Lower Semicontinuity in of a Class of Functionals Defined on with Carathéodory Integrands

Abstract

We prove lower semicontinuity in for a class of functionals of the form where , is open and bounded, for each satisfies the linear growth condition and is convex in depending only on for a.e. Here, we recall for ; the gradient measure is decomposed into mutually singular measures and . As an example, we use this to prove that is lower semicontinuous in for any bounded continuous and any Under minor addtional assumptions on , we then have the existence of minimizers of functionals to variational problems of the form for the given due to the compactness of in

1. Introduction

We prove an lower semicontinuity result for convex linear growth functionals , defined on whose integrands are radially symmetric in for a.e. that is, depends on for each and satisfies a fairly general structure condition. Our results expand the class of integrands from those of the form as presented in [1], for which lower semicontinuity in holds.

We use the conjugate function of to prove our main result, Theorem 1 in Section 3. The conjugate function is used, for example, in [2] to approximate for the given by a sequence for to prove the existence of the corresponding gradient time flow, although is assumed to be continuous in in these cases. Thus, one advantage of Theorem 1 in this paper is that we can also obtain the existence results for time flow by deriving a similar convergence result for our case, but with no continuity assumption in the variable. In fact for the results presented here, may contain singularities in as in Example 3. In addition, the corresponding convergence results in [3] assume another continuity condition on in similar to (3) which is not covered by our assumptions on .

We note that the integrands considered in this paper (of the form (1)) have been used in models in applications of image processing [2]. However, as mentioned above, the main result of this paper covers a larger class of integrands.

We assume throughout, unless otherwise stated, that is bounded and open and is radially symmetric in for a.e. and is convex in for a.e. so that for each and for each Since for a.e. is convex and real valued in it is well known that must be continuous in for a.e. ; hence, is a Carathéodory function. Furthermore, we assume the linear growth of so that where

As stated above, we make no continuity assumption for the variable for Additionally, in contrast to the works of [3–8] and [9] we do not assume to be lower semicontinuous in Also, our assumptions on are not covered by the class of integrands and in [10, 11]. The integrand class in [10] requires joint continuity of in and and and the integrands in [11] require for our case be defined on with which may not hold if it is only assumed for each as may contain singularities.

2. Mathematical Preliminaries

We recall by definition that if and only if and in which case the total variation measure is decomposed in to where and using the Lebesgue decomposition theorem [12]. Functionals defined for with Carathéodory integrands of linear growth (2) and the convex in the variable are defined [4, 5, 13, 14] by

However, it is not immediate that functionals defined by (5) are lower semicontinuous in As noted above, lower semicontinuity was was proven for certain integrands , but to the best of our knowledge, there is no general lower semicontinuity result for convex Carathéodory functions where for each ,

We will also use the conjugate function of where [8]. We note that as is convex in and is convex in

In [1], lower semicontinuity of for integrands of the form (1) is proved for convex in radially symmetric in for a.e. and a fairly general structure condition on which does not assume continuity in The proof is based on proving that where is the conjugate function of and the last equality follows from integration by parts for [12]. Lower semicontinuity in immediately follows as the final equality is the supremum of functionals, each continuous in In the next section, we use the method above to prove our main result, Theorem 1.

3. Main Results

We first define

Theorem 1. Assume with

are both radially symmetric and convex in for a.e. and if for each , is convex in and there exists such that a.e. for all where as Additionally, assume the following structure condition on that is, for some , we have where and where is in the variable open each a.e. and independent of , for each . Then, for defined by we have in fact

Thus, the functional defined on is lower semicontinuous in that is, if in , then, Moreover, if is Lipschitz, then, for the given , and hence, the functional defined on is lower semicontinuous in Here, is defined on in the sense of trace [12].

