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 study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem $-\Delta u = λf (x)/(1 − u)^2$ on a bounded domain $\Omega ⊂ R^N$ , which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain \Omega and for a large class of “permittivity profiles” f . We also show the remarkable fact that powerlike profiles f (x) ≃ |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N≥8 and as long as $α>α_N =3N−14−4√6/4+2√6$. As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp
We consider the boundary value problem Δu+u^p=0 in a bounded, smooth domain Ω in R^2 with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p → ∞. In particular, for a non simply connected domain such a solution exists for any given m\geq 1
In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact Riemannian manifold (M,g) is compact. Established in the locally conformally flat case [41,42] and for n\leq 24 [23], it has revealed to be generally false for n\geq 25 [8,9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1)Scal_g, Scal_g being the Scalar curvature of (M,g). Even tough the Yamabe equation is compact in some cases, surprisingly we show that a-priori L^\infty-bounds fail on all manifolds with n\geq 4 as well as H_1^2-bounds do in the locally conformally flat case when n\geq 7. In several situations, the results are optimal
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $R^N$,\ud with the Navier boundary condition $ u=\Delta u =0 $ on $ \partial \Omega$.\ud We establish energy estimates which show that for any non-decreasing convex and superlinear nonlinearity $f$ with $f(0)=1$, the extremal solution $ u^*$ is smooth provided $N\leq 5$.\ud If in addition $\liminf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then $u^*$ is regular for $N\leq 7$, while if $\gamma:= \limsup_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, then the same holds for $N < \frac{8}{\gamma}$.\ud It follows that $u^*$ is smooth if $f(t) = e^t$ and $ N \le 8$, or if $f(t) = (1+t)^p$ and\ud $N< \frac{8p}{p-1}$.\ud We also show that if $ f(t)=(1-t)^{-p}$, $p>1$ and $p\neq 3$, then $u^*$ is smooth for $N \leq \frac{8p}{p+1}$. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., $ u^*$ is smooth for $ N \leq 12$ when $f(t)=e^t$ [J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with \ud exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565–592], and for $N\leq8$ when $ f(t)=(1-t)^{-2}$ [C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth\ud order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763–787] (see also [A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations, 248 (2010), 594–616]
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