Nondiffusive variational problems with distributional and weak gradient constraints

1 Introduction

We begin by considering an evolutionary problem whose semi-discretization (in time) gives rise to the class of stationary problems of interest in this article. Suppose that f : ( 0 , T ) × Ω → R together with u 0 : Ω → R are given, where Ω ⊂ R d is a bounded domain with a Lipschitz boundary. Furthermore, let α be a given nonnegative function (possibly only integrable), or a nonnegative Borel measure in Ω . Suppose that u : ( 0 , T ) × Ω → R , such that u ( 0 ) = u 0 , is a solution to the following problem:

where the set K is given by

The choice of U ˜ ( 0 , T ) and K in (1.2) hinges on the type of the boundary conditions and the regularity of α . We assume that the boundary ∂ Ω is partitioned into a Dirichlet boundary part Γ D and a nonDirichlet boundary part Γ N , both composed of a finite number of connected parts, such that

Note that on Γ N , we do not necessarily prescribe Neumann boundary conditions, as it is later clarified. However, a conservation law of material is in place in the case Γ D = ∅ ; specifically, it can be inferred from (1.1) that ∫ Ω ( u ( T ) − u 0 ) d x = ∫ 0 T ∫ Ω f d x d t given that v = u ± 1 are admissible test functions as we see next. The restriction of u to the Γ D part of the boundary is assumed to be zero, and no restrictions are assumed on Γ N .

The set K is convex and it arises by a nonlinear law with a bound on the first-order derivative terms. In the most general form K is given by

with 1 ≤ p ≤ + ∞ . We briefly discuss the two possible scenarios that we consider:

1. If α is a nonnegative measurable function, then U Γ D ( Ω ) is a Sobolev-type space and G = ∇ is the weak gradient, so that ∣ ∇ v ∣ p is the ℓ p -norm of the weak gradient of v . Hence, ∣ ∇ v ∣ p ≤ α in (1.3) is considered in the almost everywhere (a.e.) in Ω sense.

2. If α is a nonnegative Borel measure, then U Γ D ( Ω ) is a subset of functions of bounded variation BV ( Ω ) . In this case, G = D is the distributional gradient, and ∣ D v ∣ p the total variation measure of D v associated with the ℓ p -norm, and the constraint ∣ D v ∣ p ≤ α is understood in the measure sense.

A possible motivation for the above class of problems is based on the study of accumulation of granular heterogeneous material on possibly discontinuous structures. This approach was pioneered by Prigozhin [1,2,3] in the case of homogeneous materials and a continuous support structure. In this vein, f : ( 0 , T ) × Ω → R represents the (density) rate of a granular material being deposited on a supporting structure u 0 : Ω → R . Moreover, ∫ 0 T ∫ Ω f d x d t is the total amount of material deposited on Ω over the time interval [ 0 , T ] . In case that α > 0 is a real number, this corresponds to the classical case of a granular cohesionless material where homogeneous piles are generated. If α : Ω → R is not constant zero, the value of α at a point determines the angle of repose of the material at that point, i.e., the steepness of a cone generated from a point source of material. This is the case for heterogenous sandpiles [4] and also a restricted case of the quasi-variational sandpile model; see [5,6, 7,8]. In a more general setting where α is a measure, using the approach in this article, it is possible to generate discontinuous structures such as cliffs by preserving discontinuities in the initial supporting structure u 0 and/or of f . Such an approach has not yet been considered in the literature to the best of our knowledge.

A description of the qualitative behavior of problem (1.1) is displayed in Figure 1. We assume two materials with different angles of repose α 1 and α 2 with α 1 > α 2 are poured on the discontinuous structure u 0 ( x ) ≔ χ ( x 1 , x 3 ) ( x ) for x ∈ Ω ≔ ( 0 , 1 ) and 0 < x 1 < x 3 < 1 . The intensity of the material being deposited is given by f ( t , x ) = f 1 χ ( x 0 , x 2 ) ( x ) + f 2 χ ( x 2 , 1 ) ( x ) for some points x 0 and x 2 , and some f 1 , f 2 > 0 , i.e., the first and second materials are poured with density rates f 1 and f 2 , respectively, during the entire time interval ( 0 , T ) . We further assume that a sharp edge can form at x 2 with maximum height of 1, and in addition discontinuities of maximum size 1 can be preserved at the locations of the discontinuities of u 0 . Finally, the the gradient constraint α is then given by d α = α 1 χ ( 0 , x 2 ) ( x ) d x + α 2 χ ( x 2 , 1 ) ( x ) d x + ∑ i = 1 3 δ ( x − x i ) , and the material is assumed to escape freely at the boundary points of Ω . On the right side of Figure 1, we see the comparison between u 0 and u ( T ) , the solution at time T > 0 ; on the left we see the depiction of f , α , and the accumulation regions of both materials.

