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46
Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity,
 Comm. Pure Appl. Math.
, 2007
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Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains
, 2010
"... We examine the regularity of the extremal solution of the nonlinear eigenvalue problem ∆ 2 u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ is a positive parameter and f is a nondecreasing nonlinearity with f(0) = 1. We give general pointw ..."
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Cited by 17 (6 self)
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We examine the regularity of the extremal solution of the nonlinear eigenvalue problem ∆ 2 u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ is a positive parameter and f is a nondecreasing nonlinearity with f(0) = 1. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity f, the extremal solution u ∗ is smooth provided N ≤ 5. f(t)f • If in addition lim inf t→+∞ ′ ′ (t) (f ′) 2 (t)> 0, then u ∗ is regular for N ≤ 7. • On the other hand, if γ: = lim sup t→+∞ f(t)f ′ ′ (t) (f ′ ) 2 (t) < +∞, then the same holds for N < 8 γ. It follows that u ∗ is smooth if f(t) = et and N ≤ 8, or if f(t) = (1 + t) p and N < 8p p−1. We also show that if f(t) = (1 − t) −p, p> 1 and p = 3, then u ∗ is smooth for N ≤ 8p p+1. We note that while these are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established for Dirichlet problems on radial domains, e.g., u ∗ is smooth for N ≤ 12 when f(t) = et [11], and for N ≤ 8 when f(t) = (1 − t) −2 [9] (see also [22]).
Compactness of a nonlinear eigenvalue problem with a singular nonlinearity
 Commun. Contemp. Math
"... We study the Dirichlet boundary value problem −∆u = λf(x) (1−u)2 on a bounded domain Ω ⊂ RN. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a byproduct, we give an uniqueness result for λ close to 0 and λ ∗ in the class of all solutions with ..."
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Cited by 17 (7 self)
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We study the Dirichlet boundary value problem −∆u = λf(x) (1−u)2 on a bounded domain Ω ⊂ RN. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a byproduct, we give an uniqueness result for λ close to 0 and λ ∗ in the class of all solutions with finite Morse index, λ ∗ being the extremal value associated to the nonlinear eigenvalue problem.
A parabolic free boundary problem modeling electrostatic
"... ABSTRACT. A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrowgap model is given by showing that ..."
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Cited by 13 (6 self)
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ABSTRACT. A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrowgap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrowgap model when the aspect ratio of the device tends to zero.
DYNAMICS OF A FREE BOUNDARY PROBLEM WITH CURVATURE MODELING ELECTROSTATIC MEMS
, 2013
"... Abstract. The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference acro ..."
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Cited by 10 (5 self)
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Abstract. The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation. The free boundary problem is shown to be wellposed, locally in time for arbitrary voltage values and globally in time for small voltages values. Furthermore, existence of asymptotically stable steadystate configurations is proved in case of small voltage values as well as nonexistence of steadystates if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the wellestablished small aspect ratio model is shown. 1.
Linear instability of entire solutions for a class of nonautonomous elliptic equations
 IN: PROCEEDINGS OF ROYAL SOCIETY EDINBURGH SECT. A
, 2008
"... We study the effect of the potential yα on the stability of entire solutions for elliptic equations on RN, N 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet no ..."
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Cited by 9 (4 self)
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We study the effect of the potential yα on the stability of entire solutions for elliptic equations on RN, N 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a byproduct of our results, we will improve the known results about the regularity of such solutions.
MULTIPLE QUENCHING SOLUTIONS OF A FOURTH ORDER PARABOLIC PDE WITH A SINGULAR NONLINEARITY Modelling A Mems Capacitor
, 2011
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Degenerate elliptic equations with singular nonlinearities
 CALCULUS OF VARIATIONS
, 2007
"... The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we prese ..."
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Cited by 7 (7 self)
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The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of pLaplace operator, for p = 2 it is crucial to define a suitable notion of semistability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blowup argument and stronger assumptions on the nonlinearity f (u) are required.
Existence and dynamic properties of a parabolic nonlocal MEMS equation
"... Let Ω ⊂ Rn be a C2 bounded domain and χ> 0 be a constant. We will prove the existence of constants λN ≥ λ ∗ N ≥ λ ∗ (1 + χ ∫ dx Ω 1−w ∗)2 for the nonlocal MEMS equation −∆v = λ/(1 − v) 2 (1+χ ∫ Ω 1/(1 − v)dx)2 in Ω, v = 0 on ∂Ω, such that solution exists for any 0 ≤ λ < λ ∗ N and no solution e ..."
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Cited by 7 (2 self)
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Let Ω ⊂ Rn be a C2 bounded domain and χ> 0 be a constant. We will prove the existence of constants λN ≥ λ ∗ N ≥ λ ∗ (1 + χ ∫ dx Ω 1−w ∗)2 for the nonlocal MEMS equation −∆v = λ/(1 − v) 2 (1+χ ∫ Ω 1/(1 − v)dx)2 in Ω, v = 0 on ∂Ω, such that solution exists for any 0 ≤ λ < λ ∗ N and no solution exists for any λ> λN where λ ∗ is the pullin voltage and w ∗ is the limit of the minimal solution of −∆v = λ/(1 − v) 2 in Ω with v = 0 on ∂Ω as λ ր λ ∗. Moreover λN < ∞ if Ω is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the parabolic nonlocal MEMS equation ut = ∆u + λ/(1 − u) 2 (1 + χ ∫ Ω 1/(1 − u)dx)2 in Ω × (0, ∞), u = 0 on ∂Ω × (0, ∞), u(x,0) = u0 in Ω. We prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λ is large.
The explosion problem in a flow
, 2008
"... We consider the explosion problem in an incompressible flow introduced in [5]. We use a novel Lp − L∞ estimate for elliptic advectiondiffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flow. We also identify the flows for which the ..."
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Cited by 5 (3 self)
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We consider the explosion problem in an incompressible flow introduced in [5]. We use a novel Lp − L∞ estimate for elliptic advectiondiffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flow. We also identify the flows for which the explosion threshold tends to infinity as their amplitude grows and obtain an effective description of the explosion threshold in the strong flow asymptotics in a twodimensional onecell flow.