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An anisotropic quasilinear problem with perturbations
Boundary Value Problems volume 2013, Article number: 147 (2013)
This work focuses on proving the existence and uniqueness of strong solutions of perturbed anisotropic total variation flow with the Neumann boundary condition when the initial data is an function.
Problems of general anisotropic total variation flow arise in a number of areas of science. The parabolic equations represent what Giga et al. called a very singular diffusivity (see ) and are a natural generalization of the total variation flow in the presence of an anisotropy. In the isotropic case, the equation becomes when the Lagrangian is given by , where is the usual -norm; i.e., . Let us recall that this PDE appears when one uses the steepest decent method to minimize the total variation. This method was introduced by Rudin and Osher (see [2, 3]) in the context of image denoising and reconstruction. In the last years, its applications have been studied by many authors (see [4–7]).
Let Ω be an open bounded subset in , , with boundary ∂ Ω of class . In this paper, we are interested in the problem
where , is a 1-homogeneous convex function with linear growth as , is the Neumann boundary operator associated to , i.e., with ν the unit outward normal on ∂ Ω, and the function satisfies the following assumptions, which we shall refer to collectively as (M):
(M1) For almost all , is continuous nondecreasing, and ;
(M2) For every , is in .
As argued in , the choice of Neumann boundary conditions is a natural choice in image processing. It corresponds to the reflection of the picture across the boundary and has the advantage of not imposing any value on boundary and not creating edges on it. For instance, in , Andreu, Caselles and Mazón considered the elliptic problem with Neumann boundary conditions. In , Andreu et al. obtained the existence and uniqueness of entropy solutions of quasilinear parabolic equation with the Neumann boundary, i.e.,
where , , and satisfies some additional assumptions. Our problem is closely related to motion under anisotropic mean curvature flow (see ) when . If we take the -distance to give a set E as an initial condition ( being the polar function of f), then each sublevel set of the anisotropic mean curvature motion behaves instantaneously as the solution of Cauchy problem (1.1) where . Recently Moll  proved the existence and uniqueness of the solutions of Dirichlet problem (1.1) with . As we all know, it is possible that the solution of (1.1) will blow up with perturbations. Therefore, in this paper, we extend the problem introduced in Moll  and obtain the existence and uniqueness of strong solutions of (1.1) when perturbation term satisfies assumption (M).
This paper is organized as follows. In Section 2 we recall some notions and basic facts. In Section 3 we define the notion of a strong solution for the Neumann problem of (1.1), and give the basic results in this paper. In Section 4 we prove the existence and uniqueness of solutions of an auxiliary equation, i.e.,
and for some we obtain the existence and uniqueness of a strong solution of problem (1.1).
To make precise our notions, let us recall some preliminary facts.
Given , Du decomposes into absolutely continuous and singular parts , where ∇u denotes the Radon-Nikodým derivative with respect to the Lebesgue measure and is its singular part. There is also the polar decomposition , where is the total variation measure of . For further information concerning functions of bounded variation, we refer to .
By we denote the space of weakly measurable functions (i.e., is measurable for every ) such that . Observe that since has separable predual, it follows easily that the map is measurable.
If and , the functional is defined by the formula
Then is a Radon measure in Ω, for all , and is absolutely continuous with respect to with the Radon-Nikodým derivative which is a measurable function from Ω to ℝ such that for any Borel set . We also have that .
In , a weak trace on ∂ Ω of the normal component of is defined. Concretely, it is proved that there exists a linear operator such that and for all if .
Next, let us introduce the concept of generalized total variation of a BV function with respect to a Finsler metric . Let be a Borel function not identically +∞. The function f will be called convex if for any , the function is convex on . We shall say that f is lower semicontinuous (in short l.s.c.) if is lower semicontinuous for any . The function will be called positively homogeneous of degree 1 (in short 1-homogeneous) if it satisfies the following property:
f is a sublinear growth if there exists a positive constant such that
Let us recall that is a Finsler metric if it is a Borel function and it satisfies (2.1) and (2.2). If f satisfies (2.1), then the dual function is defined by . It is easy to verify that is convex, l.s.c. and satisfies (2.1). Then, if we adopt the following conventions: for any , we set ; if and , we get
We say that f is coercive if there exists a positive constant such that
It is easy to see that f is convex and has a sublinear growth, then is continuous for any .
We introduce the classes of vector fields
Let , the generalized total variation of u with respect to f in Ω is defined by
As a direct consequence of the definition, we have that the generalized total variation of u with respect to f in Ω is -lower semicontinuous in Ω.