Figure 1

Accumulation of two kinds (red and blue) of granular materials on discontinuous surface. (Left) Depiction of f ( t , x ) = f 1 χ ( x 0 , x 2 ) ( x ) + f 2 χ ( x 2 , 1 ) ( x ) , the accumulation of both materials, and d α = α 1 χ ( x 0 , x 2 ) ( x ) d x + α 2 χ ( x 2 , 1 ) ( x ) d x + ∑ i = 1 3 δ ( x − x i ) . (Right) The value of the initial supporting structure u 0 and the final distribution u ( T ) .

The study of solutions to (1.1) usually makes use of the semi-discretization (in time) of the problem via an implicit Euler method. In particular, we approximate the partial time derivative ∂ t u by ( u n − u n − 1 ) / k for some time step k > 0 . The class of problems of interest in this article is then given by

where we assume f ∈ L 2 ( Ω ) . Note that ( P ) can be seen as the time-discrete version of (1.1) where the solution u to ( P ) is equal to u n when f corresponds to ∫ ( n − 1 ) k n k f ( τ ) d τ + u n − 1 with u n − 1 given. Closely related to the problem above, we consider the following class of problems:

We prove that ( P ∗ ) is the Fenchel pre-dual of problem ( P ), i.e., the Fenchel dual [9] of ( P ∗ ) under certain conditions is ( P ). Several choices for V Γ N ( Ω ) and J are explored which are directly related to the nature of α . In all cases considered, V Γ N ( Ω ) contains d -dimensional vector fields with divergences in L 2 ( Ω ) . In particular, we consider

1. If α is a nonnegative measurable function (additional assumptions are later explained but continuity is enough to guarantee what follows), then we explore two options for J :

   J ( p ) = ∫ Ω α ( x ) ∣ p ( x ) ∣ q d x , and J ( p ) = ∫ Ω α d ∣ p ∣ q .

   In the first case V Γ N ( Ω ) is a subspace of L 1 ( Ω ) d . In the second case V Γ N ( Ω ) is contained in the space of R d -valued Borel measures, so that the second functional denotes the integral of α with respect to the total variation measure of p induced by the ℓ q -norm. The two functionals are closely related, and the first can be seen as a restriction of the second one to measures that are absolutely continuous with respect to the Lebesgue measure.

2. If α is a nonnegative Borel measure, then V Γ D ( Ω ) is contained in the space of maps that are α measurable, with J given by

   J ( p ) = ∫ Ω ∣ p ∣ q d α .

The article focuses on functional analytic properties of ( P ) and ( P ∗ ) together with duality relationship properties. Additionally, we develop a mixed finite element-type method to solve the optimality conditions corresponding to ( P ) and ( P ∗ ).

Some bibliography. The structure of problems ( P ∗ ) and ( P ) and their inherent difficulties are analogous to the ones that appear in the context of plasticity; see [10,11] and references therein. In particular, the first class of applications for diffusive variational problems with gradient constraints is the elasto-plastic torsion problem. Such a problem has been thoroughly analyzed by Brézis, Caffarelli, Evans, Friedman, Gerhardt, and others; see [12,13,14, 15,16,17, 18,19]. Furthermore, a complete account of the literature can be found in [20]. A significant amount of the aforementioned studies focus on regularity of solutions, the free boundary, and the equivalence of the gradient constrained problem to a double obstacle one.

