Existence and multiplicity for Hamilton-Jacobi-Bellman equation

<p style='text-indent:20px;'>This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded regular domain with <inline-formula><tex-math id="M4">\begin{document}$ N\geq3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document}</tex-math></inline-formula> are general Hamilton-Jacobi-Bellman operators, <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is a real parameter. By using bifurcation theory, we determine the range of parameter <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the above problem which has one or multiple constant sign solutions according to the behaviors of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \infty $\end{document}</tex-math></inline-formula>, and whether <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies the signum condition <inline-formula><tex-math id="M12">\begin{document}$ f(s)s&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ s\neq0 $\end{document}</tex-math></inline-formula>.</p>

A fully nonlinear equation in relativistic Teichmüller theory

We obtain some basic estimates for a Monge–Ampère type equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmüller space of a compact oriented surface of genus [Formula: see text] is diffeomorphic to the disk of dimension [Formula: see text]. We also give another proof of properness of a certain energy function on the Teichmüller space.

On a class of fully nonlinear parabolic equations

We study the homogeneous Dirichlet problem for the fully nonlinear equation u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters {m>1} , {\sigma>1} and {d\geq 0} . At the points where {\Delta u=0} , the equation degenerates if {m>2} , or becomes singular if {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as {t\to\infty} . Sufficient conditions for exponential or power decay of {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data

We deal with existence and uniqueness of the solution to the fully nonlinear equation−F(D2u) + |u|s−1u = f(x) in ℝn,where s > 1 and f satisfies only local integrability conditions. This result is well known when, instead of the fully nonlinear elliptic operator F, the Laplacian or a divergence-form operator is considered. Our existence results use the Alexandroff-Bakelman-Pucci inequality since we cannot use any variational formulation. For radially symmetric f, and in the particular case where F is a maximal Pucci operator, we can prove our results under fewer integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary blow-up in smooth domains.