Steklov Spectrum And Elliptic Problems With Nonlinear Boundary Conditions

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Notices Of The American Mathematical Society


Problems with nonlinear boundary conditions arise naturally in many applications. For instance, in population dynamics where an impact of habitat-edges (boundary) on the dispersal pattern of species as they reach the boundary takes place in spatial ecology CC06. They occur when the biochemical reactions take place at or near the boundary, for example, in the limb bud development of a chick in which a chemical reaction produces outgrowth due to cell growth and division, and interactions between morphogens produced in several zones of the limb bud DO99. They also appear in noninvasive testing methods to locate defects in a medium by using boundary data measurements (see, e.g., CCMM16). In cryosurgery (a minimally invasive treatment used to treat some types of cancers and some conditions that may become cancer), a highly exothermic reaction takes place in a thin layer around the boundary in order to destroy abnormal tissue LOS98. These examples are not exhaustive.