Weakly differentiable functions on varifolds

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj , Duj )j for (uj )j ⊂ BV(Ω; R m ) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functionalto the space BV(Ω; R m ). Lower semicontinuity results of this type were first obtained by Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.

show abstract

“…Finally, the Poincaré inequality (28) follows directly from condition (34). The Poincaré inequality (34) for ν j implies…”

Section: Young Measuresmentioning

confidence: 99%

“…conditions (33) and (34) follow directly from the definition of ι ν , λ ν , the construction of S, and the Poincaré inequality (28). Now let µ ∈ M + (Ω × σR m × B m×d ) satisfy conditions (33), (34), and (35). It remains to show that S * µ ∈ Y(Ω × R m ; R m×d ).…”

Section: Young Measuresmentioning

confidence: 99%

Liftings, Young Measures, and Lower Semicontinuity

Rindler

Shaw

2018

Arch Rational Mech Anal

View full text Add to dashboard Cite

show abstract

“…While these papers are a good start, there is still a great deal of opportunity for the use and further development of varifolds. On the theoretical front, there is the work of Menne and collaborators [34,35,36,37,38,28,39,40,41,42]. We want to call special attention to the recent introduction to the idea of a varifold that appeared in the December 2017 AMS Notices [41].…”

Section: Further Explorationmentioning

confidence: 99%

Cubical Covers of Sets in R^n

Paxton¹,

Vixie²

2019

ntmsci

View full text Add to dashboard Cite

Wild sets in R n can be tamed through the use of various representations though sometimes this taming removes features considered important. Finding the wildest sets for which it is still true that the representations faithfully inform us about the original set is the focus of this rather playful, expository paper that we hope will stimulate interest in wild sets and their representations, as well as the other two ideas we explore very briefly: Jones' β numbers and varifolds from geometric measure theory.

show abstract

“…We notice that, by [ [9] in their statements and proofs are replaced by references to the present, more general definition in 4.2; in fact, it is sufficient to additionally replace the references to "Remark 8.6" in their proofs in [9] by references to 4.10 in the present paper and use (instead of "Example 8.7" in [9]) an approximation based on convolution, 4.5, and 4.6, to justify the second ingredient to the equality on page 1029, line 25 in [9]: namely, the equation…”

Section: Generalised Weakly Differentiable Functionsmentioning

confidence: 99%

An isoperimetric inequality for diffused surfaces

Menne¹,

Scharrer²

2018

Kodai Math. J.

Self Cite

View full text Add to dashboard Cite

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.MSC-classes 2010. 53A07 (Primary); 46E35, 49Q15 (Secondary).Keywords. Varifold, isoperimetric inequality, generalised weakly differentiable function, Sobolev inequality. IntroductionGeneral aim. The isoperimetric inequality is well established in the context of sharp surfaces (e.g., integral currents, sets, or integral varifolds) in Euclidean space, but little appears to be known for diffused surfaces (i.e., for surfaces that are not concentrated on a set of the their own dimension). General varifolds form a very flexible model for the latter case; in fact, for equations of Allen-Cahn type, their utility was established by Ilmanen, Padilla, and Tonegawa (see [6] and [12]) and, for discrete and computational geometry, their unifying use has been recently suggested by Buet, Leonardi, and Masnou (see [3]). The present paper shall contribute to this proposed development by adapting several core tools to the possibly non-rectifiable case. To outline these results, suppose m and n are positive integers, m ≤ n, V is an m dimensional varifold in R n , and, to avoid case distinctions, also m > 1; see Section 2 for the notation.Isoperimetric inequality, see Section 3. The best result up to now (see the second author [13, 6.11]) did apply to general varifolds, but controlled only their rectifiable parts:where Γ is a positive, finite number determined by m. Following the first author (see [7, 2.2]), it unified the approach of Allard in [2, 7.1] and Michael and Simon in [11, 2.1]. Clearly, if 0 < d < ∞, and Θ m ( V , x) ≥ d for V almost all x, the result impliesWe notice that δV encodes both, the total mass of the variational boundary and the integral of the modulus of the generalised mean curvature of the varifold, see Allard [2, 4.3]; in particular, a more classical form results for varifolds with vanishing mean curvature (i.e., generalised minimal surfaces) and, by Allard [2, 4.8 (4)], the isoperimetric inequality for integral currents with nonoptimal constant is a special case. In 3.5 and 3.7, we establish that, if V (R n ) < ∞, thenBy homogeneity considerations, one may not replace (A(d), d −1/m ) by (R n , 1). The sets A(d), for suitable d, naturally describe the region, where the behaviour of the diffused surface resembles the behaviour of an m dimensional sharp surface.Generalised weakly differentiable functions, see Section 4. We extend the basic theory of generalised weakly differentiable functions (see the first author [9, § § 8-9] and [10, 4.1, 2]) from rectifiable varifolds to general varifolds. This theory includes the study of closedness properties (under convergence, composition, addition, and multiplication) and a coarea formula in functional analytic form. The main differences lie in the possible non-existence of decomposit...

show abstract

Weakly differentiable functions on varifolds

Cited by 21 publications

References 30 publications

Liftings, Young Measures, and Lower Semicontinuity

Liftings, Young Measures, and Lower Semicontinuity

Cubical Covers of Sets in R^n

An isoperimetric inequality for diffused surfaces

Contact Info

Product

Resources

About