56 research outputs found
Characterization of balls through optimal concavity for potential functions
Let . If is a convex domain in \rn whose -capacitary
potential function is -concave (i.e. is
convex), then is a ball
An overdetermined problem for the anisotropic capacity
We consider an overdetermined problem for the Finsler Laplacian in the
exterior of a convex domain in , establishing a symmetry result
for the anisotropic capacitary potential. Our result extends the one of W.
Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian
capacity is considered, giving rise to an overdetermined problem for the
standard Laplace equation. Here, we replace the usual Euclidean norm of the
gradient with an arbitrary norm . The resulting symmetry of the solution is
that of the so-called Wulff shape (a ball in the dual norm )
Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations
This paper is concerned with the Minkowski convolution of viscosity solutions
of fully nonlinear parabolic equations. We adopt this convolution to compare
viscosity solutions of initial-boundary value problems in different domains. As
a consequence, we can for instance obtain parabolic power concavity of
solutions to a general class of parabolic equations. Our results apply to the
Pucci operator, the normalized -Laplacians with , the Finsler
Laplacian and more general quasilinear operators
Starshapedeness for fully-nonlinear equations in Carnot groups
In this paper we establish the starshapedness of the level sets of the
capacitary potential of a large class of fully-nonlinear equations for
condensers in Carnot groups, once a natural notion of starshapedness has been
introduced. Our main result is Theorem 1.2 below.Comment: Accepted for publication in the Journal of the London Mathematical
Societ
On space-time quasiconcave solutions of the heat equation
In this paper we first obtain a constant rank theorem for the second
fundamental form of the space-time level sets of a space-time quasiconcave
solution of the heat equation. Utilizing this constant rank theorem, we can
obtain some strictly convexity results of the spatial and space-time level sets
of the space-time quasiconcave solution of the heat equation in a convex ring.
To explain our ideas and for completeness, we also review the constant rank
theorem technique for the space-time Hessian of space-time convex solution of
heat equation and for the second fundamental form of the convex level sets for
harmonic function
To logconcavity and beyond
In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity.
In this paper, introducing a variation of concavity, we show that it preserves
in fact a stronger property than logconcavity and we identify the strongest
concavity preserved by the heat flow.Comment: We removed one result, Theorem 3.2 of the old version, due to a gap
in the proo
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