Improved regularity for the parabolic normalized \(p\)-Laplace equation. (English) Zbl 1497.35076
Hölder and Sobolev regularity of viscosity solutions of the normalized equation
\[
\frac{\partial u}{\partial t} - |\nabla u|^{2-p}\,\nabla\!\cdot\!\bigl(|\nabla u|^{p-2}\nabla u\bigr)\,=\,f(x,t)
\]
are studied when \(p \approx 2\). As \(p \to 2\) the left-hand side approaches the heat operator \(u_t-\Delta u\). Thus the restriction that \(p\) is sufficiently close to \(2\) should improve the regularity. Here \(f(x,t)\) is continuous and bounded.
First, the known local \(C^{1+\alpha,(1+\alpha)/2}\) estimate is proved again. Then a local \(W^{2,1;q}\) estimate is obtained for every finite \(q>1\). The gain of keeping \(p\) very close to \(2\) is that the range of \(q\) is unrestricted. It was previously known that this holds with \(q < 2 + \delta\) for some \(\delta > 0\), but in a much wider range of \(p\), including \(1< p < 3\).
The equation is first regularized. An approximation method due to Caffarelli is essential.
First, the known local \(C^{1+\alpha,(1+\alpha)/2}\) estimate is proved again. Then a local \(W^{2,1;q}\) estimate is obtained for every finite \(q>1\). The gain of keeping \(p\) very close to \(2\) is that the range of \(q\) is unrestricted. It was previously known that this holds with \(q < 2 + \delta\) for some \(\delta > 0\), but in a much wider range of \(p\), including \(1< p < 3\).
The equation is first regularized. An approximation method due to Caffarelli is essential.
Reviewer: Peter Lindqvist (Trondheim)
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35D40 | Viscosity solutions to PDEs |
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