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Giovanni Cimatti

Publications and source records attributed to Giovanni Cimatti.

8 recordsLinked to original sources
arXiv2021

A nonlocal formulation for the problem of microwave heating of material with temperature dependent conductivity

Microwave electromagnetic heating are widely used in many industrial processes. The mathematics involved is based on the Maxwell's equations coupled with the heat equation. The thermal conductivity is strongly dependent on the temperature, itself an unknown of the system of P.D.E. We propose here a model which simplifies this coupling using a nonlocal term as the source of heating. We prove that the corresponding mathematical initial-boundary value problem has solutions using the Schauder's fixed point theorem.

math-ph↗
arXiv2020

A variational principle for the system of P.D.E. of porous metal bearings

Porous metal bearings are widely used in small and micro devices. To compute the pressure one has to solve the Reynolds equation coupled with the Laplace equation. We show that it is possible to give to the relevant boundary value problem a variational formulation. We show that the pressure inside a porous bearing is less then that of the corresponding non-porous bearing.

math-ph↗
arXiv2019

An hyperbolic system of P.D.E. relevant in general relativity

Assuming as starting point the validity of the Einstein-Rosen metric, we study the hyperbolic system of P.D.E. to which the Einstein's field's equations can be reduced. We prove using the implicit function theorem in Banach spaces, the existence and uniqueness of gravitational waves of small amplitude. A class of solutions, not necessarily small is also constructed. In the last Section a theorem of existence and uniqueness is given for the corresponding stationary problem.

math-ph↗
arXiv2017

Functional solutions for problems of heat and mass transfer

We prove the existence and, in certain cases, the uniqueness of functional solutions for two boundary value problems of systems of P.D.E. in divergence form motivated by problems of heat and mass transfer. If $\cc_F$ and $\cc$ denote respectively the set of functional and classical solutions of these problems we settle, in simple cases, the question if $\cc_F=\cc$.

math-ph↗
arXiv2017

An application of a theorem of G. Zwirner to a class of non-linear elliptic systems in divergence form

A theorem on the solutions of the problem $U'(w)=γF(U(w),w),\ U(w_1)=u_2,\ U(w_2)=u_2$ is applied for finding the functional solutions of the system of partial differential equations \begin{equation} \nabla\cdot(a(u,w)\nabla u)=0,\ u=u_1\ on Γ_1,\ u=u_2\ on Γ_2,\ \frac{\partial u}{\partial n}=0\ on\ Γ_3 \end{equation} \begin{equation} \nabla\cdot(b(u,w)\nabla w)=0, \ w=w_1\ on Γ_1,\ w=w_2\ on\ Γ_2,\ \frac{\partial u}{\partial n}=0\ on\ Γ_3. \end{equation} The problem of existence and uniqueness of solutions is also considered.

math.AP↗
arXiv2017

Boundary value problems in general relativity

Certain theorems of existence, non-existence and uniqueness for boundary value problems modelling axial symmetric problems in general relativity are presented using the Weyl's metric. A solution related to the classical Poiseuille of non-relativistic fluid mechanics is also presented.

math-ph↗