A singular minimizer of a smooth strongly convex functional in three dimensions

By Vladimír Sverák, Xiaodong Yan. 2000

91 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/5d897f372043ed7624322cbd16c013ab9aa4e0b3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Some Open Problems in Elasticity (2002, methodology): “An instructive example (resulting from discussions with X. Yan and J. Bevan) is that of minimizing the two-dimensional energy for an incompressible material I(y) = ∫ B |Dy|2 dx, where B = B(0, 1) is the unit disc in R2, and y : B → R2, in the set of admissible mappings A = {y ∈W 1,2(B;R2) : detDy = ”
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].”
The existence of regular boundary points for non-linear elliptic systems (2007, background): “For general elliptic systems, well-known examples [13], [40], [44], [45], demonstrate that interior singularities may actually occur, unless one imposes very particular, special structure conditions for the vector field a [31], [46], [16], [1], [25], [28], [29], [48].”
Nonlinear elliptic systems with general growth (2006, background): “A recent counterexample in threedimensional case in the context of smooth strongly convex functionals has been also given by Sverak-Yan [22].”
Pointwise Calderón–Zygmund gradient estimates for the p -Laplace system (2015, methodology): “This has been demonstrated via several counterexamples, including those from [32, 50, 30, 56, 57].”
On the Dirichlet problem for variational integrals in BV (2012, background): “In contrast, in the vectorial case it is known by a series of striking examples [28], [48], [70], [52], [83], [84] that one cannot hope for everywhere regularity, not even for boundedness of minimizers without some kind of particular structure.”
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (2005, unknown): “For a more comprehensive discussion we refer to [37] and to the more recent works [105] and [65].”
Scalar Minimizers with Fractal Singular Sets (2004, background): “This result holds true even without any convexity assumption on f . Observe that this is not the case in the vectorial setting u : � → R k ,k > 1 (see for instance the counterexamples in [DG2, N, SY1 , SY2]).”
Progress and puzzles in nonlinear elasticity (2010, background): “The following such result is due to Müller, Qi & Yan [28], following [3].”
The Singular Set of Minima of Integral Functionals (2006, background): “In the vectorial case n, N 2 we cannot in general expect full (interior) regularity of minimizers, see [11, 23, 41, 48 , 49], and instead only the so-called partial regularity is available [20, 30, 22, 14].”
Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure (2009, background): “The smallest such S is called the singular set and need — by famous examples [26, 33, 38, 39] — not be empty, except for n = 2 [27, 32] or N = 1 [20, 21, 30].”
Lipschitz estimates for systems with ellipticity conditions at infinity (2016, background): “5) is a model example? The structure f (x, Du) = g(x, |Du|) is necessary to treat the general case N ≥ 1, since in the vectorial framework minimizers can be unbounded even when p = q [11,38].”