Energy minimizers to a MEMS model with an insulating layer are shown to converge in its reinforced limit to the minimizer of the limiting model as the thickness of the layer tends to zero. The proof relies on the identification of the $Gamma$-limit of the energy in this limit.
The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.
We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to volume-constrained minimizers of the Cahn-Hilliard energy. We also show how two-point rearrangements can be used to establish symmetry for the Cahn-Hilliard model. In two dimensions, the Bonnesen inequality can then be applied to quantitatively estimate the sphericity of superlevel sets.
We consider a variant of Gamows liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm and quantify the rate of convergence. We also obtain a quantitative extension of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.
We study the regularity of minimizers of the functional $mathcal E(u):= [u]_{H^s(Omega)}^2 +int_Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $Omegasubsetmathbb R^N$. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in $H^s(Omega)$ (i.e., Neumann problem), or in the case of Dirichlet condition $uin H^s_0(Omega)$ when $s>frac12$. Our main result establishes the sharp regularity of solutions in both cases: $uin C^{2s+alpha}(overlineOmega)$ in the Neumann case, and $u/delta^{2s-1}in C^{1+alpha}(overlineOmega)$ in the Dirichlet case. Here, $delta$ is the distance to $partialOmega$, and $alpha<alpha_s$, with $alpha_sin (0,1-s)$ and $2s+alpha_s>1$. We also show the optimality of our result: these estimates fail for $alpha>alpha_s$, even when $f$ and $partialOmega$ are $C^infty$.
A semilinear parabolic equation with constraint modeling the dynamics of a microelectromechanical system (MEMS) is studied. In contrast to the commonly used MEMS model, the well-known pull-in phenomenon occurring above a critical potential threshold is not accompanied by a breakdown of the model, but is recovered by the saturation of the constraint for pulled-in states. It is shown that a maximal stationary solution exists and that saturation only occurs for large potential values. In addition, the existence, uniqueness, and large time behavior of solutions to the evolution equation are studied.