This is a survey on elliptic boundary value problems on varying domains and tools needed for that. Such problems arise in numerical analysis, in shape optimisation problems and in the investigation of the solution structure of nonlinear elliptic equations. The methods are also useful to obtain certain results for equations on non-smooth domains by approximation by smooth domains.
Domain independent estimates and smoothing properties are an essential tool to deal with domain perturbation problems, especially for non-linear equations. Hence we discuss such estimates extensively, together with some abstract results on linear operators.
This handbook is the sixth and last volume in the series devoted to stationary partial differential equations. The topics covered by this volume include in particular domain perturbations for boundary value problems, singular solutions of semi-linear elliptic problems, positive solutions to elliptic equations on unbounded domains, symmetry of solutions, stationary compressible Navier-Stokes equation, Lotka-Volterra systems with cross-diffusion, fixed point theory for elliptic boundary value problems.
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