Critical 𝐿^{𝑝}-differentiability of 𝐵𝑉^{}𝔸-maps and canceling operators

Abstract

We give a generalization of Dorronsoro's Theorem [22] on critical L p -Taylor expansions for BV k -maps on R n , i.e., we characterize homogeneous linear differential operators A of k-th order such that D k−j u has j-th order L n/(n−j) -Taylor expansion a.e. for all u ∈ BV A loc (here j = 1, . . . , k, with an appropriate convention if j ≥ n). The space BV A loc , a single framework covering BV, BD, and BV k , consists of those locally integrable maps u such that Au is a Radon measure on R n .For j = 1, . . . , min{k, n − 1}, we show that the L p -differentiability property above is equivalent with Van Schaftingen's elliptic and canceling condition for A [55]. For j = n, . . . , k, ellipticity is necessary, but cancellation is not. To complete the characterization, we determine the class of elliptic operators A such that the estimateholds for all vector fields u ∈ C ∞ c . Surprisingly, the (computable) condition on A such that (1) holds is strictly weaker than cancellation.The results on L p -differentiability can be formulated as sharp pointwise regularity results for overdetermined elliptic systemswhere µ is a Radon measure, thereby giving a variant for the limit case p = 1 of a Theorem of Calderón and Zygmund in [14] which was not covered before.2010 Mathematics Subject Classification. Primary: 26B05; Secondary: 35J48.

Continuity and canceling operators of order n on $${\mathbb {R}}^n$$

“…This can be seen already from the Introduction, where the operator ∆ • (div, curl) on R 3 was mentioned. More examples illustrating the difference between the two classes can be found in [25,Sec. 4.3].…”

“…1.3] that the estimate (1.2) holds for canceling operators, while simple one-dimensional examples show that the canceling condition might not be necessary. Indeed, it was shown in [25,Thm. 1.3] that the inequality (1.2) is equivalent to a weakly canceling condition.…”

“…1.3] that the inequality (1.2) is equivalent to a weakly canceling condition. We briefly describe the difference between the two conditions from an analytic perspective: On one hand, B is canceling if and only if the equation Bu = δ 0 w for vectors w has solutions only for w = 0 [25,Lem. 2.5].…”

“…2.5]. On the other hand, B is weakly canceling if and only if solutions of Bu = δ 0 w are locally bounded [25,Lem. 4.3].…”

“…From the point of view of weak formulations (e.g., of the Euler-Lagrange equations), it is natural to investigate the spaces The emergence of the BV B -space is natural since, from a typical L 1 -bound on a minimizing/approximating sequence, one cannot infer weak L 1 -compactness, but can infer weakly- * compactness in the space of measures. Motivated by these facts, several analytical properties of W B,1 and BV B were already studied in [9,16,25].…”