In this thesis, we are interested in the study of nonlinear eigenvalue problem and the controllability of partial differential equations in a smooth bounded domain with boundary. The first part is devoted to the analysis of an eigenvalue problem for quasilinear elliptic operators involving homogeneous Dirichlet boundary conditions. We investigate the asymptotic behaviour of the spectrum of the related problem by showing on the one hand the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and bifurcation from infinity using the Leray-Schauder degree on the other hand. We also prove the existence of multiple critical points using variational methods and the Krasnoselski genus. At last, we show a stabilization result for the damped plate equation with logarithmic decay of the associated energy. The proof of this result is achieved by means of a proper Carleman estimate for the fourth-order elliptic operators involving the so-called Lopatinskii-Šapiro boundary conditions and a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions.

BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR AND HIGHER-ORDER ELLIPTIC OPERATORS AND APPLICATION TO BIFURCATION AND STABILIZATION / W.b.e. Zongo ; supervisor: R. Bernard ; supervisor: J. Le Rousseau ; coordinator: D P. Bambusi, O. Bodini. - : . Dipartimento di Matematica Federigo Enriques, 2022 Jan 10. ((34. ciclo, Anno Accademico 2021. [10.13130/zongo-wend-benedo-emmanuel_phd2022-01-10].

BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR AND HIGHER-ORDER ELLIPTIC OPERATORS AND APPLICATION TO BIFURCATION AND STABILIZATION.

W.B.E. Zongo
2022

Abstract

In this thesis, we are interested in the study of nonlinear eigenvalue problem and the controllability of partial differential equations in a smooth bounded domain with boundary. The first part is devoted to the analysis of an eigenvalue problem for quasilinear elliptic operators involving homogeneous Dirichlet boundary conditions. We investigate the asymptotic behaviour of the spectrum of the related problem by showing on the one hand the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and bifurcation from infinity using the Leray-Schauder degree on the other hand. We also prove the existence of multiple critical points using variational methods and the Krasnoselski genus. At last, we show a stabilization result for the damped plate equation with logarithmic decay of the associated energy. The proof of this result is achieved by means of a proper Carleman estimate for the fourth-order elliptic operators involving the so-called Lopatinskii-Šapiro boundary conditions and a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions.
RUF, BERNHARD
RUF, BERNHARD
quasi-linear operators; bifurcation; bifurcation from infinity; multiple solutions; Carleman estimates; stabilization; Lopatinskii-Šapiro; resolvent estimate
Settore MAT/05 - Analisi Matematica
BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR AND HIGHER-ORDER ELLIPTIC OPERATORS AND APPLICATION TO BIFURCATION AND STABILIZATION / W.b.e. Zongo ; supervisor: R. Bernard ; supervisor: J. Le Rousseau ; coordinator: D P. Bambusi, O. Bodini. - : . Dipartimento di Matematica Federigo Enriques, 2022 Jan 10. ((34. ciclo, Anno Accademico 2021. [10.13130/zongo-wend-benedo-emmanuel_phd2022-01-10].
Doctoral Thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/892091
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