On the regularity of solutions to the plastoelasticity problem

1 Introduction

We shall consider the following problem P∞: given Ω⊂ℝN open and bounded, f∈L2⁢(Ω) and u0∈W1,∞⁢(Ω), minimize

where

where |⋅| is the Euclidean norm. In the special case f≡μ>0, Ω⊂ℝ2, the solution u∞ represents the stress function in a torsion problem, with parameter μ, of an elastic bar with cross section Ω.

For the case of Ω⊂ℝ2 a disk of radius R about the origin, f≡1 and u0≡0, a solution for small R is uC⁢(x)=14⁢|x|2-14⁢R2, while, as R>2, it becomes

whose gradient is

The sets E={x∈Ω:|∇⁡u∞⁢(x)|<1} and P={x∈Ω:|∇⁡u∞⁢(x)|=1} are called the elastic and plastic subsets of Ω, and in the example one can see the occurrence of P as R becomes larger.

The investigation of the regularity properties of the solution, both for the original problem and for related problems expressed in the form of variational inequalities, has been the subject of several papers, [3, 2, 1, 7, 8, 12, 4, 6] to name a few (the more complex problem of the regularity of the stress tensor has been investigated by Seregin; see [11] and the references therein). In the example presented above, we can notice that the gradient of the solution is continuous on Ω, but that the Hessian has a jump discontinuity at |x|=2: in particular, this implies that the solution cannot possibly be more regular than being in W2,2⁢(Ω).

In [3], Brézis and Stampacchia proved that, for u0≡0 and Ω either convex with ∂⁡Ω Lipschitzian, or Ω arbitrary with ∂⁡Ω∈C2, the solution u belongs to W2,2⁢(Ω). Let d∂⁢(x) be the distance of a point x∈Ω from ∂⁡Ω. In [2], Brézis and Sibony, for the case u0≡0 and f≡μ, succeeded in proving that the minimization problem (P∞) is equivalent to the following problem: find u∈𝕂2≐{v∈W1,∞:|v⁢(x)|≤d∂⁢(x)⁢ a.e. on ⁢Ω} such that, for every v∈𝕂2, we have

i.e., they proved that the solution to (P∞) solves a variational inequality with an obstacle. Hence, as one can see in the example above, the solution to the minimization problem for f≡μ is such that the plastic subset P consists of those x∈Ω where the solution agrees with the obstacle. Since this result, the papers on the regularity of the solutions focused on the regularity of the obstacle, be it the distance function d∂ (whose regularity depends on the regularity assumptions on ∂⁡Ω) or a more general obstacle function ψ. Brézis and Kinderlehrer [1], and, independently, Frehse [6], did show that, under the condition that the obstacle ψ is C2⁢(Ω¯), the solution to a variational inequality with obstacle ψ is in C1,1⁢(Ω). Always for a variational inequality, the condition on the obstacle was relaxed to ψ∈W2,∞⁢(Ω¯) in [7]. However, the regularity conditions imposed on the obstacle and, in particular, the regularity conditions on ∂⁡Ω, prevent the application of the regularity results in the case where Ω is less regular. In fact, in the simple case where Ω⊂ℝ2 is a square centered at the origin, the function distance to the boundary has a gradient that is discontinuous along the diagonal lines. A careful analysis of the regularity of the solution to problem (P∞), with u0≡0 and f≡μ, hence in a special case of our problem (P∞), and under the condition that Ω⊂ℝ2 is such that ∂⁡Ω consists of finitely many C3 curves (hence, covering the case of Ω being a square), was carried out by Caffarelli and Friedman in [4]. In particular, they did show that the set of points of discontinuity of ∇⁡d∂ is contained in the elastic part E of Ω.

Notice, however, that the equivalence of the problem of solving a variational inequality with an obstacle and the problem of minimizing (1.1) does not hold in the general case treated here, i.e., where f∈L2 depends on x, not even in the case u0≡0. In fact, consider the problem of minimizing (1.1) on Ω=(-α,α)⊂ℝ1, with u0≡0 and f⁢(x)=χ(-α,0)⁢(x)-χ(0,α)⁢(x), i.e., the problem of minimizing

subject to the condition |x′|≤1. One can see that, for α sufficiently large, on the interval (-α,0) the solution to the minimization of (1.1) is

Consider instead the problem of minimizing the same integral, with the same boundary conditions, but subject to the obstacle |u⁢(x)|≤d∂⁢(x), i.e., subject to

On the interval (-α,0) the solution is not anymore symmetric with respect to the point x=-α2 and it is given by

We have that v2 is a solution to the variational inequality problem with obstacle (1.3), so that the two problems are not equivalent.

In Section 3 of the present paper, for Ω⊂ℝN open and bounded, f∈L2⁢(Ω) and u0 Lipschitzian, we show that the solution u∞ to problem (P∞) is in Wloc2,2⁢(Ω). We stress the point that in the conditions for our result there is neither a mention of ∂⁡Ω whatsoever nor of any other condition on Ω, besides it being open and bounded: the regularity to u∞ we prove comes entirely from it being a solution to a variational problem.

Notice that our result, when applied to the case of elastoplasticity, i.e., when f≡μ and u0≡0, but without any regularity imposed on ∂⁡Ω, contains the main property mentioned before, i.e., that the points of discontinuity of ∇⁡d∂ are contained in the elastic part E of Ω: in fact, whenever ∇⁡d∂ is discontinuous, d∂ is not in Wloc2,2⁢(Ω) and hence, by the present result, it cannot coincide (locally) with u∞, so that the points of discontinuity of ∇⁡d∂ must be in E.

The main technical difficulty to our approach comes from the fact that we are minimizing a Lagrangian that is an extended-valued convex function. In fact, although some regularity results for variational problems without upper bounds on the growth of l are known, [10, 9], we believe that there are no previous examples of a regularity result of this kind, proved for a variational problem having an extended-valued Lagrangian, a case that demands a different approach.

Our main purpose is to prove the following theorem, where we use the notation

In the case that we exclude, i.e., d∞=0, hence when u0 satisfies the condition ∥∇⁡u0∥∞≤1, we might have that u0 is the only function in u0+W01,∞⁢(Ω) satisfying the condition |∇⁡u|≤1. If this is the case, there are no non-trivial variations, a fact that prevents any regularity proof.

The Lagrangian whose restriction to the unit ball we are minimizing needs not be 12⁢|ξ|2: a general case is presented in Section 4.

2 Notations and preliminary results

For a smooth rotationally symmetric Lagrangian L, i.e., of the form L⁢(ξ)=l⁢(|ξ|) with l differentiable, we have

Whenever the assumptions on l imply that l′′⁢(0) exists, we shall consider the map g:t→l′⁢(t)/t to be extended at 0 as l′′⁢(0).

The Hessian matrix of a twice differentiable function ξ→l⁢(|ξ|) is

We set 𝕂={v∈u0+W01,∞⁢(Ω):∥v∥1,∞≤1}. For a function f:Ω→ℝ and for ω⋐Ω, we set

defined for x∈ω and for a suitably small h.

We shall approximate the function l∞ by convex functions that are C2 except at a point different from zero.

Besides (P∞), the problem of minimizing (1.1), we shall also consider solutions to the problem (Pl) of minimizing

for a convex function l, satisfying the assumptions of Lemma 2.3, and set L⁢(ξ)=l⁢(|ξ|).

3 Proof of Theorem 1.1