131 research outputs found
Partial regularity of almost minimizing rectifiable G chains in Hilbert space
We adapt to an infinite dimensional ambient space E.R. Reifenberg's
epiperimetric inequality and a quantitative version of D. Preiss' second
moments computations to establish that the set of regular points of an almost
mass minimizing rectifiable chain in is dense in its support,
whenever the group of coefficients is so that is
discrete and closed.Comment: 96 page
The Gauss–Green theorem and removable sets for PDEs in divergence form
AbstractApplying a very general Gauss–Green theorem established for the generalized Riemann integral, we obtain simple proofs of new results about removable sets of singularities for the Laplace and minimal surface equations. We treat simultaneously singularities with respect to differentiability and continuity
Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation
Letting be Borel measurable and Lipschitzian, we establish that \begin{equation*} \limsup_{r
\to 0^+} \frac{\mathcal{H}^m \left[ A \cap B(x,r) \cap (x+
W_0(x))\right]}{\alpha(m)r^m} \geq \frac{1}{2^n}, \end{equation*} for
-almost every . In particular, it follows that is
-negligible if and only if ,
for -almost every .Comment: arXiv admin note: substantial text overlap with arXiv:1904.1227
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