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Register a new userRegularity of the free boundary for a parabolic cooperative system
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In this paper we study the following parabolic system begin{equation*} Delta u -partial_t u =|u|^{q-1}u,chi_{{ |u|>0 }}, qquad u = (u^1, cdots , u^m) , end{equation*} with free boundary $partial {|u | >0}$. For $0leq q<1$, we prove optimal growth rate for solutions $u $ to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $C^{1, alpha}$ in space directions and half-Lipschitz in the time direction.
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The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Levy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when $s>frac12$, the free boundary is a $C^{1,alpha}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)in partial{u>varphi}$. Furthermore, we also prove that solutions $u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of the form [u(x,t)-varphi(x)=c_0bigl((x-x_0)cdot e+a(t-t_0)bigr)_+^{1+s}+obigl(|x-x_0|^{1+s+alpha}+ |t-t_0|^{1+s+alpha}bigr),] with $c_0>0$, $ein mathbb{S}^{n-1}$, and $a>0$.
We obtain the maximal regularity for the mixed Dirichlet-conormal problem in cylindrical domains with time-dependent separations, which is the first of its kind. The boundary of the domain is assumed to be Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,ldots,d-2$, with respect to the Hausdorff distance. We consider solutions in both $L_p$-based Sobolev spaces and $L_{q,p}$-based mixed-norm Sobolev spaces.
We investigate the regularity of the free boundary for the Signorini problem in $mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^infty$. However, even for $C^infty$ obstacles $varphi$, the set of non-regular (or degenerate) points could be very large, e.g. with infinite $mathcal{H}^{n-1}$ measure. The only two assumptions under which a nice structure result for degenerate points has been established are: when $varphi$ is analytic, and when $Deltavarphi < 0$. However, even in these cases, the set of degenerate points is in general $(n-1)$-dimensional (as large as the set of regular points). In this work, we show for the first time that, usually, the set of degenerate points is small. Namely, we prove that, given any $C^infty$ obstacle, for almost every solution the non-regular part of the free boundary is at most $(n-2)$-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $(-Delta)^s$, and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $(n-1-alpha_circ)$-dimensional for almost all times $t$, for some $alpha_circ > 0$. Finally, we construct some new examples of free boundaries with degenerate points.
We prove an $epsilon$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets. Instead, we follow the spirit of the last-named authors recent work on free boundary harmonic maps and choose a good frame directly at the free boundary. This leads to growth estimates, which, in the critical regime $p=n$, imply Holder regularity of solutions. In the supercritical regime, $p < n$, we combine the growth estimate with the geometric reflection argument: the reflected equation is super-critical, but, under the assumption of growth estimates, solutions are regular. In the case $p<n$, for stationary $p$-harmonic maps with free boundary, as a consequence of a monotonicity formula we obtain partial regularity up to the boundary away from a set of $(n-p)$-dimensional Hausdorff measure.
We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
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