Singular Perturbation Problem in Boundary/Fractional Combustion

Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is related to the fractional Laplacian. After an optimal uniform H\"older regularity is shown, we pass to the limit to study the free boundary problem it leads to

The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary

This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,\alpha}$ regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1}{2}\}}$ regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in \cite{KRS14} and the present article, we then also discuss the case of non-zero and interior thin obstacles. We obtain the optimal regularity of the solutions and the regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$ obstacles with $p\in (2(n+1),\infty]$.Comment: 62 pages, this is a slightly updated version, the arguments for the zero obstacle are strenghtened, an argument on the Hausdorff-dimension of the free boundary is adde

Microdischarge Arrays

Microhollow cathode discharges (MHCDs) are DC or pulsed gas discharges between two electrodes, separated by a dielectric, and containing a concentric hole. The diameter of the hole, in this hollow cathode configuration, is in the hundred-micrometer range. MHCDs satisfy the two conditions necessary for an efficient excimer radiation sources: (1) high energy electrons which are required to provide a high concentration of excited or ionized rare gas atoms; (2) high pressure operation which favors excimer formation (a three-body process). Flat panel excimer sources require parallel operation of MHCDs. Based on the current-voltage characteristics of MHCD discharges, which have positive slopes in the low current (Townsend) mode and in the abnormal glow mode, stable arrays of MHCD discharges in argon and xenon could be generated in these current ranges without ballasting each MHCD separately. In the Townsend range, these arrays could be operated up to pressures of 400 Torr. In the abnormal glow mode, discharge arrays were found to be stable up to atmospheric pressure. By using semi-insulating silicon as the anode material, the stable operation of MHCD arrays could be extended to the current range with constant voltage (normal glow) and also that with negative differential conductance (hollow cathode discharge region). Experiments with a cathode geometry without microholes, i.e. excluding the hollow cathode phase, revealed that stable operation of discharges over an extended area were possible. The discharge structure in this configuration reduces to only the cathode fall and negative glow, with the negative glow plasma serving to conduct the discharge current radially to the circular anode. With decreasing current, a transition from homogenous plasma to self-organized plasma filaments is observed. Array formation was not only studied with discharges in parallel, but also with MHCD discharges in series. By using a sandwich electrode configuration, a tandem discharge was generated. For an anode-cathode-anode configuration, the excimer irradiance, recorded on the axis of the discharge, was twice that of a single discharge. The extension of this basic tandem electrode structure to multiple electrode configurations permits the generation of high-irradiance excimer sources

Second order expansion for the nonlocal perimeter functional

The seminal results of Bourgain, Brezis, Mironescu and D\'avila show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the Gamma--limit of these functionals. We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function.Comment: 30 pages, 1 figur