13,000 research outputs found
Fundamental solutions of homogeneous fully nonlinear elliptic equations
We prove the existence of two fundamental solutions and
of the PDE for
any positively homogeneous, uniformly elliptic operator . Corresponding to
are two unique scaling exponents which
describe the homogeneity of and . We give a sharp
characterization of the isolated singularities and the behavior at infinity of
a solution of the equation , which is bounded on one side. A
Liouville-type result demonstrates that the two fundamental solutions are the
unique nontrivial solutions of in
which are bounded on one side in a neighborhood of the origin as well as at
infinity. Finally, we show that the sign of each scaling exponent is related to
the recurrence or transience of a stochastic process for a two-player
differential game.Comment: 35 pages, typos and minor mistakes correcte
Nonexistence of positive supersolutions of elliptic equations via the maximum principle
We introduce a new method for proving the nonexistence of positive
supersolutions of elliptic inequalities in unbounded domains of .
The simplicity and robustness of our maximum principle-based argument provides
for its applicability to many elliptic inequalities and systems, including
quasilinear operators such as the -Laplacian, and nondivergence form fully
nonlinear operators such as Bellman-Isaacs operators. Our method gives new and
optimal results in terms of the nonlinear functions appearing in the
inequalities, and applies to inequalities holding in the whole space as well as
exterior domains and cone-like domains.Comment: revised version, 32 page
Singular solutions of fully nonlinear elliptic equations and applications
We study the properties of solutions of fully nonlinear, positively
homogeneous elliptic equations near boundary points of Lipschitz domains at
which the solution may be singular. We show that these equations have two
positive solutions in each cone of , and the solutions are unique
in an appropriate sense. We introduce a new method for analyzing the behavior
of solutions near certain Lipschitz boundary points, which permits us to
classify isolated boundary singularities of solutions which are bounded from
either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as
well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure
A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions
In this paper we introduce an integer-valued degree for second order fully
nonlinear elliptic operators with nonlinear oblique boundary conditions. We
also give some applications to the existence of solutions of certain nonlinear
elliptic equations arising from a Yamabe problem with boundary and reflector
problems
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