Now, we introduce the relaxed functional, which plays a basic role in proving the existence and uniqueness of the problem.
We define the functional by
We denote by the relaxed functional of ; i.e.,
In , Amer and Belletini obtained the following result:
Moreover, in , Moll proved the representation result:
where is a representative of the equivalence class of homogeneous integrands associated to sets which are countable and sequentially weakly∗-dense in , and
for all , where . The following useful inequality holds:
The equality holds if and only if the functional defined by (2.5) is -lower semicontinuous on . By the inequality (2.9), we have the measure as follows:
for every Borel set .
In this paper, we assume that is a convex homogeneous integrand, i.e., for some constants
Let us define the functional by the formula
By on ∂ Ω, and Theorem 4 in , it is easy to obtain that the functional is the relaxed functional of F defined by
3 Strong solutions and main results
In this section we give the main concepts and results of Neumann problems (1.3) and (1.1).
Definition 3.1 A function is a strong solution of (1.3) if , there exists with a.e. in and a.e. , such that
and a.e. it holds
Next we give the main definition in this paper that is the strong solution of problem (1.1).
Definition 3.2 A function is a strong solution of (1.1) if , there exists with a.e. in and in a.e. such that
for every and a.e. on .
The main results of this paper are the following.
Theorem 3.3 Let . Assume that f satisfies (2.11), then there exists a unique strong solution of (1.3) in for every such that . Moreover, if , are the strong solutions of (1.3) corresponding to initial data , , respectively, then
Theorem 3.4 Let . Assume that f satisfies (2.11) and satisfies (M), then there exists a unique strong solution of (1.1) in for every such that . Moreover, if , are the strong solutions of (1.1) corresponding to initial data , , respectively, then
4 Proof of the main results
Let us recall the notion of completely accretive operators introduced in . Let be the space of measurable functions in Ω. Given , we shall write that if and only if for all , where . Let A be an operator (possibly multivalued) in , i.e., . We shall say that A is completely accretive if
Let . If , then A is completely accretive if and only if for any , . A completely accretive operator in is said to be m-completely accretive if for any . In that case, by Crandall-Liggett’s theorem, A generates a contraction semigroup denoted by in , which is given by the exponential formula
Let us write , then for any , and it is a mild solution (a solution in the sense of semigroups ) of
such that .
We shall use a stronger notion of the solution of (4.1). We say that is a strong solution of (4.1) on if and for almost all . If (the domain of A) and A is m-completely accretive, then and is a strong solution of (4.1) on for all .
To obtain the solution of problem (1.1), we need the result of problem (1.3). Thus, at first, we will prove the existence and uniqueness of a strong solution of problem (1.3). Let us introduce the following operator in associated to problem (1.3).
if and only if , and there exists , in such that
for all .
Proposition 4.1 The operator is m-completely accretive with dense domain. For any , the semigroup solution is a mild solution of
To prove Proposition 4.1, we need to prove the operator has some characterization, satisfies the range condition and has dense domain in .
By the results of Section 2, the relaxed functional ℱ is convex and lower semicontinuous. Therefore, the subdifferential of ℱ is a maximal monotone operator in , and consequently, if is the semigroup solution in generated by , is a strong solution of the problem (see )
Recall that the operator is defined by if and only if and , .
To prove the existence and uniqueness of a strong solution of problem (1.3), we also need the next proposition.
Proposition 4.2 The operator has dense domain in and .
The following lemmas will be used to prove Proposition 4.1 and Proposition 4.2.
Lemma 4.3 We have the following characterization of the operator , if and only if , and there exists , in such that
for all . Moreover, we have that
for all .
Proof We denote the operator by ℬ defined in the statement of the lemma. Since when , we have . Let , and there exists , in and (4.2). Let , applying results from , we have that there exists a sequence such that in , and . Using as a test function in (4.2) and letting , we obtain (4.5), then we conclude that , therefore .
Letting in (4.5), we get
Moreover, using , we obtain (i).
We take in (4.5) to obtain . Using in (4.5) and (i), we get
Thus, (ii) holds.
Using (ii) in (4.5) we have
Since the same inequality holds for , (iii) is obtained. □
We consider the following possibly multi-valued functions: and . By the convexity of f, it follows that A is a monotone function satisfying
For each , we consider the Moreau-Yosida approximation to defined by
and the Yosida approximation of the multi-valued function is defined as
We have that is a convex Fréchet differentiable function (see ) such that pointwise and a.e. in Ω when and . Moreover, when , we get the minimum in (4.6). In , Moll gave the following estimate:
We consider the operator in to prove Proposition 4.1. Let , we define if and only if , and
for all .
