Hardy inequalities and asymptotics for heat kernels
Seminar Room 1, Newton Institute
AbstractWe will discuss some Hardy inequalities and its consequences on the large time behavior of diffusion processes. Roughly speaking, the Hardy inequality ensures a further and faster decay rate. Two differente situations will be addressed. First we shall consider the heat equation with a singular square potential located both in the interior of the domain and on the boundary, following a joint work with J. L. Vázquez and a more recent one with C. Cazacu. We shall also present the main results of a recent work in collaboration with D. Krejciírk in which we consider the case of twisted domains. In this case the proof of the extra decay rate requires of important analytical developments based on the theory of self-similar scales. As we shall see, asymptotically, the twisting ends up breaking the tube and adds a further Dirichlet condition, wich eventually produces the increase of the decay rate. Some consequences in which concerns the control of these models will also be presented. References: J. L. Vázquez and E. Zuazua The Hardy inequality and the asymptotic behavior of the heat equation with an inverse square potential. J. Functional Analysis, 173 (2000), 103--153.} J. Vancostenoble and E. Zuazua. Hardy inequalities, Observability and Control for the wave and Schr\"odinger equations with singular potentials, SIAM J. Math. Anal., Volume 41, Issue 4, pp. 1508-1532 (2009) D. Krejcirik and E. Zuazua The heat equation in twisted domains, J. Math pures et appl., to appear. C. Cazacu and E. Z. Hardy inequalities with boundary singular potentials, in preparation.
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