Optimal interior partial regularity¶for nonlinear elliptic systems: the method of A-harmonic approximation

By Frank Duzaar, Joseph F. Grotowski. 2000

164 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/72ec923cdb12cdc5d9604faecc891febe5215a6b

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The boundary regularity of non-linear parabolic systems I (2010, background): “Later it was adapted to treat regularity issues for weak solutions of non-linear elliptic systems; first for the interior situation [13,14] and later on for the boundary situation [26,25,4].”
Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure (2009, methodology): “The theorem follows from techniques in [25], where a more general case was considered; compare with [5, 6, 8] and [23, Chapter 9] for the case of integrals.”
Nonlinear elliptic systems with Dini continuous coefficients (2002, methodology): “The core of our proof is the Γ –harmonic approximation technique adapted from [13], [ 5 ], [7].”
The existence of regular boundary points for non-linear elliptic systems (2007, background): “13) below extends in a natural way the well-known interior results, available for instance in [26], [30], [19], [14].”
Fractional estimates for non-differentiable elliptic systems with general growth (2008, background): “The work has been continued for example by Evans [9], Giaquinta [10], Carozza, Fusco, Mingione [6], and by Duzaar and Grotowski [8].”
Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth (2005, methodology): “The technique has been successfully applied in the framework of geometric measure theory (see [18]), and to obtain partial-regularity results for general elliptic systems (see [ 16 ], [14], [25], [26], [27]).”
The Singular Set of Solutions to Non-Differentiable Elliptic Systems (2003, background): “This fact is standard and it can be easily inferred, for instance, from [15] or [2] (see also [11] and [ 7 ] for the case of systems with linear growth p = 2; see [8] and [9] for similar results in the variational setting and for quasiconvex integrals).”
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (2005, background): “This proof, taken from [29], has its origin in Simon’s proof of the regularity theorem of Allard, see Chapter 11, compare [97] and [11], and avoids the use of the higher integrability result.”
Bounds for the singular set of solutions to non linear elliptic systems (2003, background): “it follows that Σ ⊆ Σ0 ∪ Σ1 (see for instance [20, 10 ]).”
Second order parabolic systems, optimal regularity, and singular sets of solutions (2005, methodology): “This lemma and its variant in Section 3 allow to prove partial regularity properties of solutions to non parabolic systems by linearization arguments (see Sections 6 and 7) in a particular efficient and elemen In the elliptic setting the possibility of using the harmonic approximation lemma and its ”
Harmonic type approximation lemmas (2009, methodology): “Eventually the lemma was applied for proving partial regularity of solutions to elliptic systems in [25,22], and of minimizers of quasiconvex integrals in [23,26,27]; it everywhere yielded optimal regularity results, allowing to avoid the use of heavy tools such as Gehring’s lemma [50].”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, methodology): “Finally, the technique I like most, the “A-approximation method”, once again first introduced in the setting of Geometric Measure Theory by Duzaar & Steffen [104], and applied to partial regularity for elliptic systems and functionals by Duzaar & Gastel & Grotowski [94, 92], see also the nice survey”