Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

Abstract

This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale H s−1 D (Ω) for s > 0 sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces H s (Ω). Ellipticity is enforced by assuming a Gårding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.

A priorierror estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

“…The only contribution known to the author regarding numerical analysis of a damage model is the contribution [27], which considers a linearized, time-discrete phase-field model for crack propagation in the context of optimal control. Finally, we would like to mention [18], which (among other more general results) provides regularity results for time-discrete phase-field models that are applicable in our case as well and are a key ingredient for the spatial error estimation.…”

An Optimal Control Problem Governed by a Regularized Phase-Field Fracture Propagation Model. Part II: The Regularization Limit

“…the right hand side of this equation converges to zero as an element of H −1+s (Ω) = (H 1−s D ) * (Ω) and due to [11] this shows…”

“…The regularity and stability estimate for ϕ i γ are an immediate consequence of Corollary 3.9. The bound for u i γ then follows from [11].…”

“…where the constant c depends on κ > 0 and Ω, only. Moreover, the calculations in [11,Corollary 20] show that if Ω is W 2,q regular, with q = p/2 > 1, for the homogeneous Neumann-problem −ε∆ϕ + ε −1 ϕ = f, then…”

“…From [11], we obtain the improved regularity u γ , u ∞ ∈ H 1+s (Ω) together with the bound u γ 1+s ≤ c ϕ q with a constant depending on the C 0,α norm bound of ϕ γ , only. Further, the error satisfies…”