V ×V →R associated with A by
a(u,v):=hAu,vi = (∇u·∇v+σuv) dx for u,v ∈V. (12)
V
ZΩ
Throughout this section, we assume that
σ >0 if Hn−1(Γ )=0. (13)
D
Underthecondition(13),(bythePoincar´einequalityforthecasethatσ =0andHn−1(Γ )>0),
D
a(·,·) turns out to be coercive on V ×V. Hence A is invertible, and A−1 belongs to B(V′,V).
Let f ∈V′ and ψ ∈V and define closed convex subsets K , K of V by
0 1
K :={v ∈V : v ≥ψ a.e. in Ω}, (14)
0
K :={v ∈V : Av ≥f in V′}.
1
Here the inequality Av ≥ f in V′ means that hAv−f,ϕi ≥ 0 for all ϕ ∈ V satisfying ϕ ≥ 0
V
a.e. in Ω. We also define functionals J,Jˆon V by
1
J(v):= a(v,v)−hf,vi for v ∈V, (15)
V
2
1
Jˆ(v):= a(v,v)−hfˆ,vi for v ∈V,
V
2
where fˆ:=Aψ ∈V′. Then the following equivalence holds true:
Proposition 3.1. Suppose (13) and let f ∈ V′ and ψ ∈ V. Then the following five conditions
for u∈V are equivalent to each other:
(a) u∈K , J(u)≤J(v) for all v ∈K ,
0 0
(b) u∈K , a(u,v−u)≥hf,v−ui for all v ∈K ,
0 V 0
(c) u∈K ∩K , hAu−f,u−ψi =0,
0 1 V
(d) u∈K , a(u,v−u)≥hfˆ,v−ui for all v ∈K ,
1 V 1
(e) u∈K , Jˆ(u)≤Jˆ(v) for all v ∈K .
1 1
Moreover, there exists a unique element u∈V satisfying all the conditions.
UNIDIRECTIONAL EVOLUTION EQUATIONS OF DIFFUSION TYPE 7
Proof. Since J is a coercive, continuous, strictly convex functional on the closed convex set K ,
0
J admits a unique minimizer u over K . Hence u satisfies (a).
0
We shall prove the equivalence of the conditions (a)–(e). It is well known (see, e.g., [28]) that
(a) ⇔ (b) and (d) ⇔ (e). So, let us here start with showing that (b) ⇒ (c). The condition (b) is
equivalently rewritten by
u∈K , hAu−f,v−ui ≥0 for all v ∈K . (16)
0 V 0
Foranyϕ∈V withϕ≥0a.e.inΩ,substitutingv =u+ϕ∈K to(16),wehavehAu−f,ϕi ≥0,
0 V
which yields that u ∈ K . On the other hand, substitute v = ψ ∈ K and v = 2u−ψ ∈ K to
1 0 0
(16). Then one can obtain hAu−f,u−ψi =0. Hence (c) holds.
V
To prove the inverse relation, (c) ⇒ (b), let u satisfy (c). For any v ∈K , we see that
0
a(u,v−u)−hf,v−ui =hAu−f,v−ψi −hAu−f,u−ψi .
V V V
Since the firsttermofthe right-handside isnonnegative(by u∈K andv ∈K ) andthe second
1 0
term vanishes (by the equation of (c)), the condition (b) follows.
One may prove the equivalence between (c) and (d) in a similar fashion to the above. Firstly,
suppose that u satisfies (c). For any v ∈K , one finds that
1
a(u,v−u)−hfˆ,v−ui =hAv−Au,ui −hAv−Au,ψi
V V V
=hAv−f,u−ψi −hAu−f,u−ψi .
V V
Here we used the fact that hAw,zi = hAz,wi for all w,z ∈ V. Noting that the right-hand
V V
side is non-negative by (c) and the fact that u ∈ K and v ∈ K , one can get (d). To check the
0 1
inverse relation, we also rewrite (d) as
u∈K , hAv−Au,u−ψi ≥0 for all v ∈K . (17)
1 V 1
For any ϕ ∈ L2(Ω) with ϕ ≥ 0 a.e. in Ω, substituting v = u+A−1ϕ ∈ K to (17), we have
1
(ϕ,u−ψ)L2(Ω) ≥0, whichalongwiththe arbitrarinessofϕ≥0 implies u∈K0. Moreover,letus
also substitute v = A−1f ∈ K and v = 2u−A−1f ∈ K in (17). Then we obtain hAu−f,u−
1 1
ψi =0, whence (c) follows. Consequently, all the conditions (a)–(e) are equivalent. (cid:3)
V
In the rest of this section, let p∈R satisfy
2n
1<p<∞, p≥ . (18)
n+2
Since the Ho¨lder conjugate q := p/(p−1) of p satisfies q ≤ 2∗ := 2n/(n−2) if n ≥ 3, and Ω
is a Lipschitz domain, by Sobolev’s embedding theorem, the continuous embeddings V ֒→Lq(Ω)
and Lp(Ω)∼=(Lq(Ω))′ ֒→V′ hold true. We also note that W2,p(Ω) is continuously embedded in
H1(Ω) by (18).
