On the nonlinear perturbations of self-adjoint operators

2.1 Linear operators in Hilbert spaces

Most of the basic results from the theory of Hilbert space operators can be transferred, with proper caution, from the complex to the real case. The reader is referred, for example, to [5], where special attention is drawn to the subtleties arising when working in a real Hilbert space.

By a (linear) operator T in ℋ we understand a linear mapping T : ℋ ⊃ D ( T ) ⟶ ℋ with the domain D ( T ) . We denote by B ( ℋ ) the Banach algebra of all bounded operators from the whole space ℋ into itself. Below we mainly deal with unbounded operators in ℋ , which makes the situation more delicate because their domains are in general strictly contained in ℋ . For operators S and T in ℋ , we write S ⊂ T if D ( S ) ⊂ D ( T ) and S x = T x for every x ∈ D ( S ) . We say that an operator T in ℋ is closed if its graph { ( T x , x ) : x ∈ D ( T ) } is a closed subspace of the product Hilbert space ℋ ⊕ ℋ , or equivalently D ( T ) is a Hilbert space under the graph norm ‖ ⋅ ‖ T given by

If T is a densely defined operator in ℋ (i.e., the closure of D ( T ) equals to ℋ ), then we denote by T ∗ its adjoint (being also a linear operator in ℋ ). When T = T ∗ we say that the operator T is self-adjoint. Analogously as in the complex case, each self-adjoint operator T in ℋ can be written as the spectral integral ∫ σ ( T ) t E ( d t ) with respect to a unique spectral measure E (see [5, Theorem 7.17]), i.e.,

where σ ( T ) stands for the spectrum of T . This representation together with some other basic properties of spectral integrals will be utilized in Section 3. Let us add that by ρ ( T ) we will mean the resolvent set of T , that is, ρ ( T ) = R ⧹ σ ( T ) .

Furthermore, a self-adjoint operator T in ℋ is called positive if ⟨ T x , x ⟩ ≥ 0 for every x ∈ D ( T ) . In this case, we can find a unique positive self-adjoint operator S in ℋ satisfying the equality S 2 = T ([5, Theorem 7.20a]). Such an operator S , denoted further by T 1 / 2 , is called the square root of T . Moreover, if U ∈ B ( ℋ ) , then U T ⊂ T U if and only if U T 1 / 2 ⊂ T 1 / 2 U . It is also worth noting that D ( T 1 / 2 ) is in general larger than D ( T ) .

Now let T be a closed densely defined operator in ℋ . We define its modulus as ∣ T ∣ = ( T ∗ T ) 1 / 2 . It turns out that D ( T ) = D ( ∣ T ∣ ) ⊂ D ( ∣ T ∣ 1 / 2 ) . The polar decomposition theorem ([5, Theorem 7.20b]) guarantees that there exists a unique partial isometry U T ∈ B ( ℋ ) such that T = U T ∣ T ∣ and the kernels of U T and T coincide. By a partial isometry U T we mean an operator such that ‖ U T x ‖ = ‖ x ‖ for every x belonging to the orthogonal complement of the kernel of U T in ℋ . It can be shown that if T is self-adjoint, then U T ∣ T ∣ = ∣ T ∣ U T .

2.2 Monotone operators

Following [6] or [7] we provide necessary background information on the theory of monotone operators, which pertains to the existence of nonlinear equations. We consider a (nonnecessarily linear) mapping A : ℋ ⟶ ℋ ∗ called further an operator.

We say that A is demicontinuous if for every sequence ( x n ) ⊂ ℋ and x 0 ∈ ℋ such that x n → x 0 we have A ( x n ) ⇀ A ( x 0 ) , where ⇀ denotes the weak convergence in ℋ . We say that A is monotone if

An operator A is called strongly monotone (or α -strongly monotone) if there exists α > 0 such that

In turn A is said to be relaxed monotone (or β -relaxed monotone) if there exists β ∈ R such that

Recall that the Gâteaux derivative of a mapping F : ℋ ⟶ K , where K is a real Hilbert space, at the point x ∈ ℋ is the functional F ′ ( x ) ∈ ℋ ∗ satisfying

F is called Gâteaux differentiable if it has the Gâteaux derivative at each point. Moreover, we denote by F ″ the second derivative of F , that is, the Gâteaux derivative of F ′ . An operator A is called potential if there exists a Gâteaux differentiable functional A : ℋ ⟶ R , called the potential of A , such that A ′ = A . Note that for a given operator A , a potential of A (if exists) is uniquely determined up to a constant. The Gâteaux differentiability of a functional in general does not imply any type of its continuity. However, every monotone and potential operator is necessarily demicontinuous (see [6 Lemma 5.4]). We generalize this observation to the case of relaxed monotone and potential operators.

Next we recall main existence and uniqueness tools used in this article.

It is worth noting that the proof of the above theorem presented in [6] works in separable Hilbert spaces only. However, this result can be extended to the general case following ideas described in [7, Section 7].

2.3 Analysis on the Euclidean space

In this section, we present the results that, for a given mapping, relate the relaxed monotonicity to the Lipschitz condition. These tools will be extended to the infinite dimensional case in Section 3. The symbols C ( R m ) , C 1 ( R m ) , and C ∞ ( R m ) stand for the spaces of all, respectively, continuous, continuously differentiable, and smooth functionals on R m . We denote the support of a functional f ∈ C ( R m ) by supp f = { x ∈ ℋ : f ( x ) ≠ 0 } ¯ and we put C c ∞ ( R m ) = { f ∈ C ∞ ( R m ) : supp f is compact } . The Euclidean space R m will be considered with the standard inner product ⟨ ⋅ , ⋅ ⟩ .

