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Calcul symbolique non linéaire pour une onde conormale simple

Alain Piriou (1988)

Annales de l'institut Fourier

On considère une solution u , assez régulière, d’une équation aux dérivées partielles non linéaire. Si u est conormale par rapport a une hypersurface simplement caractéristique pour l’équation linéarisée, on étudie l’équation de transport satisfaite par son symbole principal, et on en déduit la propagation de la propriété “ u est conormale classique”.

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

Carleman estimates for a subelliptic operator and unique continuation

Nicola Garofalo, Zhongwei Shen (1994)

Annales de l'institut Fourier

We establish a Carleman type inequality for the subelliptic operator = Δ z + | x | 2 t 2 in n + 1 , n 2 , where z n , t . As a consequence, we show that - + V has the strong unique continuation property at points of the degeneracy manifold { ( 0 , t ) n + 1 | t } if the potential V is locally in certain L p spaces.

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.

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