Limits of vectorial fractional Allen-Cahn solutions are stationary nonlocal minimal partitions, and minimizing 3-partitions are smooth outside a small singular set, even under a reversed triangle inequality for s close to 1/2.
Concentration sets for multiple equal-depth wells potentials in the 2D elliptic case
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abstract
The formation of codimension-one interfaces for multiwell gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The vectorial case in contrast is quite open. This lack of results and insight is to a large extend related to the absence of known appropriate monotonicity formula. In this paper, we focus on the elliptic case in two dimensions, and introduce some methods which allow to circumvent the lack of monotonicity formula. This methods lead, as expected, to concentration on one-dimensional rectifiable sets.
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Fractional multi-phase transitions and nonlocal minimal partitions
Limits of vectorial fractional Allen-Cahn solutions are stationary nonlocal minimal partitions, and minimizing 3-partitions are smooth outside a small singular set, even under a reversed triangle inequality for s close to 1/2.