Let us fix a sequence of mollifiers ( ρ n ) , that is, the functions such that

where B 1 / n is an open ball centered at the origin with radius 1 n . Given f ∈ C ( R m ) , we define the n -th approximation f n of f by

where ⋆ stands for a convolution operator. Proposition 4.21 of [8] provides an almost uniform convergence f n → f , that is, a uniform convergence f n ∣ K → f ∣ K for every compact set K ⊂ R m . Moreover, if we assume that f ∈ C 1 ( R m ) , then we obtain f n ′ → f ′ almost uniformly, because

(see [8, Proposition 4.20]). Let us recall that for a self-adjoint operator L ∈ B ( R m ) we have ‖ L ‖ = sup ‖ x ‖ = 1 ‖ L x ‖ = sup ‖ x ‖ = 1 ∣ ⟨ L x , x ⟩ ∣ . In order to extend this observation to the case of nonlinear mappings, we have to assume some analog of the symmetry L = L ∗ . For operators of the class C 1 , such a role plays the symmetry of the derivative operator at each point, which is equivalent to the operator’s potentiality. This is the subject of the following lemma.

2.4 Functional framework

Let Ω be an open bounded subset of R m with a boundary of the class C 2 (see [9] for details). We denote by H 1 ( Ω ) (resp. H 2 ( Ω ) ) the first Sobolev space (resp. the second Sobolev space) consisting of all functions from L 2 ( Ω ) , whose all first (resp. second) order weak derivatives belong to L 2 ( Ω ) . We equip H 1 ( Ω ) with the norm

Let C 0 ∞ ( Ω ) denote the space of all smooth functions with a compact support in Ω . By H 0 1 ( Ω ) we mean the closure of C 0 ∞ ( Ω ) in the norm ‖ ⋅ ‖ H 1 . For u ∈ H 2 ( Ω ) , we denote the Laplacian of u by

Following [10] we put

Here ∂ ν is the outward normal derivative defined on ∂ Ω . Moreover, the condition ∂ ν u = 0 is understood in sense of the trace operator (see [9, Section 5.5] for details). Spaces D ( Δ D ) and D ( Δ N ) corresponds to the Dirichlet condition: u = 0 on ∂ Ω and the Neumann condition: ∂ ν u = 0 on ∂ Ω , respectively.

f : Ω × R × R m ⟶ R is called a Carathéodory function if for a.e. x ∈ Ω and all u ∈ R , v ∈ R m the function f ( ⋅ , u , v ) is Lebesgue measurable, while functions f ( x , ⋅ , v ) and f ( x , u , ⋅ ) are continuous. We define (pointwisely a.e.) the Niemytskii operator N f associated with f by the formula

The domain and the codomain of N f will be specified later. In a similar way, we can define the Niemytskii operator associated with a Carathéodory function f : Ω × R ⟶ R (compare with [11, Chapter 2]). For the reader’s convenience, we recall the Krasnoselskii (see [11, Theorems 2.4 and 2.5]) and Gagliardo-Nirenberg-Sobolev (see [9, Section 5.6]) theorems in suitable forms.

3.1 Regularity of a solution

Take a self-adjoint operator T in ℋ and an operator N : D ( ∣ T ∣ 1 / 2 ) ⟶ ℋ . Let us consider the following equation:

where x ∈ D ( T ) . We define a bilinear form t : D ( ∣ T ∣ 1 / 2 ) × D ( ∣ T ∣ 1 / 2 ) ⟶ R by the formula

where T = U T ∣ T ∣ is the polar decomposition of T . Now we are in the position to formulate an abstract counterpart of regularization results.

Proposition 3.1 does not share the ambiguity about the relation between strong and weak solutions for problems driven by self-adjoint operators presented in [12, Section 3.4].

3.2 The case of potential perturbations

Take a self-adjoint operator T : ℋ ⊃ D ( T ) ⟶ ℋ , the spectral measure E of T , and a Gâteaux differentiable functional N : D ( ∣ T ∣ 1 / 2 ) ⟶ R . We assume that for every x ∈ D ( ∣ T ∣ 1 / 2 ) there exists N ( x ) ∈ ℋ satisfying

for every y ∈ D ( ∣ T ∣ 1 / 2 ) . Note the above assumption means that for every x ∈ D ( ∣ T ∣ 1 / 2 ) , the functional N ′ ( x ) has a representation with respect to the inner product in ℋ , while the differentiability of N ensures only that such a representation exists with respect to the inner product in D ( ∣ T ∣ 1 / 2 ) . The meaningness of these assumptions is well described by the following remark.

We put

For a fixed h ∈ ℋ , we define J h : D ( ∣ T ∣ 1 / 2 ) ⟶ R by

The space D ( ∣ T ∣ 1 / 2 ) will be equipped with the norm

where δ , ω > 0 . Note that, for arbitrarily chosen constants δ and ω , the norm ‖ ⋅ ‖ δ , ω is equivalent to the graph norm of ∣ T ∣ 1 / 2 . The space D ( ∣ T ∣ 1 / 2 ) becomes a Hilbert space when equipped with norm (3.8).

Let d α , β ( t ) = dist ( { t } , ( α , β ) ) for t ∈ R . The function d α , β plays only a technical role and enables us to avoid more complex calculations.