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We prove a regularity result for the unstable elliptic free boundary problem ∆u = −χ {u>0} (0.1) related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are regular, that is, {u = 0} is locally an analytic surface and u| {u>0} , u| {u<0} are locally analytic functions. Moreover we… (More)

- Henrik Shahgholian, Georg S Weiss
- 2005

For the two-phase membrane problem ∆u = λ + 2 χ {u>0} − λ − 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… (More)

We derive the precise limit of SHS in the high activation energy scaling suggested by B. In the time-increasing case the limit turns out to be the Stefan problem for supercooled water with spatially inhomogeneous coefficients. Although the present paper leaves open mathematical questions concerning the convergence, our precise form of the limit problem… (More)

In structurally disordered solids, some atoms or small groups of atoms are able to quantum mechanically tunnel between two nearly equivalent sites. These atomic tunneling systems have been identified as the cause of various low-temperature anomalies of bulk glasses and as a source of decoherence of superconducting qubits where they are sparsely present in… (More)

Parasitic two-level tunnelling systems originating from structural material defects affect the functionality of various microfabricated devices by acting as a source of noise. In particular, superconducting quantum bits may be sensitive to even single defects when these reside in the tunnel barrier of the qubit's Josephson junctions, and this can be… (More)

- Henrik Shahgholian, Nina Uraltseva, Georg S Weiss
- 2006

For the two-phase membrane problem ∆u = λ + χ {u>0} − λ − χ {u<0} , where λ + and λ − are positive Lipschitz functions, we prove in higher dimensions that the free boundary is in a neighborhood of each " branch point " the union of two C 1-graphs. The result is optimal in the sense that these graphs are in general not of class C 1,Dini , as shown by a… (More)

Consider the parabolic free boundary problem ∆u − ∂tu = 0 in {u > 0} , |∇u| = 1 on ∂{u > 0}. For a realistic class of solutions, containing for example all limits of the singular perturbation problem ∆uε − ∂tuε = βε(uε) as ε → 0, we prove that one-sided flatness of the free boundary implies regularity. In particular, we show that the topological free… (More)

Recent progress with microfabricated quantum devices has revealed that an ubiquitous source of noise originates in tunneling material defects that give rise to a sparse bath of parasitic two-level systems (TLSs). For superconducting qubits, TLSs residing on electrode surfaces and in tunnel junctions account for a major part of decoherence and thus pose a… (More)

- Sagun Chanillo, Georg S Weiss
- 2012

a r t i c l e i n f o a b s t r a c t MSC: primary 35R35 secondary 35J60 Keywords: Free boundary Star Singular point In this paper we classify the free boundary associated to equilibrium configurations of compressible, self-gravitating fluid masses, rotating with constant angular velocity. The equilibrium configurations are all critical points of an… (More)

- John Andersson, Henrik Shahgholian, Georg S Weiss
- 2009

We introduce a new method for the analysis of singularities in the unstable problem ∆u = −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… (More)