Unique Continuation Property of Solutions to General Second Order Elliptic Systems Honda

A global unique continuation property (UCP) from the boundary for solutions to a viscoelastic system with a memory term is presented. The density and elasticity tensors are assumed to be real analytic. The tensors can be anisotropic and satisfy physically natural conditions such as full symmetry and strong convexity. The global UCP is given in terms of the travel time of the slowest wave of the viscoelastic system, which is the optimal description for the global UCP in our setup.

Mathematics Subject Classification: 35B60, 35Q74.

Citation: Matthias Eller, Naofumi Honda, Ching-Lung Lin, Gen Nakamura. Global unique continuation from the boundary for a system of viscoelasticity with analytic coefficients and a memory term. Inverse Problems and Imaging, doi: 10.3934/ipi.2022049

References:

[1]	R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82 (1975), 246-251.  doi: 10.1080/00029890.1975.11993807.
[2]	T. S. Brown, S. Du, H. Eruslu and F.-J. Sayas, Analysis of models for viscoelastic wave propagation, Appl. Math. Nonlinear Sci., 3 (2018), 55-96.  doi: 10.21042/AMNS.2018.1.00006.
[3]	C. I. C$\widehat{ \rm a }$rstea, G. Nakamura and L. Oksanen, Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain, Trans. Amer. Math. Soc., 373 (2020), 3423-3443.  doi: 10.1090/tran/8014.
[4]	R. Christensen, Theory of Viscoelasticity, 2nd edition, Dover Publications, New York, 1982. doi: 10.1115/1.3408900.
[5]	J. B. Conway, A Course in Functional Analysis, Second edition, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.
[6]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.
[7]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.
[8]	M. V. de Hoop, C.-L. Lin and G. Nakamura, Holmgren-John unique continuation theorem for viscoelastic systems, time-dependent problems in imaging and parameter identification, Springer, Cham, (2021), 287-301.
[9]	M. V. de Hoop, G. Nakamura and J. Zhai, Unique recovery of piecewise analytic density and stiffness tensor from the elastic-wave Dirichlet-to-Neumann map, SIAM J. Appl. Math., 79 (2019), 2359-2384.  doi: 10.1137/18M1232802.
[10]	M. Eller and D. Toudykov, A global Holmgren Theorem for multi-dimensional hyperbolic partial differential equations, App. Anal., 91 (2012), 69-80.  doi: 10.1080/00036811.2010.538685.
[11]	W. Littman, Remarks on global uniqueness theorems for partial differential equations, Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 268 (2000), 363-371.  doi: 10.1090/conm/268/04318.
[12]	J. R. McLaughlin and J.-R. Yoon, Finite propagation speed of waves in anisotropic viscoelastic media, SIAM J. Appl. Math., 77 (2017), 1921-1936.  doi: 10.1137/16M1099959.
[13]	J. E. Muñoz Rivera and E. Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decay kernels, Comm. Math. Phys., 177 (1996), 583-602.  doi: 10.1007/BF02099539.
[14]	G. Nakamura and M. Oliva, Exponential decay of solutions to initial boundary value problem for anisotropic visco-elastic systems, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 5-19.  doi: 10.13137/2464-8728/13149.
[15]	K. Yosida, Functional Analysis, Second edition, Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York, Inc., New York, 1968.

show all references

References:

[1]	R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82 (1975), 246-251.  doi: 10.1080/00029890.1975.11993807.
[2]	T. S. Brown, S. Du, H. Eruslu and F.-J. Sayas, Analysis of models for viscoelastic wave propagation, Appl. Math. Nonlinear Sci., 3 (2018), 55-96.  doi: 10.21042/AMNS.2018.1.00006.
[3]	C. I. C$\widehat{ \rm a }$rstea, G. Nakamura and L. Oksanen, Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain, Trans. Amer. Math. Soc., 373 (2020), 3423-3443.  doi: 10.1090/tran/8014.
[4]	R. Christensen, Theory of Viscoelasticity, 2nd edition, Dover Publications, New York, 1982. doi: 10.1115/1.3408900.
[5]	J. B. Conway, A Course in Functional Analysis, Second edition, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.
[6]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.
[7]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.
[8]	M. V. de Hoop, C.-L. Lin and G. Nakamura, Holmgren-John unique continuation theorem for viscoelastic systems, time-dependent problems in imaging and parameter identification, Springer, Cham, (2021), 287-301.
[9]	M. V. de Hoop, G. Nakamura and J. Zhai, Unique recovery of piecewise analytic density and stiffness tensor from the elastic-wave Dirichlet-to-Neumann map, SIAM J. Appl. Math., 79 (2019), 2359-2384.  doi: 10.1137/18M1232802.
[10]	M. Eller and D. Toudykov, A global Holmgren Theorem for multi-dimensional hyperbolic partial differential equations, App. Anal., 91 (2012), 69-80.  doi: 10.1080/00036811.2010.538685.
[11]	W. Littman, Remarks on global uniqueness theorems for partial differential equations, Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 268 (2000), 363-371.  doi: 10.1090/conm/268/04318.
[12]	J. R. McLaughlin and J.-R. Yoon, Finite propagation speed of waves in anisotropic viscoelastic media, SIAM J. Appl. Math., 77 (2017), 1921-1936.  doi: 10.1137/16M1099959.
[13]	J. E. Muñoz Rivera and E. Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decay kernels, Comm. Math. Phys., 177 (1996), 583-602.  doi: 10.1007/BF02099539.
[14]	G. Nakamura and M. Oliva, Exponential decay of solutions to initial boundary value problem for anisotropic visco-elastic systems, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 5-19.  doi: 10.13137/2464-8728/13149.
[15]	K. Yosida, Functional Analysis, Second edition, Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York, Inc., New York, 1968.

Figure 1.

$ u = \nabla_{x, t} u = 0 $

on

$ \Gamma\times(0, T) $

Figure 2. An example of the cross-section of a slowness surface in the

$ \xi_1\xi_3 $

-plane

Figure 1. The sets

$ \Omega $

with convex

$ \Gamma\subset \partial\Omega $

, the projection of

$ F_\lambda $

at

$ t $

and the surface

$ \Sigma_{t, \lambda} $

in blue (left). Homotopy of timelike curves with curve

$ E(\cdot, t) $

for

$ 0\le \tau \le \lambda $

in red for a specific

$ t $

and

$ \lambda $

(right)

Figure A.1.

$ \phi(x, t) = \hat{\phi}(y, t) = y_n-\psi(y', t)\in C^2 $

.

$ u(y, t) = 0 $

for

$ \{(y, t):y_n\leq \psi(y', t)\} $

Figure A.2.

$ u(z, t) = 0 $

for

$ \{(z, t):z_n\leq \frac{c}{2}(|z'|^2+t^2)\} $

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