21,693 research outputs found
A Bernoulli problem with non constant gradient boundary constraint
We present in this paper a result about existence and convexity of solutions
to a free boundary problem of Bernoulli type, with non constant gradient
boundary constraint depending on the outer unit normal. In particular we prove
that, in the convex case, the existence of a subsolution guarantees the
existence of a classical solution, which is proved to be convex.Comment: 8 pages, no figure
A note on an overdetermined problem for the capacitary potential
We consider an overdetermined problem arising in potential theory for the
capacitary potential and we prove a radial symmetry result.Comment: 7 pages. This paper has been written for possible publication in a
special volume dedicated to the conference "Geometric Properties for
Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in
Palinuro in May 201
Soft congestion approximation to the one-dimensional constrained Euler equations
This article is concerned with the analysis of the one-dimensional
compressible Euler equations with a singular pressure law, the so-called hard
sphere equation of state. The result is twofold. First, we establish the
existence of bounded weak solutions by means of a viscous regularization and
refined compensated compactness arguments. Second, we investigate the smooth
setting by providing a detailed description of the impact of the singular
pressure on the breakdown of the solutions. In this smooth framework, we
rigorously justify the singular limit towards the free-congested Euler
equations, where the compressible (free) dynamics is coupled with the
incompressible one in the constrained (i.e. congested) domain
Wulff shape characterizations in overdetermined anisotropic elliptic problems
We study some overdetermined problems for possibly anisotropic degenerate
elliptic PDEs, including the well-known Serrin's overdetermined problem, and we
prove the corresponding Wulff shape characterizations by using some integral
identities and just one pointwise inequality. Our techniques provide a somehow
unified approach to this variety of problems
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