Related papers: Higher Regularity for the fractional thin obstacle…
The free boundary for the Signorini problem in $\mathbb{R}^{n+1}$ is smooth outside of a degenerate set, which can have the same dimension ($n-1$) as the free boundary itself. In [FR21] it was shown that generically, the set where the free…
We study the regularity of the free boundary in one-phase Stefan problem with nonlinear operator. Using the Hodograph transform and a linearization technique, we prove that flat free boundaries are $C^{1, \alpha}$ in space and time. When…
In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional…
The interior free boundary theory for linear elliptic operators in higher dimensions was developed by Caffarelli in the low regularity context. In these notes, the up-to-the boundary free boundary regularity is discussed for nonlinear…
We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to…
We study the regularity of the "free surface" in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a $C^{1,\alpha}$ $(n-2)$-dimensional surface in $\real^{n-1}$.
This paper investigates fractional torsional rigidity on compact, connected metric graphs, a novel extension of the classical concept to nonlocal operators. The fractional torsional rigidity is defined as the $L^1$-norm of the fractional…
We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that…
We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta \varphi\leq 0$ near the contact region. Our main result establishes that…
We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is $C^1$ near regular points.
The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$\ddot{\text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces.…
In this note we discuss the (higher) regularity properties of the Signorini problem for the homogeneous, isotropic Lam\'e system. Relying on an observation by Schumann \cite{Schumann1}, we reduce the question of the solution's and the free…
We revisit and sharpen the results from our previous work, where we investigated the regularity of the singular set of the free boundary in the nonlinear obstacle problem. As in the work of Figalli-Serra on the classical obstacle problem,…
We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…
In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the…
This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight…
In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…
The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the…
We consider the stationary Boltzmann equation with the angular cutoff cross section in a bounded convex domain under the incoming boundary condition. In this article, we discuss the fractional Sobolev regularity of the solution without…