Equations with Nonlinear Boundary Conditions
Many of the results given in the previous chapters can be extended to problems with nonlinear boundary conditions by the method of upper and lower solutions. This chapter is devoted to an extension of this method to both parabolic and elliptic boundary-value problems where the boundary function h is replaced by a nonlinear function. This extension also includes parabolic equations with nonlinear integral boundary conditions and an existence theorem for elliptic equations without the one-sided Lipshitz condition. In addition to the standard treatment of the uniqueness and positivity of a solution for elliptic boundary-value problems there is an analogous discussion on the spectrum problem where the parameter σ appears in both the internal and the boundary functions. Results are applied to three specific models in heat-conduction, biochemical reaction, and gas-liquid interaction problems.
KeywordsStrict Inequality Monotone Property Lower Solution Maximal Solution Nonlinear Boundary Condition
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