6,135 research outputs found
Geometric approach to nonvariational singular elliptic equations
In this work we develop a systematic geometric approach to study fully
nonlinear elliptic equations with singular absorption terms as well as their
related free boundary problems. The magnitude of the singularity is measured by
a negative parameter , for , which reflects on
lack of smoothness for an existing solution along the singular interface
between its positive and zero phases. We establish existence as well sharp
regularity properties of solutions. We further prove that minimal solutions are
non-degenerate and obtain fine geometric-measure properties of the free
boundary . In particular we show sharp
Hausdorff estimates which imply local finiteness of the perimeter of the region
and a.e. weak differentiability property of
.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos
Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis
201
Remarks on regularity for -Laplacian type equations in non-divergence form
We study a singular or degenerate equation in non-divergence form modeled by
the -Laplacian, We investigate local
regularity of viscosity solutions in the full range and , and
provide local estimates in the restricted cases where is close to
2 and is close to 0.Comment: 38 page
Shooting with degree theory: Analysis of some weighted poly-harmonic systems
In this paper, the author establishes the existence of positive entire
solutions to a general class of semilinear poly-harmonic systems, which
includes equations and systems of the weighted Hardy--Littlewood--Sobolev type.
The novel method used implements the classical shooting method enhanced by
topological degree theory. The key steps of the method are to first construct a
target map which aims the shooting method and the non-degeneracy conditions
guarantee the continuity of this map. With the continuity of the target map, a
topological argument is used to show the existence of zeros of the target map.
The existence of zeros of the map along with a non-existence theorem for the
corresponding Navier boundary value problem imply the existence of positive
solutions for the class of poly-harmonic systems.Comment: 19 pages, author's accepted version including corrections to a few
typographical error
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