The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution are proved by means of a suitable pseudoparabolic relaxed approximation of the equation and the corresponding passage to the limit. Our main result is monotonicity in time of the positive and negative singular parts of the distributional space derivative for bounded variation initial data. Sufficient conditions for instantaneous L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-Wloc1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,1}_{{\mathrm{loc}}}$$\end{document} or BV-Wloc1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,1}_{{\mathrm{loc}}}$$\end{document} regularizing effects are also discussed.

The time-dependent minimal surfaces equation has been widely studied since the pioneering paper by Lichnewski and Temam [21] in which the authors proved existence and uniqueness of "pseudo-solutions" for the corresponding Dirichlet problem (in a multidimensional domain).
Several generalizations are available in the literature, both for the inhomogenous Dirichlet problem and for the homogenous Neumann one. In [5,6], F. Andreu, V. Caselles and J. Mazón obtained existence and uniqueness of entropy solutions for general Lagrangians with linear growth with respect to the gradient with similar hypothesis to ours. In a previous work [22], we proved that the homogenous Neumann problem for (P u 0 α,β ) (i.e., α = β = 0) has a unique strong solution for any u 0 in L 2 (Ω) in any dimension.
On the other hand, the inhomogeneous Neumann problem for this type of parabolic equations is much less studied. To the best of our knowledge, even for the case of linear growth functionals, as in the case of the time-dependent minimal surfaces problems, the available results are limited to the study of minimizers and their properties (see [8]) and not to the evolution in time. The only result in this direction in the parabolic case is the existence and uniqueness of entropy solutions to a mixed Neumann-Dirichlet problem for the relativistic heat equation, also in the one-dimensional case ( [4]). We note that the entropy solutions constructed in [4] do not satisfy any reasonable type of time regularity. In fact, the distributional time derivative of the solutions might not be even a Radon measure.
Our purpose in this paper is twofold. First of all, to generalize the results in [22] to the nonhomogenous Neumann case; i.e. to obtain existence and uniqueness of strong solutions to (P u 0 α,β ). Concerning existence, we follow the strategy in [22]; i.e. we consider the following relaxed-pseudoparabolic regularization for problem (P u 0 α,β ) coupled with the corresponding Neumann boundary conditions. We show that they form a family of well-posed approximating problems (P u 0 α,β ) . Coupled with good a priori estimates, we are able to deduce the existence of strong solutions to problem (P u 0 α,β ) for any u 0 ∈ L 2 (Ω). On the other hand, uniqueness will be proved thanks to the monotonicity of the diffusion, via a comparison theorem between sub-and supersolutions in L 1 (Ω).
In particular, as a consequence of Theorem 3.6 below, for solutions of (P u 0 α,β ) no new jumps are created through the evolution and we observe that the size of jumps cannot increase in time. Moreover, it also leads to a control on the Cantor part of the derivative. We point out that similar results have already been reported for the case of the total variation flow. In particular, in the one-dimensional case, the total variation of the solution can be controlled by that of the initial data in a completely local way (see [20]) while in the multidimensional case, this has only been shown for the jump part (see [17] for dimension less than or equal to 7 or [23] for a related result in any dimension).
The main assumptions on the Lagrangian in (H 3 ) are regularity, monotonicity and a suitable convergence rate of the mapping ξ → a(z, ξ) to the saturation values ±1 as ξ → ±∞. First of all, regularity allows to show that the sequence formed by space derivatives of the approximating solutions and the corresponding vector fields converge a.e., respectively, to the density of the absolutely continuous part of the derivative of the solutions u and the corresponding vector field. On the other hand, finer a priori estimates in the case that u 0 ∈ BV (Ω) show that a subsequence of the positive (resp. the negative part) of the derivative of the solutions converge as measures to the positive part (resp. the negative part) of the distributional derivative of the solution. The final step will be to show that D s ± u(t) ≤ D s ± (u 0 ) holds in the sense of Radon measures. This will be possible thanks to the hypothesis on the rate of convergence towards the saturation.
For a nonlinearity a(z, ξ) ≡ a(ξ ) independent of z, this is in agreement with [10,Theorem 2.4], where instantaneous disappearance of jump-discontinuities has been proven for solutions to the Cauchy problem for the equation u t = [a(u x )] x , with bounded and increasing initial data, and suitable assumptions on the flux, which are in particular fulfilled when hypothesis (H 3 ) holds with σ i ∈ (0, 1]. Similar results concerning the regularity of the solutions to variational linear growth problems have been recently reported in the literature in this case that a(z, ξ) = a(ξ ) is independent of z. The Dirichlet problem has been widely studied since the work of Bildhauer and Fuchs [11] where a condition of μ−ellipticity on the flux was introduced for Total Variation related minimization problems. We point out that the condition of μ-ellipticity is similar to condition (H 4 ) with σ = μ + 1. This study has been then generalized to the vector valued case [12] or to Dirichlet problems for radially symmetric data [13]. Concerning the Dirichlet problem we also mention the results in [9] about global Lipschitz minimizers or [15] about boundary regularity. Finally, we mention the regularity of solutions for related parabolic problems for Lipschitz initial data and periodic boundary conditions studied in [16].
We include in the equation a smooth forcing term, F. In a subsequent work, we will use the results in here to study some qualitative properties of the solutions. In particular, we will study finite time vanishing at some point (when u 0 ≥ C > 0) or infinite time blow up to particular examples of equations satisfying hypothesis (H 2 ) − (H 3 ) and a slightly weaker condition than (H 1 ) (see hypothesis (H 1 ) in [22]).
We finish this introduction with the organization of the paper: Sect. 2 includes notations and preliminaries on bounded variation functions that we need in the paper. In Sect. 3 we collect the definitions of sub-and supersolutions and the main results of the paper. Section 4 is devoted to the well posedness of the approximating problems (P u 0 α,β ) and to show some inequalities needed for the a priori estimates that we use to pass to the limit. This a priori estimates (both for L 2 and for BV initial data) are obtained in Sect. 5. In Sect. 6, we prove the existence and comparison theorems for solutions. Finally, in Sect. 7 we prove the monotonicity properties of the positive and negative singular parts of the derivative of the solutions in the case of BV initial data under hypothesis (H 3 ) and we also prove the instantaneous regularizing effect under hypothesis (H 4 ).

