arXiv subjects

Pierpaolo Esposito

Publications and source records attributed to Pierpaolo Esposito.

23 records · Page 2Linked to original sources

arXiv2010

Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $Δ^2 u = λf(u)$ on a general bounded domain $Ω$ in $ \IR^N$, with the Navier boundary condition $ u=Δu =0 $ on $ \pOm$. Here $ λ$ is a positive parameter and $f$ is a non-decreasing 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 $N\leq 5$.

math.AP↗

arXiv2009

The critical dimension for a 4th order 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 $N \ge 9$, 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^*}{\overlineλ})^{1/3}$ and $ \barλ:= {8/9} (N-{2/3}) (N- {8/3})$.

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

Uniqueness of solutions for an elliptic equation modeling MEMS

We study the effect of the parameter $λ$, the dimension $N$, the profile $f$ and the geometry of the domain $Ω\subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem with a singular nonlinearity: $$ 180pt {{array}{ll} -Δu= \frac{λf(x)}{(1-u)^2} & \hbox{in}Ω0<u<1 &\hbox{in}Ωu=0 &\hbox{on}\partial Ω. {array}. 130pt (S)_{λ, f} $$ This equation has been proposed as a model for a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 below a rigid ground plate located at height z = 1.

math.AP↗

arXiv2005

Compactness along the Branch of Semi-stable and Unstable Solutions for an Elliptic Problem with a Singular Nonlinearity

We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-Δu=\frac{λf(x)}{(1-u)^2}$ on a bounded domain $Ω\subset \R^N$, which models --among other things-- a simple electrostatic Micro-Electromechanical System (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for $1\leq N \leq 7$ on any domain $Ω$ and for a large class of "permittivity profiles" $f$ . We also show the remarkable fact that power-like profiles $f(x) \simeq |x|^α$ can push back the critical dimension N=7 of this problem, by establishing compactness for the semi-stable branch on the unit ball, also for $N\geq 8$ and as long as $α>α_N=\frac{3N-14-4\sqrt{6}}{4+2\sqrt{6}}$ . 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.

math.AP↗