Keywords:anisotropic 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):
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.,
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 .
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:
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 introduce the classes of vector fields
Now, we introduce the relaxed functional, which plays a basic role in proving the existence and uniqueness of the problem.
In , Amer and Belletini obtained the following result:
Moreover, in , Moll proved the representation result:
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).
Next we give the main definition in this paper that is the strong solution of problem (1.1).
The main results of this paper are the following.
Theorem 3.3Let. Assume thatfsatisfies (2.11), then there exists a unique strong solution of (1.3) infor everysuch that. Moreover, if, are the strong solutions of (1.3) corresponding to initial data, , respectively, then
Theorem 3.4Let. Assume thatfsatisfies (2.11) andsatisfies (M), then there exists a unique strong solution of (1.1) infor everysuch 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
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).
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 )
To prove the existence and uniqueness of a strong solution of problem (1.3), we also need the next proposition.
The following lemmas will be used to prove Proposition 4.1 and Proposition 4.2.
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 .
Thus, (ii) holds.
Using (ii) in (4.5) we have
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:
The operator satisfies the classical Leray-Lions assumption . Hence, for every , the operator satisfies .
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
Proof We divide the proof into two steps.
By (4.15), it follows that
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 ,
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 as a test function in (4.21), taking as a test function in (4.22), and by Theorem 2 in , 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. □
Thus, to prove Theorem 3.4, we only need to obtain the following result.
By the above inequality and ℱ being lower semicontinuous in (4.25), we have 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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
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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