Radon measures and functions of bounded variation
Throughout the paper, Ω = (0, L) ⊂ R will denote an open bounded interval. For any function f : Ω → R, we let f + , f − be its positive and negative part; i.e f + := max{ f, 0}, f − := max{− f, 0}. We denote by M(Ω) the space of the finite (signed) Radon measures on Ω, and by M + (Ω) the cone of its nonnegative elements. For every μ ∈ M(Ω) and for every Borel set E ⊆ Ω, the restriction μ⌞ E of μ to E is defined by (μ⌞ E)(A) := μ(E ∩ A) for every Borel set A ⊆ Ω. Given two nonnegative measures μ and ν, we write μ << ν if μ is absolutely continuous with respect to ν (see e.g. [3,Definition 1.24]). Every μ ∈ M(Ω) has a unique decomposition μ = μ ac + μ s , with μ ac = μ r L absolutely continuous (here μ r ∈ L 1 (Ω) is the density of μ ac ) and μ s singular with respect to the Lebesgue measure L.
We denote by BV (Ω) the Banach space of functions of bounded variation in Ω: where Du is the first-order distributional derivative of u. We say that u ∈ BV loc (Ω) if u ∈ BV (Ω ) for every open subset Ω ⊂⊂ Ω. We recall that given any u ∈ BV (Ω) the measure Du can be decomposed into its absolutely continuous and singular parts, Du = u x L + D s u , u x being the density of the absolutely continuous part of Du. We use standard notations and results for BV functions as in [3] and we will always identify a BV function with its precise representative. Finally, for a BV-function u, D ± u will denote the positive (and negative, respectively) part of the measure Du.
The following lemma will be also used below: as k → ∞.
Let {ν k } ⊆ Y(Ω; R) be the sequence of Young measures associated to {u 0kx }. Then there exists a Young measure ν ∈ Y(Ω; R) such that ν k → ν narrowly, possibly up to a subsequence ( [26]). By (2.2) 3 , for a.e. x ∈ Ω the disintegration of ν satisfies in Ω. Then it can be easily In particular, this implies that the measures λ (±) are singular with respect to the Lebesgue measure and mutually singular. Therefore, we get that λ (±) = D s ± u 0 , combining the convergences u 0kx * , and using the uniqueness of the Lebesgue and Jordan decomposition of the measure Du 0 . Then (iv) follows at once.

Functionals defined on BV
For every u ∈ BV (Ω) we consider the following finite Radon measures and, setting T a,b (s) = min{a, max{s, b}} (a < b), here h and f are respectively the functions in (1.5) and in (H 1 )-(H 2 ), and we have denoted by [T a,b (u)] x the density of the absolutely continuous part of the measure DT a,b (u). We will also use the following lower semicontinuity result: Proof. Settingf (z, ξ) := f (z, ξ) + D 0 , where D 0 is the constant in (1.3), by the results in [18] (see also [1, Theorem 3.1]), we get . Therefore the conclusion follows from last inequality and the definition off (z, ξ).