The modeling of the evolution of the magnetic field in critical-state models of type II superconductors also leads to a problem like (1.1) with the addition of a diffusive operator and a state-dependent constraint in some cases; see [2,21,22, 23,24, 25,26]. See [27] for a study of evolutionary variational problems with nonconstant gradient constraints, and [28] for a complete account of evolutionary problems with derivative bounds.

Analogous problems are found in mathematical imaging involving total variation regularization [29,30,31] and more specifically in the weighted total variation version [32]. There, in contrast to the work here, the L ∞ -norm on the gradient is replaced by the L 1 -norm, leading to a pre-dual problem with a pointwise bound in its state variable.

2 Notation and preliminaries

The purpose of this section is to introduce notation involving spaces, and convergence notions that are used throughout the article; in particular, we address the well-known notions of Sobolev spaces and the space of functions of bounded variation. We refer the reader to Attouch et al. [33] that we follow closely for this introduction together with the book of Adams and Fournier [34].

For a Banach space X , we denote its corresponding norm as ‖ ⋅ ‖ X . For an element F in the topological dual X ′ of X , the duality pairing of F and an arbitrary element x ∈ X is written as ⟨ F , x ⟩ X ′ , X . Throughout the article, all Banach spaces are assumed to be real vector spaces.

The inner product on the Lebesgue space L 2 ( Ω ) of (equivalence classes of) functions that are square integrable on Ω is denoted as ( ⋅ , ⋅ ) , so that ( f , g ) ≔ ∫ Ω f ( x ) g ( x ) d x for f , g ∈ L 2 ( Ω ) where d x refers to integration with respect to the Lebesgue measure.

The Sobolev space of functions in L r ( Ω ) for 1 ≤ r < + ∞ with weak gradients in L r ( Ω ) d is denoted by W 1 , r ( Ω ) , and it is endowed with the norm

where ∇ v denotes the weak gradient of v . In the case r = 2 , we use the notation H 1 ( Ω ) ≔ W 1 , 2 ( Ω ) . Given that Ω is assumed Lipschitz, restriction of a function v ∈ W 1 , r ( Ω ) to the boundary ∂ Ω is well-defined via the continuous trace map γ 0 : W 1 , r ( Ω ) → L r ( ∂ Ω ) . Furthermore, the closed subspace of functions in W 1 , r ( Ω ) that are zero on Γ D is denoted by W Γ D 1 , r ( Ω ) , i.e.,

Similarly, we define H Γ D 1 ( Ω ) ≔ W Γ D 1 , 2 ( Ω ) .

The space of real-valued Borel measures M ( Ω ) is endowed with the norm ‖ μ ‖ M ( Ω ) ≔ ∣ μ ∣ ( Ω ) , where ∣ μ ∣ is defined for an arbitrary open set O as

Note that ⟨ μ , z ⟩ M ( Ω ) , C 0 ( Ω ) = ∫ Ω z d μ and that ∣ μ ∣ defines a Borel measure in M + ( Ω ) , the subset of nonnegative elements of M ( Ω ) , i.e., σ ∈ M + ( Ω ) if σ ( B ) ≥ 0 for every Borel set B ⊂ Ω .

We denote by BV ( Ω ) , the space of functions v in L 1 ( Ω ) , for which the total variation semi-norm

is finite and where q is the Hölder conjugate of p , i.e., 1 / p + 1 / q = 1 ; see [33, Section 10.1]. The space BV ( Ω ) is a Banach space endowed with the norm

The operator D represents the distributional gradient, and for a v ∈ BV ( Ω ) , D v is a R d -valued Borel measure. We use ∣ D v ∣ p to denote the total variation measure (associated with the ℓ p -norm) of D v , and the total mass ∣ D v ∣ p ( Ω ) is by definition

Furthermore, the Lebesgue decomposition result applied to D v implies that there exist measures D a v and D s v such that

with D a v and D s v , respectively, being absolutely continuous and singular with respect to the d -dimensional Lebesgue measure.

We define now the notions of weak and quasi-intermediate convergence of sequences in BV ( Ω ) which provide different topologies on the space BV ( Ω ) . The former is obtained by a subsequence of a bounded sequence in BV ( Ω ) . Moreover, the latter is sufficient to preserve boundary conditions in the sense of the trace as stated in Theorem 2.3.