We further suppose that
f ∈Lp(Ω), ψ ∈V, Aψ ∈Lp(Ω) (19)
and introduce a closed (in V) convex set K given by
2
K :={v∈V : f ≤Av ≤f ∨fˆin V′}⊂K .
2 1
A main result of this section is stated as follows:
Theorem 3.2. Suppose that (13), (18) and (19) are satisfied. Then each of the following condi-
tions (f)–(h) is equivalent to the conditions (a)–(e) of Proposition 3.1:
(f) u∈K , Jˆ(u)≤Jˆ(v) for all v ∈K ,
2 2
(g) u∈K , a(u,v−u)≥hfˆ,v−ui for all v ∈K ,
2 V 2
(h) u∈K ∩K , (Au−f)(u−ψ)=0 a.e. in Ω.
0 2
We first prepare a couple of lemmas, which will be also used in later sections.
8 GOROAKAGI AND MASATOKIMURA
Lemma 3.3. Let w ∈ H1(Ω) and set w (x) := (w(x)) for x ∈ Ω. Then w ∈ H1(Ω) and
+ + +
γ (w )=(γ w) Hn−1-a.e. on Γ.
0 + 0 +
Proof. SetΩ :={x∈Ω: w(x)>0}andrecallTheoremA.1.of[28]toobservethatw ∈H1(Ω)
+ +
and
w a.e. in Ω , ∇w a.e. in Ω ,
+ +
w = ∇w = (20)
+ +
(0 a.e. in Ω\Ω+, (0 a.e. in Ω\Ω+.
Set W := C(Ω)∩H1(Ω). Since W is dense in H1(Ω), there exists a sequence {w } in W such
n
that w →w strongly in H1(Ω) as n→∞. Noting that
n
k(wn)+−w+kL2(Ω) ≤kwn−wkL2(Ω) →0 as n→∞, (21)
we observe that (w ) →w strongly in L2(Ω). Applying (20) to w , we also have
n + + n
k(w ) k2 =k(w ) k2 +k∇(w ) k2 ≤kw k2 +k∇w k2 =kw k2 .
n + H1(Ω) n + L2(Ω) n + L2(Ω) n L2(Ω) n L2(Ω) n H1(Ω)
Since {w } is bounded in H1(Ω), so is {(w ) }. Hence, one can extract a (non-relabeled)
n n +
subsequence of {n} such that (w ) →w weakly in H1(Ω) Again from (20), we have
n + +
k(wn)+k2H1(Ω) =k(wn)+k2L2(Ω)+(∇(wn)+, ∇wn)L2(Ω)
→kw+k2L2(Ω)+(∇w+, ∇w)L2(Ω) =kw+k2H1(Ω),
which together with the uniform convexity of H1(Ω) also implies that (w ) → w strongly in
n + +
H1(Ω). Sinceγ ∈B(H1(Ω),H1/2(Γ)),weparticularlydeducethatγ w →γ w andγ (w ) →
0 0 n 0 0 n +
γ w strongly in L2(Γ). As in (21), one can verify that (γ w ) → (γ w) strongly in L2(Γ).
0 + 0 n + 0 +
On the other hand, since w ∈C(Ω), it is clear that
n
γ (w ) =(γ w ) Hn−1-a.e. on Γ. (22)
0 n + 0 n +
Passing to the limit as n→∞ in (22), we conclude that γ w =(γ w) Hn−1-a.e. on Γ. (cid:3)
0 + 0 +
Lemma 3.4. If v ,v ∈V, then v ∨v ∈V and v ∧v ∈V.
1 2 1 2 1 2
Proof. Applying Lemma 3.3 to w = ±(v −v ), we find that (v −v ) and (v −v ) belong
1 2 1 2 + 2 1 +
to V. Hence we obtain v ∨v ∈ V and v ∧v ∈ V, since v ∨ v = v + (v − v ) and
1 2 1 2 1 2 2 1 2 +
v ∧v =v −(v −v ) . (cid:3)
1 2 2 2 1 +
Let us move on to a proof of Theorem 3.2.
