Bounds for the singular set of solutions to non linear elliptic systems

By Giuseppe Mingione. 2003

86 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/859f3dc26943bb4fb4c23a6133ece10451893d1f

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Fractional estimates for non-differentiable elliptic systems with general growth (2008, background): “In his two articles [19] and [20] Mingione gave the answer to this question in the case pb 2, i.”
The Singular Set of Minima of Integral Functionals (2006, result): “As the functional is convex there is no difference between its minimizers and its critical points, that is, the minimizers are precisely the weak solutions of the Euler-Lagrange system, divfz(x, Du) = 0. It can be shown that conditions (1.1) also imply that the vector field x � fz(x, z)/(|z |+ 1) is”
Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems (2012, result): “Recent improvements of these results in terms of Hausdorff measures can be found in [35,50,51].”
Parabolic systems with polynomial growth and regularity (2011, background): “The relation between the two aspects has been clarified, in the elliptic case, in the papers [52, 53, 62]; see also the surveys [63, 65].”
Second order parabolic systems, optimal regularity, and singular sets of solutions (2005, background): “It has been finally settled in [31], where it is shown, among other things, that the Hausdorff dimension of solutions to general non-linear elliptic systems in R is always strictly less than n.”
The existence of regular boundary points for non-linear elliptic systems (2007, methodology): “Indeed, everything done up to now also perfectly works in the interior, hereby somewhat extending the results in [37], [38], where for instance the case 1 < p < 2 is not included.”
Boundary Regularity in Variational Problems (2010, background): “The strategy of [17] starts from the interior singular sets dimension estimates in [40, 41] asserting that in both cases (1.”
The boundary regularity of non-linear parabolic systems I (2010, result): “We remark that a similar strategy was developed for the elliptic case to get first singular sets estimates in the interior [28,29], and then at the boundary [15], and [16,18] for the interior parabolic case; we refer to [25–27] for results concerning the stationary variational case.”
A borderline case of Calder\'on-Zygmund estimates for non-uniformly elliptic problems (2019, background): “Let us remark that fractional differentiability properties of solutions to various types of potentially degenerate elliptic equations are a powerful tool in regularity theory (see for instance [28] where these are employed to estimate singular sets of solutions), and have been recently the object of”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, unknown): “The first results in this direction can be found in my papers [248] [249], that I will summarize now, also keeping into account of some later improvements in [206] [98].”
New perturbation methods for nonlinear parabolic problems (2012, background): “Let us remark that asymptotic regularity results of the type just described are often crucial in establishing dimension estimates for singular sets of solutions to elliptic system (see for instance [18,19,27] and the recent survey [28] for a general overview) and in several problems coming from math”
The Singular Set of Lipschitzian Minima of Multiple Integrals (2007, result): “1 obviously applies to elliptic systems, that are a particular case of strictly quasimonotone systems as considered above, and in some cases improves the results in [29], where the second author proved the estimate dimH(Σu) ≤ n − 2α.”
Regularity for Double Phase Variational Problems (2015, result): “The previous theorem - involving the Nikolski spaceN - sharply extends previously known differentiability and integrability theorems valid in the p-Laplacean case [8], [24], [40], [41].”
Local behavior of fractional $p$-minimizers (2015, unknown): “By the fractional Poincaré type inequality (see, e.g., Proposition 5.1, Formula (6.3) in [20]) and the assumption in (2.1) for K we get with a constant c ≡ c(n, p, s, λ, (cid:6)) Now observe that v is a truncation of the sum of a constant and log (u + d) and hence it follows that At this stage, in…”
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (2005, methodology): “In the case of solutions to nonlinear elliptic systems similar estimates were obtained by Mingione in [73, 74].”
Harmonic type approximation lemmas (2009, background): “For estimates on the Hausdorff dimension of the singular set Ω \Ωu we refer to [67,68]; in general the dimension of Ω \Ωu is always strictly less than the ambient dimension n, as proved in [68].”