The obstacle problem revisited.

*(English)*Zbl 0928.49030The 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 |

##### References:

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