Nonlinear Robin Boundary Conditions Research Articles

Homogenization of a boundary‐value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

AbstractWe consider a boundary‐value problem for the Poisson equation in a thick junction Ωε, which is the union of a domain Ω0 and a large number of ε‐periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂νuε + εκ(uε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow‐up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H1(Ωε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.

Analysis of the Impact of Nonlinear Heat Transfer Laws on Temperature Distribution in Irradiated Biological Tissues: Mathematical Models and Optimal Controls

In this paper, we consider nonlinear control problems governed by some generalized transient bioheat transfer-type models with the nonlinear Robin boundary conditions. The control estimates the blood perfusion rate, the heat transfer parameter, the distributed energy source terms, and the heat flux due to the evaporation, which affect the effects of thermal physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics in medical practices, for example, for laser surgery, photo and thermotherapy for regional hyperthermia often used in treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness, and regularity of a solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Afterwards, the optimal control problem is formulated in order to control the online temperature given by radiometric measurement. We prove that an optimal solution exists and obtain necessary optimality conditions. Some strategy for numerical realization based on the adjoint variables are provided.

Fully nonlinear parabolic boundary value problems in one space dimension, I

Of concern is the parabolic equation ∂u ∂t (x,t) = F (x,u, ∂u ∂x , ∂ 2u ∂x 2 ) for t > 0, x ϵ [0, 1], and with the initial condition u( x, 0) = u 0( x) and the nonlinear Robin boundary condition ( ∂u ∂x )(0, t) ϵ β 0(u(0, t)), ( −∂u dx )(1, t) ϵ β 1(u(1, t)) . Here β 0 and β 1 are maximal monotone graphs in R × R with 0 ϵ β 0(0)∩ β 1(0). We solve it by using nonlinear semigroup theory in the space C[0, 1].