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1. CJM 2012 (vol 65 pp. 927)
Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$ We consider the following prescribed boundary mean curvature problem
in $ \mathbb B^N$ with the Euclidean metric:
\[
\begin{cases}
\displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N,
\\[2ex]
\displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1},
\end{cases}
\]
where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb
S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$.
We show that if $\widetilde K(x)$ has a local maximum point,
then the above problem has infinitely many positive solutions
that are not rotationally symmetric on $\mathbb S^{N-1}$.
Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction Categories:35J25, 35J65, 35J67 |
2. CJM 2002 (vol 54 pp. 1121)
Fully Nonlinear Elliptic Equations on General Domains By means of the Pucci operator, we construct a function $u_0$, which plays
an essential role in our considerations, and give the existence and regularity
theorems for the bounded viscosity solutions of the generalized Dirichlet
problems of second order fully nonlinear elliptic equations on the general
bounded domains, which may be irregular. The approximation method, the accretive
operator technique and the Caffarelli's perturbation theory are used.
Keywords:Pucci operator, viscosity solution, existence, $C^{2,\psi}$ regularity, Dini condition, fully nonlinear equation, general domain, accretive operator, approximation lemma Categories:35D05, 35D10, 35J60, 35J67 |
3. CJM 2002 (vol 54 pp. 945)
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications |
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jet-distributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$-meromorphic and $L$-analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 |