The operator satisfies the classical Leray-Lions assumption . Hence, for every , the operator satisfies .
Moreover, we need the following characterization of the operator .
Lemma 4.4 For every , the operator is completely accretive in .
Proof Let and . Since , taking as a test function in (4.10), we get
Similarly, , we take as a test function in (4.10) and obtain
Using (4.11) + (4.12), we may write that
According to (4.6) and , we obtain that . Moreover, by Lemma 4.3 and Theorem 2 in , we have that
It follows that the operator is completely accretive in . □
Lemma 4.5 The operator satisfies , and is dense in .
Proof We divide the proof into two steps.
Step 1. We first prove . Let , we shall find such that , i.e., there exists such that , -a.e. , and
for all .
Using (4.10) and , we have that for every there is such that and
for all . Since is completely accretive, it is obtained that
Now taking and in (4.14), respectively, we get that
Using the estimate (4.9), (4.8) and Ω being a bounded subset in , we have that
By (4.15), it follows that
where depends on . Moreover, we obtain that for all ,
Thus, is bounded in and we may extract a subsequence such that converges in . Now, by (4.15) and (4.16), we know that in and .
Observe that by (4.6) and (4.17), is bounded in and weakly relatively compact in . We may assume that
By (4.17) and in , we also have that
Given and taking in (4.14), we have
and letting , it follows that
that is, in , and
From the proof of Proposition 4 in , we obtain that . Moreover, by (4.17) and (4.18), it implies that . Now, we prove that u, v and z verify (4.13). Applying the Lebesgue convergence theorem in (4.14), there exists for every ,
To prove (4.13), we assume that there exists . Let and let be such that in as . Using as a test function in (4.20) and letting , we obtain (4.13). That is, .
Step 2. Now let us prove that is dense in . We only need to prove that . Let . By Step 1, we have that for all . Thus, for each , there exists such that and, in consequence, there exists some such that in and
for every . Taking in the above inequality, we get
Letting , it follows that in . This implies that . □
Proof of Proposition 4.1 Let and . Let be such that , and
for every . Taking as a test function in (4.21), taking as a test function in (4.22), and by Theorem 2 in , we have that
We get the operator is completely accretive in .
Now, we prove that is closed. Let such that in . Since , there exists with in such that
for every . Since , we may assume that
Working as before, it is easy to see that . Moreover, since in , we have in , and
Letting in (4.23), and having in mind the lower semicontinuity of the functional ℱ defined in (2.13), we obtain that
Consequently, . By Lemma 4.5, it follows that is m-completely accretive in . By Crandall-Liggett’s theorem, generates a contraction semigroup in given by the exponential formula
The function is a mild solution of
with . □
Proof of Proposition 4.2 We first prove that . Let and , there exists with in such that
for every . Thus, , that is, .
Next, by the proof of Proposition 4.1, we have that the operator is closed. Since and , we have that . □
Proof of Theorem 3.3 As a consequence of Proposition 4.2, the semigroups generated by and by coincide, and therefore is a strong solution of
with , i.e., and for a.e. . Then we have in . By the characterization (i) in Lemma 4.3, we have (3.2) and (3.3) hold. The contractivity estimate (3.5) follows directly from the nonlinear semigroup theory. □
Let us define several operators that will be needed in this section. The single-valued operator is defined in as
Take defined by
where . It is easy to see that and H is convex. Moreover, by Fatou’s lemma, H is lower semicontinuous. Hence, ∂H is a maximal monotone graph in .
Thus, to prove Theorem 3.4, we only need to obtain the following result.
Lemma 4.6 .
Proof From , we have , and there exists , in such that
for all . Since satisfies (M1), there exists such that
By the above inequality and ℱ being lower semicontinuous in (4.25), we have that
We have proved that .
By Proposition 14 in , we have that , and the operator is closed. Hence, . □
Using Crandall-Liggett’s theorem and a similar proof of Theorem 3.3 again, we obtain that Theorem 3.4 holds.
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We would like to thank the anonymous referees for their constructive comments, which were very helpful for improving this paper. The authors acknowledge the financial support of this research by the National Natural Science Foundation of China (Grant No. 10871117), NSFSP (Grant No. ZR2010AM013) and Fundamental Research Funds for the Central Universities (12CX04081A, 11CX04058A).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
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Rui, J., Si, J. An anisotropic quasilinear problem with perturbations. Bound Value Probl 2013, 147 (2013). https://doi.org/10.1186/1687-2770-2013-147
- anisotropic total variation flow
- strong solution