Results
In this section we collect the definitions of sub-and supersolutions to Problem (P u 0 α,β ) and the main results of the paper. The proof of these results is the content of the rest of the paper.
Strong solutions to problem (P u 0 α,β ) exhibit more regularity in the case of BV -initial data.
We finish this section with a stronger regularity result if further assumptions on the flux a(z, ξ) are assumed. We point out that this result is quite technical, and that Theorem 4.1-(ii) and Proposition 5.5 below are only used for proving it.
Then, for every strong solution u of problem It is worth observing that the conditions a(u, [27,Lemma 7.19]).

Monotonicity properties and regularizing effects for D s ± u(t)
Besides (H 1 )-(H 2 ), relying on hypothesis (H 3 ), we shall prove that for any strong hold true as well. We note that (1.9) is easy to prove. We proceed then with (1.10). Since the computations for the two different examples are very similar, we only give the proof for the former case. In this case, (1.10) reads as .
Let us show that strong solutions of (P u 0 α,β ) exhibit a regularizing effect in timenamely, the singular measures D s ± u(t) disappear instantaneously-if the convergence rate of the mapping ξ → a(z, ξ) to the saturation values ±1 (see ( Finally, the following theorem provides sufficient conditions for instantaneous BV (Ω)-W 1,1 loc (Ω) regularizing effects.

Well-posedness of the approximating problems
For any ε > 0 and u 0 ∈ C 1 (Ω), let us consider the initial-boundary value problem Definition 4.1. For every u 0 ∈ C 1 (Ω), by a solution to problem (4.1)-(4.2) we mean ) and the couple (u, v) satisfies (4.1) in the strong sense.
By rephrasing (4.1) as an abstract differential equation in the Banach space X = C 1 (Ω), we get the following well-posedness result.
Proof. (i) Let us consider the following abstract ODE in the Banach space where L : Since a ∈ Lip(R 2 ), a routine proof shows that L is continuous in and globally Lipschitz continuous with respect to Deriving the equation u t = v x + F with respect to x and using (4.5), we get This proves that u is a solution to problem (4.1)-(4.2) in the sense of Definition 4.1, whereas the uniqueness part follows by observing that every solution to (4.1)-(4.2) in the sense of Definition 4.1 is also a solution to problem (4.4), which is uniquely solvable.
(ii) In order to show that v ∈ C 1 (Q), it is enough to prove that for every (x, t) ∈ Q T there exists the partial derivative v t (x, t) and v t ∈ C([0, T ]; C 2 (Ω) ∩ C 0 (Ω)). Fix any t ∈ (0, T ) (the cases t = 0 and t = T can be dealt with similarly). Observe that the function belongs to C 2 (Ω) and is the unique solution to the problem where, for every x ∈ Ω, . Therefore the conclusion easily follows letting h → 0 in the elliptic problems (4.6) (we omit the standard details).
For every a, b ∈ R, a < b, set We define r α,β to be the linear interpolation between α and β in Ω; i.e.
We are ready to prove the following equalities that will be used for obtaining the a priori estimates needed to pass to the limit.
Proof. It suffices to multiply the equation u t = v x + F by the test function η = u ζ and then to integrate by parts.
Let us address (4.14). For every τ ∈ (0, T ] we have and the conclusion follows since, as before, We finish this section with some useful inequalities satisfied by the approximating solutions. For every g ∈ C 1 (R) and (z, ξ) ∈ R 2 , set Proposition 4.5. For every u 0 ∈ C 1 (Ω), let u be the solution to (4.1)-(4.2). Then, for every τ ∈ (0, T ]; here G ε is the function in (4.16) and as g is nondecreasing. Moreover, we have where H ε,g is the function in (4.18). Combining the previous inequalities, the conclusion follows.

A priori estimates for L 2 -initial data
Let us begin by following elementary technical lemma.
Lemma 5.1. Let r α,β be the function in (4.8). Then, for all z, ξ ∈ R and x ∈ Ω there holds Proof. Inequalities and here R is the function in (4.11) and C 0 , C 1 , D 0 are the constants in assumption (H 2 ).

(5.45)
Proof. The proof is analogous to that of Proposition 5.3, by using (5.38) instead of (5.12) (we omit the details). Proof. The proof is completely analogous to that of Proposition 5.4, relying on (4.9) instead of (4.10) (we omit the details).

