arXiv subjects

Craig Cowan

Publications and source records attributed to Craig Cowan.

22 records · Page 2Linked to original sources

arXiv2009

Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems

Motivated by certain mathematical models for Micro-Electro-Mechanical Systems (MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions of nonlinear eigenvalue problems of the form $-Δu = λf(x) F(u)$ on a smooth bounded domain $ Ω$ in $\IR^N$. We are mainly interested in the {\it pull-in distance}, that is the $L^\infty-$norm of the extremal solution $u^*$ and how it depends on the geometry of the domain, the dimension of the space, and the so-called {\it permittivity profile} $f$. In particular, our results provide mathematical proofs for various observed phenomena, as well as rigorous derivations for several estimates obtained numerically by Pelesko \cite{P}, Guo-Pan-Ward \cite{GPW} and others in the case of the MEMS non-linearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^α$.

math.AP↗

arXiv2008

The critical dimension for a fourth order elliptic problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_νu=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0 \la^*$. Our main result asserts that the extremal solution $u_{λ^*}$ is regular $(\sup_B u_{λ^*} <1)$ provided $ N \le 8$ while $u_{λ^*} $ is singular ($\sup_B u_{λ^*} =1$) for $N \ge 17$, in which case $1-C_0|x|^{4/3}\leq u_{λ^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= <(\frac{λ^*}{\barλ}>)^{1/3}$ and $ \barλ:= \frac{8 (N-{2/3}) (N- {8/3})}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\leq N\leq 16$) requires a computer assisted proof and will not be addressed in this paper.

math.AP↗

arXiv2008

Regularity of the extremal solution in a MEMS model with advection

We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_λ\qquad {rcr} -Δu + c(x) \cdot \nabla u &=& \fracλ{(1-u)^2} \qquad {in $ Ω$}, u &=& 0 \qquad {on $ \pOm$}, where $ Ω$ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth bounded vector field on $\bar Ω$. We show that, just like in the advection-free model ($c\equiv 0$), all semi-stable solutions are smooth if (and only if) the dimension $N\leq 7$. The novelty here comes from the lack of a suitable variational characterization for the semi-stability assumption. We overcome this difficulty by using a general version of Hardy's inequality. In a forthcoming paper \cite{CG2}, we indicate how this method applies to many other nonlinear eigenvalue problems involving advection (including the Gelfand problem), showing that they all essentially have the same critical dimension as their advection-free counterparts.

math.AP↗

arXiv2008

Hardy inequalities for general elliptic operators with improvements

We obtain optimal generalized versions of Hardy inequalities, which as special cases contain Hardy's inequality and Hardy's inequality involving the distance function to the boundary of $ Ω$. In addition we obtain neccesary and sufficient conditions to add improvements in the form of non negative potentials.

math.AP↗