A note on elliptic type boundary value problems with maximal monotone relations.
In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as “coefficients”. A reformulation of the respective problems is constructed such that they turn out to be unitary equivalent to inverting a maximal monotone relation in a Hilbert space. The method is based on the idea of “tailor-made” distributions as provided by the construction of extrapolation spaces, see e.g. [Picard, McGhee: Partial Differential Equations: A unified Hilbert Space Approach. DeGruyter, 2011]. The abstract framework is illustrated by various examples.
Institut für Analysis
A note on elliptic type boundary value problems with maximal monotone relations.
lliptic Differential Equations in Divergence form; maximal monotone relations; Gelfand triples
Mathematics subject classification 2010: 35J60, 35J15, 47H04, 47H05
In mathematical physics elliptic type problems play an important role, in analyzing various equilibria as for example in potential theory, in stationary elasticity and many other types of stationary boundary value problems. Classical monographs, focusing mainly on linear problems, are for instance [1, 11, 13, 15]. We also refer to [27, Chapter VIII] for a survey of the literature. Also non-linear elliptic type problems have been studied intensively. The authors of [3, 4] study non-linear perturbations of a selfadjoint operator and obtain existence of a solution. Later-on, uniqueness results could be proved, see [2, 26]. Operators in divergence form with non-linear coefficients are studied in [5, 6, 7, 8, 29], where some monotonicity condition is imposed on the coefficients to obtain existence. This monotonicity condition might also be very weak, cf. . The case of divergence form operators with multi-valued coefficients is treated among other things in , where also existence results could be obtained. In this note, we restrict ourselves to the Hilbert space setting and study conditions under which abstract divergence form operators with possibly multi-valued coefficients lead to well-posed operator inclusions. The restriction to the Hilbert space case enables us to show continuity estimates also for inhomogeneous boundary value problems of elliptic type, cf. the Corollaries 3.1.3 and 3.1.5, where – to the best of the authors’ knowledge – the first one is new. The main topic is the discussion of the structure of the following type of problem: Let , and , be Hilbert spaces and let be given. Moreover, let be a relation such that becomes a Lipschitz-continuous mapping (the main focus will be laid on -maximal monotone relations, which will be defined below), densely defined closed linear. We study the problem of finding such that the inclusion
holds true, i.e. there exists such that
We want to find the “largest” space to allow for existence results and the “largest” space to yield uniqueness. Endowing and with suitable topologies, we seek a solution theory for these type of inclusions. We will give a framework in order to cover inhomogeneous boundary value problems with Dirichlet or Neumann boundary data. Compatibility conditions such as in [10, Theorem 4.22] arise naturally in our approach, cf. also Remark 3.
Our approach consists in rewriting ( ‣ 1) as an inclusion in “tailor-made” distributions spaces by introducing suitable extrapolation spaces, which are also known as Sobolev chains or Sobolev towers, see e.g. [12, 22] and the references therein. The core idea is to generalize extrapolation spaces to the non-selfadjoint operator case. This was also done and extensively used in  for studying time-dependent problems. Using this extrapolation spaces the abstract problem ( ‣ 1) turns out to be unitary equivalent to the problem of inverting the relation in a suitable space. Since elliptic type problems are not well-posed in general, one has to develop a suitable framework in order to determine possible right-hand sides. We discuss some preliminary facts in Section 2 used in Section 3.1, which are particularly needed for the Theorems 3.1.1, 3.1.2, 3.1.4 and the Corollaries 3.1.3, 3.1.5. These theorems and corollaries are the main results of this paper. We discuss extrapolation spaces in Section 2.1. Section 2.2 contains some results in the theory of maximal monotone relations. Most importantly, the following problem is discussed: When is a composition of a orthogonal projection with a maximal monotone relation again maximal monotone? This question was also addressed in [5, 14, 20, 25]. Particularly in , this question was, at least for our purposes, satisfactorily answered. For easy reference, we also state some well-known results in the theory of maximal monotone relations in Section 2.2. In Section 3.1 we apply the results of the previous ones to give an abstract solution theory for both homogeneous and inhomogeneous boundary value problems of elliptic type. In Section 3.2 we will give some examples, how the abstract theory could be employed to study boundary value problems in potential theory, stationary elasticity and magneto- and electro-statics.
The underlying scalar field of any vector space discussed here is the field of complex numbers and the scalar product of any Hilbert space in this paper is anti-linear in the first component.
2 Functional analytic preliminaries
2.1 Operator-theoretic framework
Definition (modulus of , cf. [17, Vi 2.7]).
