Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space

Roberto  Triggiani

doi:10.3934/eect.2016016

Article Contents

2016, Volume 5, Issue 4: 489-514. Doi: 10.3934/eect.2016016

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Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space

Roberto Triggiani^1,

1.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152

Received: July 31, 2016

Revised: July 31, 2016

Published: November 2016

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Abstract

The purpose of this paper is to complement available literature on sharp regularity theory of second order mixed hyperbolic problem of Neumann type [13,15,26] with a series of new results in the case--so far rather unexplored--where the Neumann boundary term (input, control) possesses a regularity below $L^2(\Gamma)$ in space on the boundary $\Gamma$. We concentrate on the cases $H^{-\frac{1}{2}}(\Gamma))$, $H^{-\beta}(\Gamma))$, $H^{-1}(\Gamma))$, $\beta$ being a distinguished parameter of the problem. Our present results are consistent with the sharp result of [13,15,26] (obtained through a pseudo-differential/micro-local analysis approach), whose philosophy is expressed by a gain of $\beta$ in space regularity in going from the boundary control to the position in the interior. A number of physically relevant illustrations are given.

Keywords:

Sharp regularity,
Neumann boundary control.

Mathematics Subject Classification: Primary: 35M13, 93D20.

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