arXiv subjects

Alvis Zahl

Publications and source records attributed to Alvis Zahl.

3 recordsLinked to original sources

arXiv2026

Free Boundary Regularity for Non-Convex Fully Nonlinear Alt-Phillips Problems

This paper studies the fully nonlinear Alt-Phillips free boundary problem \begin{equation} F(D^2u) = u^\gamma \chi_{\{u>0\}}, \end{equation} for $\gamma \in (-1/3,1)$ without any convexity assumption on $F$. We establish that flat free boundaries are smooth. This result is new even for the fully nonlinear obstacle problem when $\gamma = 0$. Our approach is based on a partial hodograph transform, which converts the free boundary into a fixed boundary and produces a fully nonlinear degenerate equation. For this equation we prove a Harnack inequality, yielding an improvement of flatness, and a Schauder estimate for the associated linearized equation. Both estimates appear to be new and are of independent interest.

math.AP↗

arXiv2025

Minimizing Eigenvalues of the Fractional Laplacian

We study the minimizers of \begin{equation} \lambda_k^s(A) + |A| \end{equation} where $\lambda^s_k(A)$ is the $k$-th Dirichlet eigenvalue of the fractional Laplacian on $A$. Unlike in the case of the Laplacian, the free boundary of minimizers exhibit distinct global behavior. Our main results include: the existence of minimizers, optimal H\"older regularity for the corresponding eigenfunctions, and in the case where $\lambda_k$ is simple, non-degeneracy, density estimates, separation of the free boundary, and free boundary regularity. We propose a combinatorial toy problem related to the global configuration of such minimizers.

math.AP↗

arXiv2021

First Order Hyperbolic Boundary Value Problems With a Large Oscillatory Zero Order Term

We study the weakly stable hyperbolic boundary value problem with a large zero order oscillatory coefficient. This problem is related to linearized problems in the study of Mach stem and vortex sheets. We wish to establish a uniform energy estimate with respect to {\epsilon}, which is needed in mentioned applications and justification of geometric optics solutions, but the zero order oscillatory term gives rise to great obstacles. In this paper we obtain positive results in the small/medium frequency region by adapting the approach of Williams to a more general situation than the one treated there. We also show that it is possible to construct high order approximate solutions by the method of geometric optics for those systems without any restriction on frequencies.

math.AP↗