Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

<abstract><p>Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in <italic>conformal dimensions</italic>, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.</p></abstract>

Critical polyharmonic systems and optimal partitions

<p style='text-indent:20px;'>We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to <inline-formula><tex-math id="M2">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^N $\end{document}</tex-math></inline-formula>. We give a detailed description of the shape of these domains.</p>

Boundary value problems for systems of inhomogeneous polyharmonic equations with applications in the theory of thin shells and plates

A number of problems in the theory of elasticity, the theory of heterogeneous media, the theory of thin shells and plates is reduced to solving boundary value problems for systems of inhomogeneous polyharmonic equations. The paper proposes a numerical algorithm for solving systems of polyharmonic equations of the form in single-connected and multi-connected areas with a piecewise smooth contour with specified boundary conditions. Two cases are considered when the function is a known polyharmonic function and when the function is also the desired polyharmonic function. The boundary conditions can have the form similar to Dirichlet conditions, Neiman conditions, and can have a mixed form when on one part of the boundary conditions of the Dirichlet type are given, and on the other - hand, the conditions of the Neiman type. On the basis of multiple applications of the Laplace operator and the boundary element method, which is based on the green integral identity, the given system is reduced to a system of integral identities. After approximating the boundary by an inscribed n-gon and discretizing the system of integral identities, the latter is reduced to a system of linear algebraic equations, which is conveniently represented as a system of matrix equations. The existence and uniqueness of the solution follows from the existence of a unique solution of a system of linear algebraic equations. Special attention is paid to the application of the algorithm to the solution of problems on the bending of thin plates, and the bending load can be a known function, and can be an unknown polyharmonic function of an arbitrary order with given boundary conditions. The problem of bending a thin plate of elliptic shape with a known load on the surface is solved, as well as the problem of bending a thin square plate with an unknown load, which is the solution of a harmonic equation with given boundary conditions. The level lines are constructed and the forms of curved plates are given.