arXiv subjects

Teo Kukuljan

Publications and source records attributed to Teo Kukuljan.

5 recordsLinked to original sources

arXiv2022

Regularity theory for fully nonlinear parabolic obstacle problems

We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension $n-1$ which is also $\varepsilon$-flat in space, for any $\varepsilon>0$.

math.AP↗

arXiv2022

$C^{2,α}$ regularity of free boundaries in parabolic non-local obstacle problems

We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian $(-Δ)^s$ (and more general integro-differential operators) in the regime $s>\frac{1}{2}$. We prove that once the free boundary is $C^1$ it is actually $C^{2,α}$. To do so, we establish a boundary Harnack inequality in $C^1$ and $C^{1,α}$ (moving) domains, providing that the quotient of two solutions of the linear equation, that vanish on the boundary, is as smooth as the boundary. As a consequence of our results we also establish for the first time optimal regularity of such solutions to nonlocal parabolic equations in moving domains.

math.AP↗

arXiv2021

Higher order parabolic boundary Harnack inequality in $C^1$ and $C^{k,α}$ domains

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in $C^1$ and $C^{k,α}$ domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary. As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.

math.AP↗

arXiv2020

The fractional obstacle problem with drift: higher regularity of free boundaries

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-Δ)^s +b\cdot\nabla$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free boundary is $C^1$ then it is $C^\infty$, whenever $s\not\in\mathbb{Q}$. In order to achieve this, we establish a fine boundary expansion for solutions to linear nonlocal equations with drift in terms of the powers of distance function. Quite interestingly, due to the drift term, the powers do not increase by natural numbers and the fact that $s$ is irrational plays al important role. Such expansion still allows us to prove a higher order boundary Harnack inequality, where the regularity holds in the tangential directions only.

math.AP↗

arXiv2019

On the absolutely continuous spectrum of generalized indefinite strings II

We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-\infty,-1/2]\cup [1/2,\infty)$.

math.SP↗