Let be Hilbert spaces. Let be a densely defined closed linear operator. The operator is non-negative and selfadjoint in . We define the modulus of . It holds .
Definition (extrapolation spaces, Sobolev chain).
Let be a Hilbert space. Let be a densely defined closed linear operator and such that is contained in the resolvent set of . Define to be the Hilbert space endowed with the norm . Define and let be the completion of with respect to the norm . The triple is called (short) Sobolev chain.
It can be shown that is unitary. Moreover, the operator has a unique unitary extension, cf. [23, Theorem 2.1.6].
Sometimes it is useful to identify with , the dual space of (cf. [23, Theorem 2.2.8]). This can be done by the following unitary mapping
where denotes the Riesz-mapping of . Its inverse is given by
By this unitary mapping we can identify for with the functional
We apply the above to the following particular situation. It should be noted that at least for selfadjoint operators a similar strategy has been presented in . Let be Hilbert spaces and let be a densely defined closed linear operator such that the range of , , is closed in . Recall that and . The main idea of formulating elliptic type problems is to use the Sobolev chain of the modulus of
and the modulus of the respective adjoint.
The following holds
Let . Then we have111Occasionally, we will identify an operator with its graph, i.e., .
We note that since is closed, the operator is continuous by the closed graph theorem. We may show a similar property for .
It holds and .
The first equality is clear. Moreover, we deduce that is continuous and closed. Hence, . ∎
The operators and are continuously invertible. Moreover, the operator
is unitary and the operator
can be extended to a unitary operator from to .
As and are continuously invertible, so is . Thus, the spectral theorem for self-adjoint operators implies the continuous invertibility of . Interchanging the roles of and , we get the continuous invertibility of . Now, let . Then we have
Since the operator is clearly onto and hence unitary. Now, for it suffices to show that the norm is preserved for . Let . Using [23, Lemma 2.1.16] for the transmutation relation , we conclude that
We can construct the Sobolev chains of the operators and , respectively. The operator can then be established as a bounded linear operator for (cf. [23, Lemma 2.1.16]). In virtue of Remark 2.1.1(b), the element for can be interpreted as a bounded linear functional on . If denotes the partial isometry such that (cf. [17, VI 2.7, formula (2.23)]), we compute for
We clearly have and . Since is defined as the completion of with respect to the norm and since this norm is equivalent to the norm , we also get . Clearly the analogue results hold for the Sobolev chains of and .
2.2 Maximal monotone relations
We begin to introduce some notions for the treatment of relations.
For a binary relation and an arbitrary subset we denote by
the post-set of under and by
the pre-set of under .
The relation is called monotone if for all pairs the following holds
and maximal monotone, if for ever monotone relation with it follows that .
Finally we define for a constant the relation by
and is called -maximal monotone if is maximal monotone.
A reason for the treatment of maximal monotone relations as natural generalization of positive semi-definite linear operators is the following theorem.
Theorem 2.2.1 ([19, Theorem 1.3]).
Let be monotone, . Then the resolvent of is Lipschitz continuous with . If in addition is maximal monotone, then . In particular, if is maximal monotone then is Lipschitz continuous with .
In Section 3, in particular in the Theorems 3.1.2 and 3.1.4, we want to deduce from the maximal monotonicity of a relation in the Hilbert space the respective property for the relation , where denotes the orthogonal projection onto a closed subspace . The question whether a product of the type , for some continuous , is again maximal monotone is addressed in various publications, cf. e.g. [5, 14, 20, 25] and the references therein. In particular, in  conditions are given for the case of real Hilbert spaces. The author of  uses the theory of convex analysis in his proof. The methods carry over to the complex case. We gather some results concerning maximal monotone relations without proof.
Theorem 2.2.2 ([25, Theorem 4]).
Let be a Hilbert space, a closed subspace and let be a maximal monotone relation. Moreover, assume that . Denote by the orthogonal projection onto . Then the relation is maximal monotone.
Let be a Hilbert space, a closed subspace. Denote by the orthogonal projection onto . If and is -maximal monotone with , then is -maximal monotone.
Let be a Hilbert space, such that is Lipschitz-continuous. For we . Then is Lipschitz-continuous with the same Lipschitz-constant as .
The proof is straight-forward and we omit it.
If is maximal monotone, then is also maximal monotone (cf. [28, Lemma 3.37]).
3 Solution theory for elliptic boundary value problems
3.1 Abstract theorems
The first theorem comprises the essential observation of the whole article. It may be regarded as an abstract version of homogeneous boundary value problems for both the Dirichlet and the Neumann case.
Theorem 3.1.1 (solution theory for homogeneous elliptic boundary value problems).
