On the porosity of free boundaries in degenerate variational inequalities. (English) Zbl 0956.35054

Let us define \[ K_{\theta}=\{ v\in W^{1,p}(\Omega): \;v-\theta \in W^{1,p}_{0}(\Omega), \;v\geq 0 \text{ a.e. in }\Omega \} \] for a given bounded domain \( \Omega \) and \(\theta \in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). A function \(v\in K_{\theta}\) is a solution to the obstacle problem if \[ \int\limits_{\Omega}(|\nabla u|^{p-2}\nabla u (\nabla v-\nabla u)+f(v-u))dx\geq 0 \] whenever \(v\in K_{\theta}\). The authors obtained exact estimates for the solution to the obstacle problem near the free boundary and prove that the free boundary is porous and therefore its Hausdorff dimension is less than \(N\) and hence it is of Lebesgue measure zero.


35J70 Degenerate elliptic equations
35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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[1] Caffarelli, L. A., Compactness methods in free boundary problems, Comm. Partial Differential Equations, 5, 427-448 (1980) · Zbl 0437.35070
[2] Choe, H. J.; Lewis, J. L., On the obstacle problem for quasilinear elliptic equations of \(p\) Laplacian type, SIAM J. Math. Anal., 22, 623-638 (1991) · Zbl 0762.35035
[3] Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations (1993), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0780.31001
[4] Fuchs, M., Hölder continuity of the gradient for degenerate variational inequalities, Nonlinear Anal., 15, 85-100 (1990) · Zbl 0701.49019
[5] Martio, O.; Vuorinen, M., Whitney cubes, \(p\)-capacity, and Minkowski content, Exposition. Math., 5, 17-40 (1987) · Zbl 0632.30023
[6] Mu, J.; Ziemer, W. P., Smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations, Comm. Partial Differential Equations, 16, 821-843 (1991) · Zbl 0742.35010
[7] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math., 111, 247-302 (1964) · Zbl 0128.09101
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