Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in 𝑅²

We study the problem of finding functions, defined within and on an ellipse, whose Laplacian is -1 and which satisfy a homogeneous Robin boundary condition on the ellipse. The parameter in the Robin condition is denoted by β. The integral of the solution over the ellipse, denoted by Q, is a quantity of interest in some physical applications. The dependence of Q on β and the ellipse's geometry is found. Several methods are used.• To find the general solution the boundary value problem is formulated in elliptic cylindrical coordinates. A Fourier series solution is then derived. Results concerning the difference equation which the Fourier coefficients satisfy are presented.• Variational methods have given simple and accurate lower bounds.• Various asymptotic approximations are found directly from the pde formulations, this being far easier than from our series solution. The pde for large β asymptotics again leads to difference equations.It is intended that this arXiv preprint will be referenced by the journal version, which will be submitted soon, as the arXiv contains material, e.g. codes for calculating Q, not in the very much shorter journal version.1

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Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in 𝑅²

Abstract: ABSTRACT. We consider some two-dimensional semilinear elliptic boundary value problems over a bounded convex domain in R2 and show the uniqueness of the critical point of the solutions.

Cited by 3 publications

References 21 publications

On the location of critical point for the Poisson equation in plane

On the location of critical point for the Poisson equation in plane

Nonlinear Eigenvalue Problem with Quantization

Functions with constant Laplacian satisfying homogeneous Robin boundary conditions

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