Let be Hilbert spaces and let be a densely defined closed linear operator and such that is closed. Define and let such that is Lipschitz-continuous. Then for all there exists a unique such that the following inclusion holds
Here stands for the continuous extension of . Moreover, the solution depends Lipschitz-continuously on the right-hand side with Lipschitz constant . 222For a Lipschitz continuous mapping between two metric spaces and , we denote by
the best Lipschitz constant.
In other words, the relation defines a Lipschitz-continuous mapping with .
It is easy to see that if and only if . Hence, the assertion follows from , Theorem 2.1.4 and the fact that is Lipschitz-continuous on . ∎
In view of Theorem 2.2.2, there are many maximal monotone relations such that their respective projections to is maximal monotone. In order to apply Theorem 3.1.1 one encounters the difficulty to show that is closed. By the closed graph theorem, the closedness of is equivalent to the following Poincare-type estimate
A sufficient condition on the operator to have closed range is that the domain is compactly embedded into the underlying Hilbert space . Indeed, in this case, it is possible to derive an estimate of the form (3.1) and therefore our solution theory is applicable.
The latter theorem also gives a possibility to solve the inverse problem, i.e., to determine the “coefficients” from the solution mapping “”. If is thought to be a -maximal monotone relation in such that then it is only possible to reconstruct the part , where denotes the orthogonal projection onto .
Now, we introduce an abstract setting for dealing with inhomogeneous boundary value problems. For this purpose we need a second operator which is in the Dirichlet-type case an extension and in the Neumann-type case a restriction of our operator . For simplicity we just treat the case where is -maximal monotone and .
Theorem 3.1.2 (solution theory for inhomogeneous Dirichlet-type problems).
Let be two Hilbert spaces and , be two densely defined closed linear operators with and closed. Furthermore, let be -maximal monotone for some with . Then for each there is a unique with
where is again the restriction of .
Denote by the orthogonal projector onto . We set , and obtain again a -maximal monotone relation with . We show that is a solution of (3.2) if and only if is the solution of
Indeed, if satisfies this inclusion, then we find such that and . By definition of this implies and since we get . This means solves the problem (3.2). If, on the other hand, satisfies (3.2), then we find such that and . Since this implies and hence solves the problem (3.3). Since (3.3) has a unique solution in by Theorem 3.1.1 and Corollary 2.2.3, we get the assertion. ∎
We may now show a continuity estimate. The proof for this estimate is adopted from [28, Section 2.5].
Corollary 3.1.3 (continuity estimate, Dirichlet case).
Let be as in Theorem 3.1.2. Then there exists such that for all , with333Here, for a relation for Hilbert spaces the adjoint relation is defined as where the orthogonal complement is with respect to the scalar product of . and the respective solutions of
the following estimate holds
From the proof of Theorem 3.1.2, we know that satisfies
Hence, there exists such that
and the respective property for , where is replaced by . Let such that for all we have . Then we compute with the help of :
Thus, it suffices to estimate . To this end, let be such that . Using the monotonicity of and the definition of , we conclude that
Applying the Cauchy-Schwarz-inequality to the left-hand side, we get for
For small enough, this yields an estimate for in terms of , and . ∎
The norm in the above corollary can be interpreted as the “graph-norm” of . We also shall briefly discuss two extreme cases of the above corollary. Since is a linear relation, . Thus, we have a continuous dependence result for varying right-hand sides and fixed boundary data. If is a bounded linear mapping, then . Therefore the condition is trivially satisfied and the term can be estimated by , where is the operator norm of .
Theorem 3.1.4 (solution theory for inhomogeneous Neumann-type problems).
Let be two Hilbert spaces and be two densely defined closed linear operators with and closed in . Furthermore, let be -maximal monotone for some with . Then for each with 444This means that we find an element such that in the sense of Remark 2.1.1(b) there exists a unique such that
Consider the following problem of finding such that
holds, where and satisfies with on . Note that such a choice for is possible, since is closed. Indeed, and are both closed linear operators restricting . Hence, and are closed subspaces of . Thus, the norms of the spaces and are equivalent on and therefore is closed.
We show that the problem (3.6) is equivalent to (3.4). Then the assertion follows from Theorem 3.1.1 and Corollary 2.2.3. So let be a solution of (3.6). That means that we find such that and . This, however, implies and for we compute
Moreover, for , we have and . Thus, is a solution of (3.4) in the stated sense. If, on the other hand, solves problem (3.4), then we find with and . It suffices to show that and coincide on . For this purpose let . Then we compute
Hence, by the definition of we derive that solves (3.6) with