34,887 research outputs found
Perturbation results for some nonlinear equations involving fractional operators
By using a perturbation technique in critical point theory, we prove the
existence of solutions for two types of nonlinear equations involving
fractional differential operators.Comment: 14 page
On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data
We study reaction-diffusion equations in cylinders with possibly nonlinear
diffusion and possibly nonlinear Neumann boundary conditions. We provide a
geometric Poincar\'e-type inequality and classification results for stable
solutions, and we apply them to the study of an associated nonlocal problem. We
also establish a counterexample in the corresponding framework for the
fractional Laplacian
Regularity of stable solutions up to dimension 7 in domains of double revolution
We consider the class of semi-stable positive solutions to semilinear
equations in a bounded domain of
double revolution, that is, a domain invariant under rotations of the first
variables and of the last variables. We assume . When
the domain is convex, we establish a priori and bounds for each
dimension , with when . These estimates lead to the
boundedness of the extremal solution of in every
convex domain of double revolution when . The boundedness of extremal
solutions is known when for any domain , in dimension
when the domain is convex, and in dimensions in the radial case.
Except for the radial case, our result is the first partial answer valid for
all nonlinearities in dimensions
On the Geroch-Traschen class of metrics
We compare two approaches to semi-Riemannian metrics of low regularity. The maximally 'reasonable' distributional setting of Geroch and Traschen is shown to be consistently contained in the more general setting of nonlinear distributional geometry in the sense of Colombea
Supersonic flow onto a solid wedge
We consider the problem of 2D supersonic flow onto a solid wedge, or
equivalently in a concave corner formed by two solid walls. For mild corners,
there are two possible steady state solutions, one with a strong and one with a
weak shock emanating from the corner. The weak shock is observed in supersonic
flights. A long-standing natural conjecture is that the strong shock is
unstable in some sense.
We resolve this issue by showing that a sharp wedge will eventually produce
weak shocks at the tip when accelerated to a supersonic speed. More precisely
we prove that for upstream state as initial data in the entire domain, the
time-dependent solution is self-similar, with a weak shock at the tip of the
wedge. We construct analytic solutions for self-similar potential flow, both
isothermal and isentropic with arbitrary .
In the process of constructing the self-similar solution, we develop a large
number of theoretical tools for these elliptic regions. These tools allow us to
establish large-data results rather than a small perturbation. We show that the
wave pattern persists as long as the weak shock is supersonic-supersonic; when
this is no longer true, numerics show a physical change of behaviour. In
addition we obtain rather detailed information about the elliptic region,
including analyticity as well as bounds for velocity components and shock
tangents.Comment: 105 pages; 22 figure
The classification of complete stable area-stationary surfaces in the Heisenberg group
We prove that any complete, orientable, connected, stable
area-stationary surface in the sub-Riemannian Heisenberg group
is either a Euclidean plane or congruent to the hyperbolic paraboloid .Comment: 32 pages, no figures, added reference missed in version
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=\f{\lambda}{(1-u)^2}, which models a simple
Micro-Electromechanical System (MEMS) device on a ball B\subset\IR^N, under
Dirichlet boundary conditions on . 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 exists for \la\in (0,\la^*), while there is
none of any kind when \la>\la^*. Our main result asserts that the extremal
solution is regular provided while is singular () for , in which case
on the unit ball, where
and .Comment: 19 pages. This paper completes and replaces a paper (with a similar
title) which appeared in arXiv:0810.5380. Updated versions --if any-- of this
author's papers can be downloaded at this http://www.birs.ca/~nassif
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