Proof of comparison
Let us prove the comparison principle for sub-and supersolutions to (P u 0 α,β ).
For every j ∈ N large enough, let ρ j ∈ W 1,∞ (Ω) ∩ C c (Ω) be defined by setting Then Definition 3.1-(iii) and the condition u, u ∈ L 1 w (0, T ; BV loc (Ω)) ensure that, for every K > 0 and 0 < t 0 < τ < T there holds On the other hand, since α 1 ≥ α 2 and β 1 ≤ β 2 , we have Let us address the first term in the right-hand side of (6.1). To this aim, let Ω j ⊂⊂ Ω be any open set such that supp ρ j ⊂ Ω j . Then u, u ∈ L 1 w (0, T ; BV (Ω j )), and by [2, Corollary 3.1] there holds for every K > 0 and for a.e. t ∈ (0, T ). By (6.3) and the nondecreasing character of the map ξ → a(z, ξ), we get constant of a).

Existence for L 2 -initial data
Let {ε k } be any sequence such that ε k → 0 + as k → ∞. For every u 0 ∈ L 2 (Ω) let (such a sequence {u 0k } can be constructed by convolution). For every ε k > 0, we denote by u k the solution to (4.1)-(4.2) with ε = ε k and initial datum u 0k , and we set The following propositions rely on the a priori estimates in Propositions 5.2, 5.3 and 5.4.
and, for every Proof. Multiplying the equation v k − a(u k , u kx ) = √ ε k u kx + ε k u kxt by the test function ζ ∈ C 1 ([0, T ]; C(Ω)) and integrating by parts gives and the right-hand side of the above equality converges to zero as k → ∞ by (5.11) and (6.8). By (5.13) and (6.8) there exists C > 0 independent of ε k such that Hence, for every τ ∈ (0, T ), we have (6.13) which plainly gives (6.11). Finally, in order to prove (6.12), let us observe that for every θ ∈ (0, 1) there holds the latter convergence in the above inequality being a direct consequence of (6.13).
in Ω, such that for every τ ∈ (0, T ) and a, b ∈ R, a < b, Moreover, possibly up to a subsequence (not relabeled), there holds u kt u t in L 2 (Q τ,T ) for all τ ∈ (0, T ) , (6.14) and for a.e. t ∈ (0, T ) (ii) By (5.24), for every Ω ⊂⊂ Ω and τ ∈ (0, T ), there exists C Ω ,τ > 0 independent of k such that (6.20) (recall that Ω is one dimensional and see also (5.8)), whence since by (6.18), up to a subsequence (not relabeled), we may assume that u k → u a.e. in Q 0,T . (iii) By (6.19), (6.17) and the Dominated Convergence theorem, we get Let us also explicitly observe that the limiting function u given in Proposition 6.2 belongs to L ∞ (Q τ,T ) for every Ω and τ as above.
for every τ ∈ (0, T ), by the definition of the measures h(u, Du) and h(u, DT a,b (u)) in (2.3), (2.5), it follows that for all τ ∈ (0, T ) and a, b ∈ R, a < b. Moreover we explicitly notice that (6.39) plainly implies item (iv) in Definition 3.1.
(ii) The proof is essentially the same as the one of item (i). The only difference is that we have to take Ω ⊂⊂ Ω and τ ∈ (0, T ) such that suppρ ⊆ Ω and supp η ⊆ (τ, T ). Therefore, after repeating the same computations we obtain for every nonnegative η ∈ C 1 c (0, T ) and ρ ∈ C 1 c (Ω), ρ ≥ 0. From here, the proof finishes as in (i). (iii) From (6.47), we get Therefore (6.41) and (6.42) follow from the above equalities, as every ζ ∈ C 1 (R 2 ) ∩ L ∞ (R 2 ) can be uniformly approximated in bounded subsets of R 2 by finite sums whence the conclusion immediately follows (see also (6.8)).
Proof of Theorem 3.4. Claim (i) follows from Proposition 6.6, whereas claim (ii) is a direct consequence of the a priori estimate (5.45) and (2.1). The proof of equality (3.1) relies on (6.53) and (6.56), letting k → ∞ in the weak formulation of the equality

Monotonicity properties of D s ± u(t): proofs
We begin by proving the convergence a.e. in Q 0,T of the sequence {u kx } to the density u x of the absolutely continuous part of the measure Du.
where {v k } is the sequence in (6.9).
Proof of Proposition 7.1. The claim will follow by a diagonal argument if we prove that, for every Ω ⊂⊂ Ω and τ ∈ (0, T ), possibly up to a subsequence (not relabeled), there holds u kx → u x a.e. in Q τ,T .
Notice that the estimate in the right-hand side of (7.25) is independent of z ∈ [−M, M] and k ∈ N.