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Free Boundary Regularity for Almost Every Solution to the Signorini Problem.

Xavier Fernández-Real1, Xavier Ros-Oton2,3

  • 1EPFL SB, Station 8, 1015 Lausanne, Switzerland.

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This study reveals that for the Signorini problem, the set of non-regular free boundary points is typically small, specifically at most 1-dimensional. This finding applies to almost every solution, offering a significant advancement in understanding degenerate points in free boundary problems.

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Area of Science:

  • Partial Differential Equations
  • Geometric Measure Theory
  • Free Boundary Problems

Background:

  • The regularity of free boundaries in the Signorini problem is a key area of research.
  • Existing literature shows regular points are typically 1-dimensional, but degenerate points can be large and complex.
  • Previous structural results for degenerate points were limited to specific assumptions (analytic obstacles or specific dimensions).

Purpose of the Study:

  • To investigate the 'usual' size of the set of non-regular (degenerate) points for the Signorini problem.
  • To establish new structural results for degenerate points under general conditions.
  • To extend these findings to related problems like the fractional Laplacian obstacle problem and the parabolic Signorini problem.

Main Methods:

  • Analysis of the regularity of free boundaries in the context of the Signorini problem.
  • Application of techniques from geometric measure theory to quantify the size of degenerate sets.
  • Development of new methods to handle 'almost every' solution, rather than specific cases.

Main Results:

  • For the Signorini problem, it is proven that for almost every solution, the non-regular part of the free boundary is at most 1-dimensional.
  • Analogous results are established for the obstacle problem with the fractional Laplacian and for the parabolic Signorini problem.
  • New examples of free boundaries exhibiting degenerate points are constructed.

Conclusions:

  • This work demonstrates that degenerate points in the Signorini problem are 'small' in a probabilistic sense, a significant departure from previous understanding.
  • The findings provide a more comprehensive picture of free boundary regularity across different types of elliptic and parabolic problems.
  • The construction of new examples enriches the understanding of potential geometric configurations of degenerate free boundaries.