An anisotropic quasilinear problem with perturbations
© Rui and Si; licensee Springer. 2013
Received: 7 December 2012
Accepted: 30 April 2013
Published: 18 June 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.
Keywordsanisotropic total variation flow semigroups strong solution
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]).
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 .
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).
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.
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 .
It is easy to see that f is convex and has a sublinear growth, then is continuous for any .
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.
for every Borel set .
3 Strong solutions and main results
In this section we give the main concepts and results of Neumann problems (1.3) and (1.1).
Next we give the main definition in this paper that is the strong solution of problem (1.1).
for every and a.e. on .
The main results of this paper are the following.
4 Proof of the main results
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).
for all .
To prove Proposition 4.1, we need to prove the operator has some characterization, satisfies the range condition and has dense domain in .
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.
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 .
Moreover, using , we obtain (i).
Thus, (ii) holds.
Since the same inequality holds for , (iii) is obtained. □
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 .
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.
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 .
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, .
Letting , it follows that in . This implies that . □
We get the operator is completely accretive in .
with . □
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 . □
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. □
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 .
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.
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).
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