We consider two-dimensional attractive Bose-Einstein condensates in a trap V (x) rotating at the velocity Ω, which can be described by the complex-valued Gross-Pitaevskii energy functional. We prove that there exists a critical rotational velocity 0 < Ω * := Ω * (V ) ≤ ∞, depending on the general potential V (x), such that for any small rotational velocity 0 ≤ Ω < Ω * , minimizers exist if and only if the attractive interaction strength a satisfies a < a * = w 2 2 , where w is the unique positive radial solution of ∆w − w + w 3 = 0 in R 2 . For any 0 < Ω < Ω * , we also analyze the limit behavior of the minimizers as a ր a * , where all the mass concentrates at a global minimum point of VΩ(x) := V (x) − Ω 2 4 |x| 2 . Specially, if V (x) = |x| 2 and 0 < Ω < Ω * (= 2), we further prove that, up to the phase rotation, all minimizers are real-valued, unique and free of vortices as a ր a * .

Part 1. Second-Order Equations Modeling Stationary MEMS Chapter 2. Estimates for the Pull-In Voltage 2.1. Existence of the Pull-In Voltage 2.2. Lower Estimates for the Pull-In Voltage 2.3. Upper Bounds for the Pull-In Voltage 2.4. Numerics for the Pull-In Voltage Further Comments Chapter 3. The Branch of Stable Solutions 3.1. Spectral Properties of Minimal Solutions 3.2. Energy Estimates and Regularity of Solutions 3.3. Linear Instability and Compactness 3.4. Effect of an Advection on the Minimal Branch Further Comments Chapter 4. Estimates for the Pull-In Distance 4.1. Lower Estimates on the Pull-In Distance in General Domains 4.2. Upper Estimate for the Pull-In Distance in General Domains 4.3. Upper Bounds for the Pull-In Distance in the Radial Case 4.4. Effect of Power-Law Profiles on Pull-In Distances 4.5. Asymptotic Behavior of Stable Solutions near the Pull-In Voltage Further Comments Chapter 5. The First Branch of Unstable Solutions 5.1. Existence of Nonminimal Solutions 5.2. Blowup Analysis for Noncompact Sequences of Solutions 5.3. Compactness along the First Branch of Unstable Solutions 5.4. Second Bifurcation Point Further Comments vii viii CONTENTS Chapter 6. Description of the Global Set of Solutions 6.1. Compactness along the Unstable Branches 6.2. Quenching Branch of Solutions in General Domains 6.3. Uniqueness of Solutions for Small Voltage in Star-Shaped Domains 6.4. One-Dimensional Problem Further Comments Chapter 7. Power-Law Profiles on Symmetric Domains 7.1. A One-Dimensional Sobolev Inequality 7.2. Monotonicity Formula and Applications 7.3. Compactness of Higher Branches of Radial Solutions 7.4. Two-Dimensional MEMS on Symmetric Domains Further Comments Part 2. Parabolic Equations Modeling MEMS Dynamic Deflections Chapter 8. Different Modes of Dynamic Deflection 8.1. Global Convergence versus Quenching 8.2. Quenching Points and the Zero Set of the Profile 8.3. The Quenching Set on Convex Domains Further Comments Chapter 9. Estimates on Quenching Times 9.1. Comparison Results for Quenching Times 9.2. General Asymptotic Estimates for Quenching Time 9.3. Upper Estimates for Quenching Times for all > 9.4. Quenching Time Estimates in Low Dimension Further Comments Chapter 10. Refined Profile of Solutions at Quenching Time 10.1. Integral and Gradient Estimates for Quenching Solutions 10.2. Refined Quenching Profile 10.3. Refined Quenching Profiles in Dimension N D 1 10.4. Refined Quenching Profiles in the Radially Symmetric Case 10.5. More on the Location of Quenching Points Further Comments

We analyze the nonlinear elliptic problem ∆u = λf (x) (1+u) 2 on a bounded domain Ω of R N with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. When a voltage -represented here by λ-is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value λ * (pull-in voltage). This creates a so-called "pull-in instability" which greatly affects the design of many devices. The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers [11] and [12]. For now, we focus on the stationary equation where the challenge is to estimate λ * in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile f . Applying analytical and numerical techniques, the existence of λ * is established together with rigorous bounds. We show the existence of at least one steady-state when λ < λ * (and when λ = λ * in dimension N < 8) while none is possible for λ > λ * . More refined properties of steady states -such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results-are shown to depend on the dimension of the ambient space and on the permittivity profile.

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