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Carleman estimates for the non-stationary Lamé system and the application to an inverse problem

Oleg Yu. Imanuvilov, Masahiro Yamamoto (2005)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over ( 0 , T ) × ω , where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

Carleman estimates for the non-stationary Lamé system and the application to an inverse problem

Oleg Yu. Imanuvilov, Masahiro Yamamoto (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress

Victor Isakov, Nanhee Kim (2008)

Applicationes Mathematicae

We derive Carleman type estimates with two large parameters for a general partial differential operator of second order. The weight function is assumed to be pseudo-convex with respect to the operator. We give applications to uniqueness and stability of the continuation of solutions and identification of coefficients for the Lamé system of dynamical elasticity with residual stress. This system is anisotropic and cannot be principally diagonalized, but it can be transformed into an "upper triangular"...

Catastrophes and partial differential equations

John Guckenheimer (1973)

Annales de l'institut Fourier

This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed....

Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

Vladyslav Litovchenko, Iryna Dovzhytska (2012)

Open Mathematics

In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.

Cauchy problems in weighted Lebesgue spaces

Jan W. Cholewa, Tomasz Dłotko (2004)

Czechoslovak Mathematical Journal

Global solvability and asymptotics of semilinear parabolic Cauchy problems in n are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over n , n . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients

Dian K. Palagachev, Maria A. Ragusa, Lubomira G. Softova (2003)

Bollettino dell'Unione Matematica Italiana

Let Q T be a cylinder in R n + 1 and x = x , t R n × R . It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator u t - i , j = 1 n a i j x D i j u = f x q.o. in  Q T , u x = 0 su  Q T , in the Morrey spaces W p , λ 2 , 1 Q T , p 1 , , λ 0 , n + 2 , supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.

Changing blow-up time in nonlinear Schrödinger equations

Rémi Carles (2003)

Journées équations aux dérivées partielles

Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is L 2 -critical. On the other...

Characteristic Cauchy problems and solutions of formal power series

Sunao Ouchi (1983)

Annales de l'institut Fourier

Let L ( z , z ) = ( z 0 ) k - A ( z , z ) be a linear partial differential operator with holomorphic coefficients, where A ( z , z ) = j = 0 k - 1 A j ( z , z ' ) ( z 0 ) j , ord . A ( z , z ) = m > k and z = ( z 0 , z ' ) C n + 1 . We consider Cauchy problem with holomorphic data L ( z , z ) u ( z ) = f ( z ) , ( z 0 ) i u ( 0 , z ' ) = u ^ i ( z ' ) ( 0 i k - 1 ) . We can easily get a formal solution u ^ ( z ) = n = 0 u ^ n ( z ' ) ( z 0 ) n / n ! , bu in general it diverges. We show under some conditions that for any sector S with the opening less that a constant determined by L ( z , z ) , there is a function u S ( z ) holomorphic except on { z 0 = 0 } such that L ( z , z ) u S ( z ) = f ( z ) and u S ( z ) u ^ ( z ) as z 0 0 in S .

Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness

Jarmila Ranošová (1996)

Commentationes Mathematicae Universitatis Carolinae

Let T be a positive number or + . We characterize all subsets M of n × ] 0 , T [ such that inf X n × ] 0 , T [ u ( X ) = inf X M u ( X ) i for every positive parabolic function u on n × ] 0 , T [ in terms of coparabolic (minimal) thinness of the set M δ = ( x , t ) M B p ( ( x , t ) , δ t ) , where δ ( 0 , 1 ) and B p ( ( x , t ) , r ) is the “heat ball” with the “center” ( x , t ) and radius r . Examples of different types of sets which can be used instead of “heat balls” are given. It is proved that (i) is equivalent to the condition sup X n × + u ( X ) = sup X M u ( X ) for every bounded parabolic function on n × + and hence to all equivalent conditions given in the article [7]....

Clarke critical values of subanalytic Lipschitz continuous functions

Jérôme Bolte, Aris Daniilidis, Adrian Lewis, Masahiro Shiota (2005)

Annales Polonici Mathematici

The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawłucki's extension of the Puiseux lemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

Classical global solutions of the initial boundary value problems for a class of nonlinear parabolic equations

Guo Wang Chen (1994)

Commentationes Mathematicae Universitatis Carolinae

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation u t = - A ( t ) u x 4 + B ( t ) u x 2 + g ( u ) x 2 + f ( u ) x + h ( u x ) x + G ( u ) with the initial boundary value conditions u ( - , t ) = u ( , t ) = 0 , u x 2 ( - , t ) = u x 2 ( , t ) = 0 , u ( x , 0 ) = ϕ ( x ) , or with the initial boundary value conditions u x ( - , t ) = u x ( , t ) = 0 , u x 3 ( - , t ) = u x 3 ( , t ) = 0 , u ( x , 0 ) = ϕ ( x ) , are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

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