バナッハ空間に於ける劣微分作用素の和の極大単調性
赤木 剛朗 (芝浦工業大学)
g-akagi@shibaura-it.ac.jp
Abstract
This talk is devoted to providing a suﬃcient condition for the maximality of thesum of subdiﬀerential operators deﬁned on reﬂexive Banach spaces and ﬁnally prov0
ing the maximal monotonicity in Lp (Ω) × Lp (Ω) of the nonlinear elliptic operator
u 7→ −∆m u + β(u(·)) with a maximal monotone graph β.

1Introduction

Let E and E ∗ be a reﬂexive Banach space and its dual space, respectively, and let φ1 , φ2 :E → (−∞, ∞] be proper (i.e., φ1 , φ2 6≡ ∞) lower semicontinuous convex functionals withthe eﬀective domains D(φi ) := {u ∈ E; φi (u) < ∞} for i = 1, 2. Then the subdiﬀerential∗operator ∂E φi : E → 2E of φi is deﬁned by∂E φi (u) := {ξ ∈ E ∗ ; φi (v) − φi (u) ≥ hξ, v − uiE for all v ∈ D(φi )} ,where h·, ·iE denotes the duality pairing between E and E ∗ , with the domain D(∂E φi ) ={u ∈ D(φi ); ∂E φi (u) 6= ∅} for i = 1, 2. This talk provides a new suﬃcient condition
for the maximal monotonicity of the sum ∂E φ1 + ∂E φ2 in E × E ∗ and an application tononlinear elliptic operators in Lp -spaces.
This research is motivated by the question of whether the following operator M is0maximal monotone in Lp (Ω) × Lp (Ω) with p ∈ [2, ∞), p0 = p/(p − 1) and a boundeddomain Ω of RN :0

M : D(M) ⊂ Lp (Ω) → Lp (Ω); u 7→ −∆m u + β(u(·)),(1.1)

where β is a maximal monotone graph in R such that β(0) 3 0, and ∆m is a modiﬁed
Laplacian given by()∆m u = ∇ · |∇u|m−2 ∇u , 1 < m < ∞equipped with the homogeneous Dirichlet boundary condition, i.e., u|∂Ω = 0. The operator
M can be divided into two parts: u 7→ −∆m u and u 7→ β(u(·)), and they are maximal0
monotone in Lp (Ω) × Lp (Ω). Indeed, set E = Lp (Ω) and put∫ 1|∇u(x)|m dx if u ∈ W01,m (Ω),φ1 (u) :=(1.2)
m Ω ∞otherwise, ∫j(u(x))dx if j(u(·)) ∈ L1 (Ω),φ2 (u) :=(1.3)
 ∞Ωotherwise,
where j : R → (−∞, ∞] is a proper lower semicontinuous convex function such that ∂j =β. Then ∂E φ1 (u) and ∂E φ2 (u) coincide with −∆m u equipped with u|∂Ω = 0 and β(u(·)),
respectively, and moreover, every subdiﬀerential operator is maximal monotone. However,these facts are not suﬃcient to ensure the maximal monotonicity of M = ∂E φ1 + ∂E φ2 inE × E ∗.
1

The maximality for the sum of two maximal monotone operators was well studied inHilbert space settings (see [1] and [2]). As for Banach space settings, a couple of suﬃcientconditions are proposed by Brézis, Crandall and Pazy [3]. Let A and B be maximal
monotone operators from E into E ∗ . Their results ensure the maximal monotonicity ofA + B in E × E ∗ if one of the following conditions is at least satisﬁed:(i) D(A) ∩ (IntD(B)) 6= ∅,(ii) B is dominated by A, i.e., D(A) ⊂ D(B) and kB(u)kE ∗ ≤ kkA(u)kE ∗ + `(|u|E ) forall u ∈ D(A) with k ∈ (0, 1) and a non-decreasing function ` in R,(iii) B = ∂E φ with a proper, lower semicontinuous convex function φ : E → (−∞, +∞],andφ(Jλ u) ≤ φ(u) + Cλ for u ∈ D(φ) and λ > 0,(1.4)
where Jλ denotes the resolvent of A in E.
Here we write kCkE ∗ := inf{|c|E ∗ ; c ∈ C} for each non-empty subset C of E ∗ , andfurthermore, the resolvent Jλ : E → D(A) is given such that uλ := Jλ u is a uniquesolution of FE (uλ − u) + A(uλ ) 3 0, where FE stands for the duality mapping between Eand E ∗ , for each u ∈ E.
However, these results could not be applicable directly to our setting for (1.1). Asfor (i), neither D(∂E φ1 ) nor D(∂E φ2 ) might have any interior points in E (= Lp (Ω)).
Condition (ii) cannot be checked unless we impose an appropriate growth condition onβ. Condition (iii) is available for the case that p = 2, because the duality mapping FEof E = L2 (Ω) is the identity and the resolvent Jλ for ∂E φ2 has a simple representation
formula,(Jλ u)(x) = (1 + λβ)−1 (u(x)) for a.e. x ∈ Ω,(1.5)
which enables us to check (1.4). However, it is somewhat diﬃcult to check (1.4) for thecase that p 6= 2. Actually, the relation between the resolvents of ∂E φ2 and β is unclear,since the duality mapping FE is severely nonlinear whenever p 6= 2.
In this talk we propose a new suﬃcient condition for the maximality of ∂E φ1 + ∂E φ2in E × E ∗ such that the representation formula (1.5) in L2 (Ω) can be eﬀectively used inapplications to nonlinear elliptic operators such as (1.1). More precisely, we introduce aHilbert space H as a pivot space of the triplet E ,→ H ≡ H ∗ ,→ E ∗ and an extension φH
2of φ2 to H, and moreover, we give a suﬃcient condition for the maximality in terms ofthe resolvent and the Yosida approximation in H for ∂H φH
2 .

References[1] H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dansles Espaces de Hilbert, Math Studies, Vol.5 North-Holland, Amsterdam/New York,1973.
[2] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial diﬀerential equations, in: E. Zarantonello (Eds.), Contributions toNonlinear Functional Analysis, Academic Press, New York-London, 1971, pp.101–156.
[3] H. Brézis, M.G. Crandall, A. Pazy, Perturbations of nonlinear maximal monotone
sets in Banach space, Comm. Pure. Appl. Math. 23 (1970), 123–144.

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