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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 $$M(X_0)$$.
Reviewer: D.Silin (Berkeley)

##### MSC:
 49Q05 Minimal surfaces and optimization 49Q10 Optimization of shapes other than minimal surfaces
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##### References:
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