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On a class of elliptic operators with unbounded coefficients in convex domains

Giuseppe Da Prato, Alessandra Lunardi (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study the realization A of the operator A = 1 2 - ( D U , D ) in L 2 Ω , μ , where Ω is a possibly unbounded convex open set in R N , U is a convex unbounded function such that lim x Ω , x Ω U x = + and lim x + , x Ω U x = + , D U x is the element with minimal norm in the subdifferential of U at x , and μ d x = c exp - 2 U x d x is a probability measure, infinitesimally invariant for A . We show that A , with domain D A = u H 2 Ω , μ : D U , D u L 2 Ω , μ is a dissipative self-adjoint operator in L 2 Ω , μ . Note that the functions in the domain of A do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow...

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

Jérôme Le Rousseau, Gilles Lebeau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations∗∗∗

Jérôme Le Rousseau, Gilles Lebeau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...

On definitions of superharmonic functions

Seizô Itô (1975)

Annales de l'institut Fourier

Let A be an elliptic differential operator of second order with variable coefficients. In this paper it is proved that any A -superharmonic function in the Riesz-Brelot sense is locally summable and satisfies the A -superharmonicity in the sense of Schwartz distribution.

On Dittmar's approach to the Beltrami equation

Ewa Ligocka (2002)

Colloquium Mathematicae

We recall an old result of B. Dittmar. This result permits us to obtain an existence theorem for the Beltrami equation and some other results as a direct consequence of Moser's classical estimates for elliptic operators.

On Hölder regularity for elliptic equations of non-divergence type in the plane

Albert Baernstein II, Leonid V. Kovalev (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

This paper is concerned with strong solutions of uniformly elliptic equations of non-divergence type in the plane. First, we use the notion of quasiregular gradient mappings to improve Morrey’s theorem on the Hölder continuity of gradients of solutions. Then we show that the Gilbarg-Serrin equation does not produce the optimal Hölder exponent in the considered class of equations. Finally, we propose a conjecture for the best possible exponent and prove it under an additional restriction.

On Kelvin type transformation for Weinstein operator

Martina Šimůnková (2001)

Commentationes Mathematicae Universitatis Carolinae

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator W k : = Δ + k x n x n on n is proved. In this note there is shown that in the cases k 0 , k 2 no other transforms of this kind exist and for case k = 2 , all such transforms are described.

On Neumann elliptic problems with discontinuous nonlinearities

Nikolaos Halidias (2001)

Archivum Mathematicum

In this paper we study a class of nonlinear Neumann elliptic problems with discontinuous nonlinearities. We examine elliptic problems with multivalued boundary conditions involving the subdifferential of a locally Lipschitz function in the sense of Clarke.

On radial limit functions for entire solutions of second order elliptic equations in R2.

André Boivin, Peter V. Paramonov (1998)

Publicacions Matemàtiques

Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R2, we consider entire solutions of the equation Lu = 0 for whichlimr→∞ u(reiφ) =: U(eiφ)exists for all φ ∈ [0; 2π) as a finite limit in C. We characterize the possible "radial limit functions" U. This is an analog of the work of A. Roth for entire holomorphic functions. The results seems new even for harmonic functions.

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