Proof of Theorem 3.2. It is obvious that (f) ⇔ (g). As in Proposition 3.1, one can uniquely
choose u ∈ K which satisfies (f) and (g). Next, we shall prove the equivalence between (a)–(e)
2
and(f),(g). Letu betheuniqueelementofV satisfying(a)–(e)andletu betheuniqueelement
1 2
of V satisfying (f) and (g).
We claim that u =u . Indeed, note that Au ∈Lp(Ω) by u ∈K . Set w :=u −ψ ∈V and
1 2 2 2 2 2
h:=Aw ∈Lp(Ω). Since u satisfies (g), it follows that
2
0≤a(u ,v−u )−hfˆ,v−u i =hAu −Aψ,v−u i
2 2 2 V 2 2 V
=hAv−Au ,wi for all v ∈K . (23)
2 V 2
We set a measurable set
N :={x∈Ω: w(x)<0}.
Define g ∈Lp(Ω) by
f(x)∨fˆ(x) if x∈N,
g(x):=
Au (x) if x∈Ω\N.
(cid:26) 2
UNIDIRECTIONAL EVOLUTION EQUATIONS OF DIFFUSION TYPE 9
Then by definition one can observe that A−1g ∈ K . Hence substituting v = A−1g to (23), we
2
have
0≤hg−Au ,wi = (f ∨fˆ)−Au wdx.
2 V 2
ZN(cid:16) (cid:17)
Since (f ∨fˆ)−Au ≥0 and w<0 a.e. in N, one can derive the relation Au =f ∨fˆa.e. in N.
2 2
It follows that
h=Au −Aψ =(f ∨fˆ)−fˆ=(f −fˆ) ≥0 a.e. in N.
2 +
On the other hand, we recall that w ∈V solves the equation Aw =h in V′, that is,
a(w,v)=hh,vi for all v ∈V. (24)
V
We define w :=w∧0. Then by Lemma 3.4 and Theorem A.1.of [28], we have
−
w in N, ∇w a.e. in N,
w ∈V, w = ∇w =
− − −
(0 in Ω\N, (0 a.e. in Ω\N.
Substituting v =w into (24), we have
−
(24)
0≤a(w ,w )=a(w,w ) = hh,w i = hwdx. (25)
− − − − V
ZN
Since h ≥ 0 and w < 0 a.e. in N, the right-hand side of (25) is non-positive. Hence we deduce
that a(w ,w ) = 0, which along with the coercivity of a(·,·) implies that w = 0 (i.e., w ≥ 0)
− − −
a.e. in Ω. Therefore u belongs to K .
2 0
Substitute v =A−1f ∈K to the condition (23). Then we obtain
2
hAu −f,u −ψi ≤0. (26)
2 2 V
Moreover, noting that Au −f ≥ 0 (by u ∈ K ) and u −ψ ≥ 0 (by u ∈ K ), we derive
2 2 2 2 2 0
hAu −f,u −ψi = 0 by (26). Hence, u = u satisfies the condition (c). By uniqueness, we
2 2 V 2
obtain u =u . Thus we have proved that all the conditions (a)–(g) are equivalent.
1 2
Finally,wenote that(h) immediatelyimplies (c), sinceK ⊂K . Conversely,letusatisfy (c).
2 1
Then u belongs to K by (f), and hence, Au∈Lp(Ω) and
2
0=hAu−f,u−ψi = (Au−f)(u−ψ)dx.
V
ZΩ
Thus we obtain (Au−f)(u−ψ)=0 a.e. in Ω, since Au≥f and u≥ψ a.e. in Ω. Therefore (h)
holds. (cid:3)
Thanks to Theorem 3.2, for each solution u of the variational inequality of obstacle type,
u∈K , a(u,v−u)≥hf,v−ui for all v ∈K , (27)
0 V 0
we have obtained an additional information, u ∈ K . In order to more explicitly clarify the
2
feature of the additional information, let us introduce the following assumption:
A−1g ∈W2,p(Ω) for all g ∈Lp(Ω) in case σ >0 or p≤2∗,
1 (28)
(A−11g ∈W2,ρ(Ω) for all g ∈Lρ(Ω) and ρ∈[2∗,p] in case σ =0 and p>2∗,
where 2∗ = 2n/(n−2) (here we note that (10) is a special case of (28) with p = 2). Then we
+
find that
Proposition 3.5. Assume that (28) holds. Then
K ⊂W2,p(Ω).