Proof. From the above assumptions on , we have Similarly, we have giving
From the above estimate for we have if and only if if and only if and hence, ☐

Now, with and from the above and is defined by

We now show as . In fact, from Taking supremum over both sides gives

Similarly, we have giving

Now, let in (17) to get

For the second claim in the theorem, as is Lipschitz, the continuous trace operator [12]) exists and (13) follows as in the proof above. Finally, lower semicontinuity of follows from Theorem 5 in [1].

Remark 2. We note that the condition may be modified if in the expression , one of the corresponds to That is, if, e.g., , we may only require that as each is a Lebesgue point of by continuity In fact, we have, noting that

The last term is bounded by which approaches 0 as on the common Lebesgue set of The next to last term approaches a.e. as since is continuous in wherever are defined. The Lebesgue set of thus contains the Lebesgue set of independent of The rest follows exactly as in the proof of Theorem 4 in [15], but with and replacing and and Remark 2 in [1].

Example 3. For , , the functional, is lower semicontinuous on

Proof. Letting we have where Letting , we have for and hence, as by Lebesgue’s dominated convergence theorem, as a.e. Note that is convex in since by defining We see the left derivative at is while the right derivative at is Thus, is convex in As is also increasing in , we have that is convex in ☐

The conditions for Theorem 1 are thus satisfied for and with noting Remark 2. Hence,

For the case , we note that for a.e. and for each As in the proof of Lemma 2 in [16] and Theorem 1 above, we have for a.e all and similar to the above estimates (17), for each , we have

Letting gives

Lower semicontinuity of immediately follows.

We finally note that a version of Theorem 1, along with Remark 2, holds for nonradially symmetric integrands but with the additional smoothness assumption that for a.e.

Theorem 4. Assume that with

where is convex and in for a.e. and if for each , is both convex and in and there exists such that a.e. for all where as Additionally assume the following structure condition on that is, for some we have where and where is in the variable open each a.e. and independent of , for each . Then

Thus, the functional defined on is lower semicontinuous in Moreover, if is Lipschitz, then, for the given , and hence, the functional defined on is lower semicontinuous in

Proof. The proof is the same as the proof of Theorem 1, noting Theorem 4 in [15] and Remark 2 in [1].☐

We immediately have from standard theory the following existence result:

Corollary 5. Let satisfy the assumptions of Theorems 1 and 4 or Remark 2. If in addition we have

Then, for the given the functionals have a minimizer in Furthermore, the minimizer is unique if is strictly convex in

Proof. For this follows from lower semicontinuity of in and standard compactness results for , noting that assumptions (42) and (43) imply using (5) that Thus, minimizing sequences of are bounded in the norm so that there is with in [12], and hence, The proof is essentially the same for using (13). Finally, if is strictly convex, then, so is and Thus, if there are minimizers then, we have a contradiction.☐

We finally remark, as noted in [1], that Theorems 1 and 4 of this paper may be extended to vector-valued functions where is an matrix with for each and is defined by writing as a vector of length with and depending on for the case of Theorem 1. We may also consider integrands with appropriate assumptions on , such as Lipschitz continuity in using similar methods as presented here and in [1, 15, 16].

4. Conclusion

In this paper, we have expanded the class of functionals defined on the space which are lower semicontinuous to include certain integrands which, for each are only assumed to be in The structure condition for which lower semicontinuity holds is fairly general and is for many cases not difficult to verify. Furthermore, as mentioned above, using the method presented here, we may expand the main theorem of this paper to include functionals of the form for vector-valued functions Finally, as noted above, lower semicontinuity is essential to proving the existence of minimizers of functionals of the form over

Data Availability

As this is a theoretical paper, there is no data but all references used throughout the manuscript are found in the bibliography at the end of the paper (see below) and can be easily accessed, although I do not believe that they are publicly archived. The references list all the necessary information for accessing the papers. Also, I have the full papers recently published by the author (Thomas Wunderli) as listed in the bibliography and will be provided by the author upon request.

Conflicts of Interest

The author(s) declare(s) that they have no conflicts of interest.