Papers by Claudia Garetto
G- and G ∞ -hypoellipticity of partial differential operators with constant Colombeau coefficients
Linear and Non-Linear Theory of Generalized Functions and its Applications, 2010
Monatshefte für Mathematik, 2005
We study the topological duals of the Colombeau algebras Gc(Ω), G(Ω) and G S (R n ), discussing s... more We study the topological duals of the Colombeau algebras Gc(Ω), G(Ω) and G S (R n ), discussing some continuous embeddings and the properties of generalized delta functionals.
We study pseudodifferential operators with amplitudes aε(x, ξ) depending on a singular parameter ... more We study pseudodifferential operators with amplitudes aε(x, ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 0, refined versions of estimates for classical pseudodifferential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudodifferential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations.
Lp and Sobolev boundedness of pseudodifferential operators with non-regular symbol: A regularisation approach
Journal of Mathematical Analysis and Applications, Sep 1, 2011
We provide sufficient conditions of local solvability for partial differential operators with var... more We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators which are a bounded perturbation of a differential operator with constant Colombeau coefficients. The local solutions are intended in the Colombeau algebra G(Ω) as well as in the dual L(Gc(Ω), e C).
Topological Structures in Colombeau Algebras: Investigation of the Duals of , and
Monatsh Math, 2005
C-modules: structural properties and applications to variational problems
We develop a theory of Hilbert e C-modules by investigating their structural and functional ana- ... more We develop a theory of Hilbert e C-modules by investigating their structural and functional ana- lytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for e C-linear functionals and e C-sesquilinear forms. By making use of a gen- eralized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear
Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients
Mathematische Nachrichten, 2014
On hyperbolic equations and systems with non-regular time dependent coefficients
Journal of Differential Equations, 2015
The aim of this work is to develop a global calculus for pseudo-differential operators acting on ... more The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on R n .
Wave equation for sums of squares on compact Lie groups
Journal of Differential Equations, 2015
ABSTRACT In this paper we investigate the well-posedness of the Cauchy problem for the wave equat... more ABSTRACT In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the Cauchy problem in local Sobolev spaces depending on the order to which the H\"ormander condition is satisfied, but no loss in globally defined spaces. We also establish Gevrey well-posedness for equations with irregular coefficients and/or multiple characteristics. As in the Sobolev spaces, if formulated in local coordinates, we observe well-posedness with the loss of local Gevrey order depending on the order to which the H\"ormander condition is satisfied.
Archive for Rational Mechanics and Analysis, 2014
In this paper we study weakly hyperbolic second order equations with time dependent irregular coe... more In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means to assume that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.
New Developments in Pseudo-Differential Operators, 2008
Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is b... more Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data.
We study the topological duals of the Colombeau algebras $\Gc(\Om)$, $\G(\Om)$ and $\GS(\R^n)$, d... more We study the topological duals of the Colombeau algebras $\Gc(\Om)$, $\G(\Om)$ and $\GS(\R^n)$, discussing some continuous embeddings and the properties of generalized delta functionals.
Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems
Transactions of the American Mathematical Society, 2011
ABSTRACT
Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities
Proceedings of the Edinburgh Mathematical Society, 2014
Proceedings of the Edinburgh Mathematical Society, 2005
We characterize microlocal regularity, in the G ∞ -sense, of Colombeau generalized functions by a... more We characterize microlocal regularity, in the G ∞ -sense, of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic regularity result for distributional solutions and consider propagation of G ∞ -singularities for homogeneous first-order hyperbolic equations.
Generalized oscillatory integrals and Fourier integral operators
Proceedings of the Edinburgh Mathematical Society, 2009
In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase fun... more In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distribution data.
We develop a theory of Hilbert $\widetilde{\C}$-modules by investigating their structural and fun... more We develop a theory of Hilbert $\widetilde{\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\widetilde{\C}$-linear functionals and $\widetilde{\C}$-sesquilinear forms. By making use of a generalized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form.
Mathematische Nachrichten, 2009
We present closed graph and open mapping theorems for C-linear maps acting between suitable class... more We present closed graph and open mapping theorems for C-linear maps acting between suitable classes of topological and locally convex topological C-modules. This is done by adaptation of De Wilde's theory of webbed spaces and Adasch's theory of barrelled spaces to the context of locally convex and topological C-modules respectively. We give applications of the previous theorems to Colombeau theory as well to the theory of Banach C-modules. In particular we obtain a necessary condition for G ∞ -hypoellipticity on the symbol of a partial differential operator with generalized constant coefficients. * Supported by FWF (Austria), grants T305-N13 and Y237-N13, and TWF (Tyrol), grant UNI-0404/305.
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Papers by Claudia Garetto