1,892 research outputs found
Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation
Let be a complete smooth metric measure space with
-Bakry-\'Emery Ricci tensor bounded from below. We derive elliptic
gradient estimates for positive solutions of a weighted nonlinear parabolic
equation \begin{align*} \displaystyle \Big(\Delta_f - \frac{\partial}{\partial
t}\Big) u(x,t) +q(x,t)u^\alpha(x,t) = 0, \end{align*} where and is an arbitrary constant. As
Applications we prove a Liouville-type theorem for positive ancient solutions
and Harnack-type inequalities for positive bounded solutions.Comment: 18 page
Liouville properties and critical value of fully nonlinear elliptic operators
We prove some Liouville properties for sub- and supersolutions of fully
nonlinear degenerate elliptic equations in the whole space. Our assumptions
allow the coefficients of the first order terms to be large at infinity,
provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We
give two applications. The first is a stabilization property for large times of
solutions to fully nonlinear parabolic equations. The second is the solvability
of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique
critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation
The Theory of Quasiconformal Mappings in Higher Dimensions, I
We present a survey of the many and various elements of the modern
higher-dimensional theory of quasiconformal mappings and their wide and varied
application. It is unified (and limited) by the theme of the author's
interests. Thus we will discuss the basic theory as it developed in the 1960s
in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore
the connections with geometric function theory, nonlinear partial differential
equations, differential and geometric topology and dynamics as they ensued over
the following decades. We give few proofs as we try to outline the major
results of the area and current research themes. We do not strive to present
these results in maximal generality, as to achieve this considerable technical
knowledge would be necessary of the reader. We have tried to give a feel of
where the area is, what are the central ideas and problems and where are the
major current interactions with researchers in other areas. We have also added
a bit of history here and there. We have not been able to cover the many recent
advances generalising the theory to mappings of finite distortion and to
degenerate elliptic Beltrami systems which connects the theory closely with the
calculus of variations and nonlinear elasticity, nonlinear Hodge theory and
related areas, although the reader may see shadows of this aspect in parts
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