2
10 GOROAKAGI AND MASATOKIMURA
Proof. Let v ∈ K . In case σ > 0 or p ≤ 2∗ (hence V ֒→ Lp(Ω)), since Av ∈ Lp(Ω) and v ∈ V,
2
one can check that v ∈Lp(Ω). Hence A v =Av+(1−σ)v ∈Lp(Ω), which along with (28) gives
1
v ∈W2,p(Ω). In case σ =0 andp>2∗, noting that A v ∈L2∗(Ω) as above,we find by (28) that
1
v ∈W2,2∗(Ω). By iteration argumentalong with Sobolev’s embeddings, one can finally conclude
that v ∈Lp(Ω). Therefore we deduce that v ∈W2,p(Ω) as in the former case. (cid:3)
Condition(28)canberegardedaselliptic regularity ofweaksolutionsfortheellipticboundary
value problem,
−∆u+u=f in Ω, u=0 on Γ , ∂ u=0 on Γ ,
D ν N
and it holds true in many cases, e.g., smooth domains with Γ = ∅ or Γ = ∅ (see, e.g., [24]).
N D
However, the validity of (28) is more delicate, if Ω is not smooth or mixed boundary conditions
are imposed. So we here explicitly made the assumption.
To take account of boundary conditions, we further define a subspace of W2,p(Ω) by
Xp :={v∈W2,p(Ω): γ (∇v)·ν =0 Hn−1-a.e. on Γ }.
0 N
Prior to stating a regularity result for (27), we prepare a proposition, which provides equivalent
forms of the assumption (28).
Proposition 3.6. Under the assumption (18), the operator Aλ|Xp∩V restricted onto Xp∩V is
injective and bounded linear from Xp∩V into Lp(Ω), and it coincides with the operator −∆+λ,
where ∆ means the Laplace operator from D(∆) = Xp ∩V into Lp(Ω), that is, the Laplacian
equipped with the Dirichlet and Neumann boundary conditions on Γ and Γ , respectively, in a
D N
strong form.
Moreover, the following conditions for Ω and Γ , Γ are equivalent to each other:
D N
(i) there exists λ>0 such that A−1g ∈W2,p(Ω) for all g ∈Lp(Ω);
λ
(ii) for any λ>0, it holds that A−1g ∈W2,p(Ω) for all g ∈Lp(Ω);
λ
(iii) there exists λ > 0 such that A−1g ∈ Xp for all g ∈ Lp(Ω), and (−∆+λ) ∈ Isom(Xp ∩
λ
V,Lp(Ω));
(iv) for any λ > 0, it holds that A−1g ∈ Xp for all g ∈ Lp(Ω), and (−∆+λ) ∈ Isom(Xp ∩
λ
V,Lp(Ω)).
Proof. Denote Bλ :=Aλ|Xp∩V for λ>0. Then, for u∈Xp∩V, we observe by Green’s formula,
which is valid for Lipschitz domains, that
hB u,vi = (∇u·∇v+λuv) dx= (−∆u+λu)vdx for all v ∈V ∩W1,q(Ω),
λ V
ZΩ ZΩ
which implies that B u=−∆u+λu and B ∈B(Xp∩V,Lp(Ω)), since V ∩W1,q(Ω) is dense in
λ λ
Lq(Ω). Thus we obtain B =(−∆+λ). Moreover, B is injective, since so is A .
λ λ λ
As for the equivalence of (i)–(iv), we shall show (ii) ⇒ (i) ⇒ (iii) ⇒ (iv) ⇒ (ii). It is clear
that (ii) ⇒ (i) (and also (iv) ⇒ (iii)). We show (i) ⇒ (iii). Assume (i), let g ∈ Lp(Ω) and set
u:=A−1g ∈W2,p(Ω)∩V. For all v ∈V ∩W1,q(Ω), we have
λ
gvdx=hA u,vi = (∇u·∇v+λuv) dx= (∂ u)vdHn−1+ (−∆u+λu)vdx,
λ V ν
ZΩ ZΩ ZΓN ZΩ
which implies that u ∈ Xp and −∆u(x)+ λu(x) = g(x) for a.e. x ∈ Ω. Hence u = B−1g.
λ
Therefore, B is surjective from Xp∩V into Lp(Ω). By the open mapping theorem, we obtain
λ
(−∆+λ)=B ∈Isom(Xp∩V,Lp(Ω)).
λ
We next show (iii) ⇒ (iv). Under the condition (iii), it holds that B ∈Isom(Xp∩V,Lp(Ω)),
λ
in particular, B : Xp∩V → Lp(Ω) is a Fredholm operator of index zero. For arbitrary µ > 0,
λ
we find that B = B +(µ−λ) is a Fredholm operator of index zero from Xp∩V to Lp(Ω) as
µ λ
well,since Xp∩V is compactlyembeddedinLp(Ω). Since B is injective (i.e.,dimker(B )=0),
µ µ
we infer that B is surjective, and hence B also belongs to Isom(Xp∩V,Lp(Ω)). Furthermore,
µ µ