Recall that if { μ n } is a sequence of measures in M ( Ω ) then μ n ⇀ μ in M ( Ω ) for some μ ∈ M ( Ω ) , that is, μ n weakly converges to μ , if

for all g ∈ C 0 ( Ω ) .

Definition 2.1 is understood in light of the following fact: If { u n } is a bounded sequence in BV ( Ω ) , there exists u ∗ ∈ BV ( Ω ) such that along a subsequence u n ⇀ u ∗ in BV ( Ω ) . The latter follows since the embedding BV ( Ω ) ↪ L 1 ( Ω ) is compact (see Attouch et al. [33, Theorem 10.1.4.]) for Lipschitz domains, and since a bounded sequence of measures admits a weakly convergent subsequence.

We shall use the direct method of calculus of variations to establish existence of solutions to problems in BV ( Ω ) and with Dirichlet homogeneous boundary conditions on Γ D . The space of interest is BV Γ D ( Ω ) defined as

where γ 0 is a trace operator; see [33, section 10.2]. Note that we use the same notation for the trace operator in Sobolev spaces W 1 , p ( Ω ) . There is a fundamental issue with the trace in BV ( Ω ) and the application of the direct method as we show next with a standard example adapted from [33].

Consider a bounded sequence { u n } in BV Γ D ( Ω ) . Then, we can extract a subsequence (not relabeled) of { u n } such that u n ⇀ u ∗ in BV ( Ω ) . The problem is that in general it is not possible to say that u ∗ ∈ BV Γ D ( Ω ) : Let Ω = ( 0 , 1 ) with Γ D = { 0 } , and consider { v n } defined as

Then, v n ∈ BV Γ D ( Ω ) and v n ⇀ v ∗ ∈ BV ( Ω ) ⧹ BV Γ D ( Ω ) , with v ∗ = 1 . The underlying reason is that the trace operator in BV ( Ω ) is not continuous with respect to weak convergence, but it is with respect to the intermediate (and quasi-intermediate) convergence subsequently defined. We further note that ∣ D v n ∣ ( 0 , 1 ) = 1 and ∣ D v ∗ ∣ ( 0 , 1 ) = 0 , this discrepancy is central to the issue we are considering.

The name quasi-intermediate convergence arises since it describes a stronger topology than the one of weak convergence, but not as strong as the intermediate one in which the second convergence in the above definition is exchanged to ∫ Ω ∣ D u n ∣ → ∫ Ω ∣ D u ∗ ∣ . The importance of the intermediate convergence is that the trace map γ 0 : BV ( Ω ) → L ℋ d − 1 1 ( ∂ Ω ) is intermediate-strong continuous. We refer to Attouch et al. [33, Theorem 10.2.2] for its proof. Similarly, we have

The proof of Theorem 2.3 follows by direct observation of the generalized Green’s formula, see [33, Theorem 10.2.1].

3 Existence theory for ( P )

In this section, we discuss the existence and uniqueness of solution to the problem ( P ). We start with the case when α is a measure, and the case when α is a function follows as a special one. In particular, existence of solutions is studied in the function spaces U Γ D ( Ω ) = BV Γ D ( Ω ) and U Γ D ( Ω ) = W Γ D 1 , 1 ( Ω ) . Both of these spaces share the same difficulty: Bounded sequences do not necessarily admit convergent (in some sense) subsequences that preserve the zero boundary condition on Γ D in the limit. The main purpose of this section is to overcome this obstacle.

4 Existence theory for the pre-dual problem ( P ∗ )

The focus of this section is on existence and uniqueness of solutions of problem ( P ∗ ) under different functional analytic settings. In particular, we focus on two cases where p is either (i) a function or (ii) a Borel measure. In the first case, we let α be either a function or a measure; here, existence results are limited to d = 1 . On the other hand, in the second case we establish an existence and uniqueness result for p with arbitrary d ∈ N , for a specific class of α ’s (to be specified later). Furthermore, this second case requires a nonstandard space of vector measures with divergences in L 2 ( Ω ) . Remarkably, a version of the integration-by-parts formula still holds in this general setting; such a construct is rather recent [38]. We start with the case when p is a function.