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This book contains the contributions resulting from the 6th
Italian-Japanese workshop on Geometric Properties for Parabolic and
Elliptic PDEs, which was held in Cortona (Italy) during the week of
May 20-24, 2019. This book will be of great interest for the
mathematical community and in particular for researchers studying
parabolic and elliptic PDEs. It covers many different fields of
current research as follows: convexity of solutions to PDEs,
qualitative properties of solutions to parabolic equations,
overdetermined problems, inverse problems, Brunn-Minkowski
inequalities, Sobolev inequalities, and isoperimetric inequalities.
This book collects recent research papers by respected specialists
in the field. It presents advances in the field of geometric
properties for parabolic and elliptic partial differential
equations, an area that has always attracted great attention. It
settles the basic issues (existence, uniqueness, stability and
regularity of solutions of initial/boundary value problems) before
focusing on the topological and/or geometric aspects. These topics
interact with many other areas of research and rely on a wide range
of mathematical tools and techniques, both analytic and geometric.
The Italian and Japanese mathematical schools have a long history
of research on PDEs and have numerous active groups collaborating
in the study of the geometric properties of their solutions.
This book collects recent research papers by respected specialists
in the field. It presents advances in the field of geometric
properties for parabolic and elliptic partial differential
equations, an area that has always attracted great attention. It
settles the basic issues (existence, uniqueness, stability and
regularity of solutions of initial/boundary value problems) before
focusing on the topological and/or geometric aspects. These topics
interact with many other areas of research and rely on a wide range
of mathematical tools and techniques, both analytic and geometric.
The Italian and Japanese mathematical schools have a long history
of research on PDEs and have numerous active groups collaborating
in the study of the geometric properties of their solutions.
This book contains the contributions resulting from the 6th
Italian-Japanese workshop on Geometric Properties for Parabolic and
Elliptic PDEs, which was held in Cortona (Italy) during the week of
May 20-24, 2019. This book will be of great interest for the
mathematical community and in particular for researchers studying
parabolic and elliptic PDEs. It covers many different fields of
current research as follows: convexity of solutions to PDEs,
qualitative properties of solutions to parabolic equations,
overdetermined problems, inverse problems, Brunn-Minkowski
inequalities, Sobolev inequalities, and isoperimetric inequalities.
In this paper the authors 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, they 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 their ideas and for completeness, the authors 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.
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