共查询到20条相似文献,搜索用时 15 毫秒
1.
Craig Cowan Pierpaolo Esposito Nassif Ghoussoub Amir Moradifam 《Archive for Rational Mechanics and Analysis》2010,198(3):763-787
We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation
D2 u=\fracl(1-u)2{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball
B ì \mathbbRN{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u
λ with 0 < u
λ < 1 exists for l ? (0,l*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution ul*{u_{\lambda^*}} is regular (supB ul* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided
N \leqq 8{N \leqq 8} while ul*{u_{\lambda^*}} is singular (supB ul* = 1){({\rm sup}_B u_{\lambda^*} =1)} for
N \geqq 9{N \geqq 9}, in which case
1-C0|x|4/3 \leqq ul* (x) \leqq 1-|x|4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where
C0:=(\fracl*[`(l)])\frac13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and
[`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}. 相似文献
2.
W. Czernous 《Nonlinear Oscillations》2011,13(4):595-612
We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the
first order
$ {*{20}{c}} {{\partial_t}{z_i}\left( {t,x} \right) + \sum\limits_{j = 1}^n {{\rho_{ij}}\left( {t,x,V\left( {z;t,x} \right)} \right){\partial_{{x_j}}}{z_i}\left( {t,x} \right) = {G_i}\left( {t,x,V\left( {z;t,x} \right)} \right),} } \hfill & {1 \leq i \leq m,} \hfill \\ $ \begin{array}{*{20}{c}} {{\partial_t}{z_i}\left( {t,x} \right) + \sum\limits_{j = 1}^n {{\rho_{ij}}\left( {t,x,V\left( {z;t,x} \right)} \right){\partial_{{x_j}}}{z_i}\left( {t,x} \right) = {G_i}\left( {t,x,V\left( {z;t,x} \right)} \right),} } \hfill & {1 \leq i \leq m,} \hfill \\ \end{array} 相似文献
3.
G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form
of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation,
G-equations are Hamilton–Jacobi equations with convex (L
1 type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by
small diffusion. The nonlinear eigenvalue [`(H)]{\bar H} from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s
T. An important problem in turbulent combustion theory is to study properties of s
T, in particular how s
T depends on the flow amplitude A. In this paper, we study the behavior of [`(H)]=[`(H)](A,d){\bar H=\bar H(A,d)} as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that
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