Universidad del País Vasco/Euskal Herriko Unibertsitatea, Departamento de Matemática Aplicada, Escuela Universitaria Politécnica de Donostia-San Sebastián, Plaza de Europa 1, 20018 Donostia-San Sebastián, Spain. jone.apraiz@ehu.es
Luis Escauriaza
Affiliation:
Universidad del País Vasco/Euskal Herriko Unibertsitatean, Dpto. de Matemáticas, Apto. 644, 48080 Bilbao, Spain; luis.escauriaza@ehu.es
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.
Ahlfors, L. and Bers, L., Riemann’s mapping theorem for
variable metrics. Ann. Math.72 (1960)
265–296. Google Scholar
[2]
Alessandrini, G. and
Escauriaza, L., Null-controllability of
one-dimensional parabolic equations. ESAIM :
COCV14 (2008)
284–293. Google Scholar
[3]
Astala, K., Area distortion under
quasiconformal mappings. Acta Math.173 (1994) 37–60.
Google Scholar
[4]
Benabdallah, A. and
Naso, M.G., Null controllability of a
thermoelastic plate. Abstr. Appl.
Anal.7 (2002) 585–599.
Google Scholar
[5]
Benabdallah, A., Dermenjian, Y. and
Le Rousseau, J., On the controllability of
linear parabolic equations with an arbitrary control location for stratified
media. C. R. Acad. Sci. Paris, Sér. 1344 (2007)
357–362. Google Scholar
[6]
L. Bers and L. Nirenberg, On a representation
theorem for linear elliptic systems with discontinuous coefficients and applications, in
Convegno Internazionale sulle Equazioni alle Derivate Parziali.
Cremonese, Roma (1955) 111–138.
[7]
L. Bers, F. John and M. Schechter, Partial
Differential Equations. Interscience. New York (1964).
[8]
Cho, S., Dong, H. and Kim, S., Global estimates for Green’s matrix
of second order parabolic systems with application to elliptic systems in two
dimensional domains. Potential Anal.36 (2012)
339–372. Google Scholar
[9]
L.C. Evans, Partial differential
equations. American Mathematical Society, Providence, RI (1998).
[10]
A. Fursikov and O.Yu. Imanuvilov, Controllability
of Evolution Equations. Seoul National University, Korea. Lect. Notes Ser.34 (1996).
[11]
M. Giaquinta, Multiple integrals in the
calculus of variations and nonlinear elliptic systems. Princeton University
Press (1983).
[12]
D. Gilbarg and N.S. Trudinger, Elliptic
Partial Differential Equations of Second Order, 2nd edition. Springer-Verlag
(1983).
[13]
F. John, Plane Waves and Spherical Means
Applied to Partial Differential Equations. Interscience Publishers, Inc., New
York (1955).
[14]
F. John, Partial Differential
Equations. Springer-Verlag, New York (1982).
[15]
Léautaud, M., Spectral inequalities for
non-selfadjoint elliptic operators and application to the null-controllability of
parabolic systems. J. Funct. Anal.258 (2010)
2739–2778. Google Scholar
[16]
Lebeau, G. and Robbiano, L., Contrôle exact de l’équation
de la chaleur. Commun. Partial Differ.
Equ.20 (1995)
335–356. Google Scholar
[17]
Lebeau, G. and Zuazua, E., Null controllability of a system
of linear thermoelasticity. Arch. Rational Mech.
Anal.141 (1998)
297–329. Google Scholar
[18]
Malinnikova, E., Propagation of smallness
for solutions of generalized Cauchy-Riemann systems.
Proc. Edinb. Math. Soc.47 (2004)
191–204. Google Scholar
[19]
A.I. Markushevich, Theory of Functions of
a Complex Variable. Prentice Hall, Englewood Cliffs, NJ (1965).
[20]
Miller, L., On the controllability of
anomalous diffusions generated by the fractional laplacian.
Math. Control Signals Syst.3 (2006) 260–271.
Google Scholar
[21]
C.B. Morrey, Multiple Integrals in the
Calculus of Variations. Springer (1966).
[22]
Morrey, C.B. and Nirenberg, L., On the analyticity of the
solutions of linear elliptic systems of partial differential equations.
Commun. Pure Appl. Math.X (1957) 271–290.
Google Scholar
[23]
Nadirashvili, N.S., A generalization of
Hadamard’s three circles theorem. Mosc. Univ. Math.
Bull.31 (1976) 30–32.
Google Scholar
[24]
Nadirashvili, N.S., Estimation of the
solutions of elliptic equations with analytic coefficients which are bounded on some
set. Mosc. Univ. Math. Bull.34 (1979) 44–48.
Google Scholar
[25]
J. Le Rousseau and G. Lebeau, On Carleman
estimates for elliptic and parabolic operators. Applications to unique continuation and
control of parabolic equations. ESAIM : COCV, doi:10.1051/cocv/2011168.
CrossRef
[26]
Le Rousseau, J. and
Robbiano, L., Local and global Carleman
estimates for parabolic operators with coefficients with jumps at
interfaces. Invent. Math.183 (2011)
245–336. Google Scholar
[27]
Russel, D.L., A unified boundary
controllability theory for hyperbolic and parabolic partial differential
equations. Stud. Appl. Math.52 (1973)
189–221. Google Scholar
[28]
Vessella, S., A continuous dependence result
in the analytic continuation problem. Forum
Math.11 (1999)
695–703. Google Scholar
[29]
H.F. Weinberger, A first course in partial
differential equations with complex variables and transform methods. Dover
Publications, New York (1995).