We study the obstacle problem for parabolic operators of the type $partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-Delta)^s$, in the supercritical regime $s in (0,frac{1}{2})$. The best result in this
context was due to Caffarelli and Figalli, who established the $C^{1,s}_x$ regularity of solutions for the case $L = (-Delta)^s$, the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually textit{more} regular than in the elliptic case. More precisely, we show that they are $C^{1,1}$ in space and time, and that this is optimal. We also deduce the $C^{1,alpha}$ regularity of the free boundary. Moreover, at all free boundary points $(x_0,t_0)$, we establish the following expansion: $$(u - varphi)(x_0+x,t_0+t) = c_0(t - acdot x)_+^2 + O(t^{2+alpha}+|x|^{2+alpha}),$$ with $c_0 > 0$, $alpha > 0$ and $a in mathbb R^n$.
We study the regularity of minimizers of the functional $mathcal E(u):= [u]_{H^s(Omega)}^2 +int_Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $Omegasubsetmathbb R^N$. More precisely, we are interested
on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in $H^s(Omega)$ (i.e., Neumann problem), or in the case of Dirichlet condition $uin H^s_0(Omega)$ when $s>frac12$. Our main result establishes the sharp regularity of solutions in both cases: $uin C^{2s+alpha}(overlineOmega)$ in the Neumann case, and $u/delta^{2s-1}in C^{1+alpha}(overlineOmega)$ in the Dirichlet case. Here, $delta$ is the distance to $partialOmega$, and $alpha<alpha_s$, with $alpha_sin (0,1-s)$ and $2s+alpha_s>1$. We also show the optimality of our result: these estimates fail for $alpha>alpha_s$, even when $f$ and $partialOmega$ are $C^infty$.
In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n-1$. - The solution admits a $C^infty$-expansion at all singular points, up to a set o
f parabolic Hausdorff dimension at most $n-2$. - In $mathbb R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $mathcal Ssubset mathbb R$ has Hausdorff dimension at most $1/2$. These results provide us with a refined understanding of the Stefan problems singularities and answer some long-standing open questions in the field.
We study solutions to $Lu=f$ in $Omegasubsetmathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Levy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Omega$, $u=0$ in $Omega^c$, in $C^{1,alpha}$ d
omains~$Omega$. We show that solutions $u$ satisfy $u/d^gammain C^{varepsilon_circ}big(overlineOmegabig)$, where $d$ is the distance to $partialOmega$, and $gamma=gamma(L, u)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ u$ to the boundary $partialOmega$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,alpha}$ domains. We do it via a new efficient approximation argument, which exploits the Holder regularity of $u/d^gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.
We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coeff
icients, whereas the right hand side is in $L^q$ with $q > n$. Our approach is based on the scaling and comparison arguments of cite{DS20}, and we show that all our assumptions are sharp. As a consequence of our results, we deduce the $mathcal{C}^{1,alpha}$ regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.
The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $ngeq3$ is completely open. In this contex
t, axially symmetric solutions are expected to play the same role as Simons cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of large solutions for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions $nle 5$ is one-dimensional, independently of the parameter $sin (0,1)$.
In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N geq 3$. We do this i
n terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.
We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Delta)^s u=f$ in $Omega$, $mathcal N_s u=0$ in $Omega^c$, then $u$ is $C^alpha$ up tp the bound
ary for some $alpha>0$. Moreover, in case $s>frac12$, we then show that $uin C^{2s-1+alpha}(overlineOmega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
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 degen
erate) 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.
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^infty$ outside a set of singular points. Explicit examples show that th
e singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $nleq4$, the free boundary is generically a $C^infty$ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions $nleq4$.
