Electron. J. Differential Equations,
Vol. 2016 (2016), No. 266, pp. 116.
Convergence of exterior solutions to radial Cauchy solutions for
Helge Kristian Jenssen, Charis Tsikkou
Abstract:
Consider the Cauchy problem for the 3D linear wave equation
with radial initial data
,
. A standard result states that U belongs
to
whenever
.
In this article we are interested in the question of how U can be realized
as a limit of solutions to initialboundary value problems on the
exterior of vanishing balls
about the origin. We note that,
as the solutions we compare are defined on different domains, the answer
is not an immediate consequence of
wellposedness for the wave equation.
We show how explicit solution formulae yield convergence and optimal
regularity for the Cauchy solution via exterior solutions, when the latter are
extended continuously as constants on
at each time.
We establish that for s=2 the solution U can be realized
as an
limit
(uniformly in time) of exterior solutions on
satisfying vanishing Neumann conditions
along
,
as
.
Similarly for s=1:
U is then an
limit
of exterior solutions satisfying vanishing Dirichlet
conditions along
.
Submitted July 22, 2016. Published September 30, 2016.
Math Subject Classifications: 35L05, 35L15, 35L20.
Key Words: Linear wave equation; Cauchy problem; radial solutions;
exterior solutions; Neumann and Dirichlet conditions.
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Helge Kristian Jenssen
Department of Mathematics
Penn State University, University Park
State College, PA 16802, USA
email: jenssen@math.psu.edu


Charis Tsikkou
Department of Mathematics
West Virginia University
Morgantown, WV 26506, USA
email: tsikkou@math.wvu.edu

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