How to approximate the solutions of certain free boundary problems for the Laplace equation by using the contraction principle

Boundary value problems for second-order elliptic equations (35J25) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Numerical methods for partial differential equations, boundary value problems (65N99)

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A trial-free-boundary method for computing Batchelor flows ⋮ On the Convexity and on the Successive Approximation of Solutions in a Free–Boundary Problem with Two Fluid Phases ⋮ Nonlinear integral equations for Bernoulli's free boundary value problem in three dimensions ⋮ Improved trial methods for a class of generalized Bernoulli problems ⋮ On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions ⋮ On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem

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