On the Existence of Integrable Solutions to Nonlinear Elliptic Systems and Variational Problems with Linear Growth

Abstract

We investigate the properties of certain elliptic systems leading, a priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously as in the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.1.2. Formulation of the problems. We first consider the following problem: for a bounded, connected, Lipschitz domain Ω ⊂ R d , d ≥ 2, with Dirichlet boundary Γ D and Neumann boundary Γ N , which are relatively open subsets of ∂Ω such that Γ D ∩ Γ N = ∅ and Γ D ∪ Γ N = ∂Ω, a given vector field f : Ω → R N , with N ∈ N, a given g : Γ N → R N , a given boundary datum u 0 : Ω → R N , and a given bounded mapping D D D :where n denotes the unit outward normal vector on Γ N . When Γ D = ∅, f and g will be assumed to satisfy a standard compatibility condition (cf. (D3) below).1 a , a > 0.Problem (1.8) is then an almost direct analogue of problem (1.1) with N = d; the only aspect in which the latter model differs from (1.1) (and is therefore considerably more difficult) is that, in contrast with (1.1), one is forced to operate in the space of symmetric matrices and function spaces of symmetric gradients. We refer the interested reader to [17,18,19,10,9] for a detailed overview of limiting strain models, their theoretical justification stemming from implicit constitutive theory, a discussion of their importance in modeling the responses of materials near regions of stress-concentration, where |T T T| is large, and their mathematical analysis (see in particular the survey paper [9] for more details).Analogously to problem (1.1), we adopt the following natural assumptions associated with limiting strain models (see [9]): there exist constants C 0 ≥ 0 and C 1 , C 2 > 0 such that, for all T T T ∈ R d×d sym , ε ε ε *

Integrability of the derivative of solutions to a singular one-dimensional parabolic problem

“…(1.2) is the time semidiscretization of (1.1), then integrability of the derivative of solutions following from integrability of the derivative of f is not surprising. A similar statement for a domain in R N is proved by Beck et al in [4], but for smooth nonlinearities corresponding to functionals with linear growth. In the setting of [4] the smooth dependence of W on p is important for the argument.…”

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

“…where a is a uniformly monotone operator (2.16) with at most linear growth at infinity (2.15). Hence, the existence and uniqueness of the solution T ∈ L 1 (Ω) and u ∈ H 1 0 (Ω) (or, in higher regularity, T ∈ L 2 (Ω) and u ∈ W 1,2 0 (Ω)) to (2.20) -(2.22), or u ∈ W 1,2 0 (Ω) to (2.18), is guaranteed by [1].…”

Finite element approximation of a strain-limiting elastic model

“…When the Neumann part of the boundary ∂ N Ω is nonempty, the structure of the solution is potentially much more complicated. It was shown in [3] that, in general, the solution in that case belongs to the space of Radon measures, but if the problem is equipped with a so-called asymptotic radial structure, then the solution can in fact be understood as a standard weak solution, with one proviso: the attainment of the boundary value is penalized by a measure supported on ∂ N Ω. For simplicity, in this initial effort to construct a provably convergent numerical algorithm for the problem under consideration, we shall therefore suppose henceforth that ∂ D Ω = ∂Ω (i.e., ∂ N Ω = ∅) and that the Dirichlet boundary datum is g = 0 on ∂Ω.…”