Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals

By Frank Duzaar, K. Steffen. 2002

165 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/46e344855dd57f34a85c26917637322f28f1760a

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Citations in the corpus

Existence and uniqueness of minimizers of general least gradient problems (2013, background): “Once this is known, our claim about the existence of functions vk, for k large enough, satisfying (59) follows from [4, 21].”
Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law (2014, methodology): “8 reduces to a local almost-minimality notion analogous to the ones considered in [5,7,13,35] and [27, Section 21].”
Nonlinear elliptic systems with Dini continuous coefficients (2002, methodology): “In [ 7 ] the first author and Steffen prove an optimal boundary regularity result for almost minimizing rectifiable currents of general elliptic integrands.”
The quantitative isoperimetric inequality and related topics (2015, background): “The regularity results needed in the proof were obtained in [20] and [54] .”
The boundary regularity of non-linear parabolic systems I (2010, methodology): “Originally, this technique was used in the setting of geometric measure theory in order to prove regularity result for almost minimizing currents of elliptic integrands in the interior and at the boundary; see [21].”
Second order parabolic systems, optimal regularity, and singular sets of solutions (2005, methodology): “In the elliptic setting the possibility of using the harmonic approximation lemma and its non-isotropic variant, the A-harmonic approximation lemma, was already exploited both in the setting of Geometric Measure Theory to prove optimal regularity results for minimizing currents to elliptic integrand”
The Singular Set of ω-minima (2005, methodology): “It was introduced in the setting of Geometric Measure Theory (see [3, 4, 8] and the references therein) and has subsequently been studied in the non-parametric case by several authors.”
Regularity of minimizers of W1,p-quasiconvex variational integrals with (p,q)-growth (2008, methodology): “Proceeding with the proof we will use the A-harmonic approximation method developed in [ 12 ].”
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]).”
Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth (2009, methodology): “Actually, to prove Main Theorem 6.4 we can argue exactly as demonstrated in [41] by means of theA-harmonic approximation method of [ 18 ].”
Boundary Regularity in Variational Problems (2010, background): “By now standard references on partial regularity include [25, 23, 20, 2, 18, 31, 45, 46, 50, 51]; an up-dated presentation is in [42].”
Parabolic systems with polynomial growth and regularity (2011, methodology): “In turn, such a technique builds on the elliptic counterpart of the A-harmonic approximation method, developed in [37] in the context of geometric measure theory and then transferred to the non-parametric case; all such compactness methods obviously find their roots in the original harmonic approxim”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, methodology): “At this stage the linearization is achieved via the A-harmonic approximation method of Duzaar & Steffen [104], which is a variant of Theorem 4.14 for p = 2.”
Harmonic type approximation lemmas (2009, background): “The previous lemma still works—see [32]—when considering a bilinear form satisfying, instead of (2.”
Bounds for the singular set of solutions to non linear elliptic systems (2003, methodology): “One of the main points of these papers, apart from the important technical simplifications introduced (the A-approximation method, see also [ 11 ]), is the proof that the regularity of solutions is, in a sharp way, sensitive with respect to the regularity of the lower order terms, “the coefficients””
Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory (2012, background): “Concerning the regularity theory for minimizing currents, the paper by Duzaar and Steffen [DS02] is a valuable source for both its clarity and its completeness.”