##
**The obstacle problem revisited.**
*(English)*
Zbl 0928.49030

The problem is to minimize the Dirichlet integral
\[
D(u)=\int_D \left( \nabla u\right)^2dX
\]
subject to the boundary condition \(u| _{\partial D} =f(X)\) and a constraint imposed by an obstacle: \(u\geq \varphi\) on entire \(D\). It is the linearized version of a minimal surface problem with an obstacle.

It is assumed that the functions \(f\) and \(\varphi\) and the domain \(D\) are smooth. The author reviews his earlier results concerning regularity of the solution and the boundary of the set \(\{u=\varphi\}\) appearing on the obstacle [see, e.g., L. A. Caffarelli, Acta Math. 139, 155-184 (1978; Zbl 0386.35046)]. Also, a result characterizing the structure of the set of singular points of the solution is obtained. The main theorem is formulated in terms of normalized solutions, i.e., functions \(w\in C^{1,1}\) which are nonnegative throughout the domain \(D\) and whose Laplacian is identically equal to one where the function is positive (clearly, normalized solutions are “inspired” by the difference \(u-\varphi\)). It is obtained that a normalized solution is behaving like a quadratic function at each point \(X_0\) of the boundary of the set \(w>0\). Moreover, there exists a universal modulus of continuity \(\sigma\) such that \[ \left| w(X)-1/2\left( (X-X_0)^T M(X_0) (X-X_0) \right) \right| \leq \left| X-X_0 \right| ^2 \sigma(\left| X-X_0\right |). \] The matrix of the quadratic form is continuous with respect to \(X_0\) and its trace is identically equal to one. Further, the singular set of the normalized solution \(w\) is a \(k\)-dimensional manifold in a neighborhood of \(X_0\), and the dimension \(k\) of this manifold is equal to the defect of the matrix \(M(X_0)\). Finally, the size of this neighborhood is determined by the smallest non-zero eigenvalue of \(M(X_0)\).

It is assumed that the functions \(f\) and \(\varphi\) and the domain \(D\) are smooth. The author reviews his earlier results concerning regularity of the solution and the boundary of the set \(\{u=\varphi\}\) appearing on the obstacle [see, e.g., L. A. Caffarelli, Acta Math. 139, 155-184 (1978; Zbl 0386.35046)]. Also, a result characterizing the structure of the set of singular points of the solution is obtained. The main theorem is formulated in terms of normalized solutions, i.e., functions \(w\in C^{1,1}\) which are nonnegative throughout the domain \(D\) and whose Laplacian is identically equal to one where the function is positive (clearly, normalized solutions are “inspired” by the difference \(u-\varphi\)). It is obtained that a normalized solution is behaving like a quadratic function at each point \(X_0\) of the boundary of the set \(w>0\). Moreover, there exists a universal modulus of continuity \(\sigma\) such that \[ \left| w(X)-1/2\left( (X-X_0)^T M(X_0) (X-X_0) \right) \right| \leq \left| X-X_0 \right| ^2 \sigma(\left| X-X_0\right |). \] The matrix of the quadratic form is continuous with respect to \(X_0\) and its trace is identically equal to one. Further, the singular set of the normalized solution \(w\) is a \(k\)-dimensional manifold in a neighborhood of \(X_0\), and the dimension \(k\) of this manifold is equal to the defect of the matrix \(M(X_0)\). Finally, the size of this neighborhood is determined by the smallest non-zero eigenvalue of \(M(X_0)\).

Reviewer: D.Silin (Berkeley)

### MSC:

49Q05 | Minimal surfaces and optimization |

49Q10 | Optimization of shapes other than minimal surfaces |

### Citations:

Zbl 0386.35046
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\textit{L. A. Caffarelli}, J. Fourier Anal. Appl. 4, No. 4--5, 383--402 (1998; Zbl 0928.49030)

### References:

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[2] | Athanasopoulos, O. and Caffarelli, L.A. (1985). A theorem of real analysis and its application to free boundary problems.Comm. Pure Appl. Math.,38(5), 499–502. · Zbl 0593.35084 |

[3] | Caffarelli, L.A. (1977). The regularity of free boundaries in higher dimensions.Acta Math.,139(3–4), 155–184. · Zbl 0386.35046 |

[4] | Caffarelli, L.A. (1980). Compactness methods in free boundary problems.Comm. Partial Differential Equations,5(4), 427–448. · Zbl 0437.35070 |

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[6] | Caffarelli, L.A. and Kinderlehrer, D. (1980). Potential methods in variational inequalities.J. Analyse Math.,37, 285–295. · Zbl 0455.49010 |

[7] | Caffarelli, L.A., Fabes, E., Mortola, S., and Salsa, S. (1981). Boundary behavior of nonnegative solutions of elliptic operators in divergence form.Indiana Univ. Math. J.,30(4), 621–640. · Zbl 0512.35038 |

[8] | Frehse, J. (1972). On the regularity of solutions of a second order variational inequality.Boll. Un. Mat. Ital.,6(4). · Zbl 0261.49021 |

[9] | Friedman, A. (1982).Variational Principles and Free Boundary Problems. Wiley, New York. · Zbl 0564.49002 |

[10] | Jerison, D. and Kenig, C. (1982). Boundary behaviour of harmonic functions in nontangentially accessible domains.Adv. in Math.,46(1), 80–147. · Zbl 0514.31003 |

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