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Pointwise regularity of the free boundary for the parabolic obstacle problem. (English) Zbl 1327.35455

In this article, the authors consider the parabolic obstacle problem \[ \Delta u - u_t = f \chi_{\{u>0\}}, \quad u \geq 0, \] in \(\mathbb{R}^{n}\times (-1,0]\) under the (normalization) conditions that \[ 0\in \{u=0\}\cap \partial\{u>0\}, \quad u,f \in L^{p}(\mathbb R^{n}\times (-1,0]), \quad f(0)=1, \quad \mathrm{supp}(u)\subset B_1 \times (-1,0]. \] Assuming only (Dini type) continuity of the inhomogeneity \(f\) with respect to different moduli of continuity, the authors derive a classification of free boundary points, prove a Taylor expansion of second order at the singular free boundary, show a regularity result for the regular free boundary and give a structure theorem for the singular set of the free boundary.
All these results rely on the derivation and analysis of two central monotonicity formulae. The first can be regarded as a generalization of a parabolic “Weiß-type” functional including a rough (Dini type) inhomogeneity \(f\). This leads to the classification of the free boundary points in terms of degenerate, regular and singular free boundary points. Second, the authors introduce a parabolic analogue of the monotonicity formula from [R. Monneau, J. Fourier Anal. Appl. 15, No. 3, 279–335 (2009; Zbl 1178.35142)] at the singular free boundary by measuring the closeness to the “singular blow-up polynomials” in the presence of a rough (Dini type) inhomogeneity. This second monotonicity formula is crucial in analyzing the structure of the singular free boundary, as it for instance leads to both the uniqueness proof of blow-ups at the singular free boundary and to quantitative error estimates for the second-order Taylor expansion at the singular free boundary.

MSC:

35R35 Free boundary problems for PDEs

Citations:

Zbl 1178.35142
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References:

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