626 research outputs found
Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions
We consider a second-order selfadjoint elliptic operator with an anisotropic
diffusion matrix having a jump across a smooth hypersurface. We prove the
existence of a weight-function such that a Carleman estimate holds true. We
moreover prove that the conditions imposed on the weight function are
necessary.Comment: 43 page
Control in moving interfaces and deep learning
Tesis Doctoral inĂ©dita leĂda en la Universidad AutĂłnoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 14-05-2021This thesis has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
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