Time periodic solutions to a nonhomogeneous Dirichlet periodic problem.
We study boundary value problems for quasilinear parabolic equations when the initial condition is replaced by periodicity in the time variable. Our approach is to relate the theory of such problems to the classical theory for initial-boundary value problems. In the process, we generalize many previously known results.
We consider the problem of localizing an inaccessible piece of the boundary of a conducting medium , and a cavity contained in , from boundary measurements on the accessible part of . Assuming that is the given thermal flux for , and that the corresponding output datum is the temperature measured at a given time for , we prove that and are uniquely localized from knowledge of all possible pairs of input-output data . The same result holds when a mean value of the temperature...
We consider the problem of localizing an inaccessible piece I of the boundary of a conducting medium Ω, and a cavity D contained in Ω, from boundary measurements on the accessible part A of ∂Ω. Assuming that g(t,σ) is the given thermal flux for (t,σ) ∈ (0,T) x A, and that the corresponding output datum is the temperature u(T0,σ) measured at a given time T0 for σ ∈ Aout ⊂ A, we prove that I and D are uniquely localized from knowledge of all possible pairs of input-output data . The same result...
We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the norm of the heat sources for exponents close to one. Such estimates are...
Let L be a nonsymmetric second order uniformly elliptic operator with generalWentzell boundary conditions. We show that a suitable version of L generates a quasicontractive semigroup on an Lp space that incorporates both the underlying domain and its boundary. This extends the earlier work of the authors on the symmetric case.