arXiv subjects

Georg S. Weiss

Publications and source records attributed to Georg S. Weiss.

21 records · Page 2Linked to original sources

arXiv2009

Uniform Regularity close to Cross Singularities in an Unstable Free Boundary Problem

We introduce a new method for the analysis of singularities in the unstable problem $$Δu = -χ_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of $u$ are unbounded, the level set $\{u=0\}$ consists of two $C^1$-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

math.AP↗

arXiv2007

The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions

For the parabolic obstacle-problem-like equation $$Δu - \partial_t u = λ_+ χ_{\{u>0\}} - λ_- χ_{\{u<0\}} ,$$ where $λ_+$ and $λ_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partial\{u<0\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.

math.AP↗

arXiv2005

The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to the Regularity at Branch Points

For the two-phase membrane problem $ Δu = {λ_+\over 2} χ_{\{u>0\}} - {λ_-\over 2} χ_{\{u<0\}} ,$ where $λ_+> 0$ and $λ_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection. In higher dimensions we show that the free boundary has finite $(n-1)$-dimensional Hausdorff measure.

math.AP↗