Interpolating estimates with applications to some quantitative symmetry results

<abstract><p>We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.</p></abstract>

On fourth order accuracy stable difference scheme for a multi-point overdetermined elliptic problem

In this paper fourth order of accuracy difference scheme for approximate solution of a multi-point elliptic overdetermined problem in a Hilbert space is proposed. The existence and uniqueness of the solution of the difference scheme are obtained by using the functional operator approach. Stability, almost coercive stability, and coercive stability estimates for the solution of difference scheme are established. These theoretical results can be applied to construct a stable highly accurate difference scheme for approximate solution of multi-point overdetermined boundary value problem for multidimensional elliptic partial differential equations.

Recovery of harmonic functions from partial boundary data respecting internal pointwise values

We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary

Singular critical points for variational free boundary problem from isoparametric hypersurfaces

We construct three families of singular critical points for a variational free boundary problem. These critical points are homogeneous solutions of degree one to some overdetermined boundary value problem. The intersections of the level sets of these solutions with the unit sphere are isoparametric hypersurfaces and their focal submanifolds.

Diffraction of a Plane Elastic Wave by a Gradient Transversely Isotropic Layer

The problem of diffraction of a plane elastic wave by a gradient transversely isotropic layer is considered. Using the method of overdetermined boundary value problem in combination with the Fourier transform method, the system of ordinary differential equations of the second order with boundary conditions of the third type is obtained which is solved by the grid method. Results of calculations obtained using the above-mentioned technique for the case of piecewise linear profiles for the Young modulus of the layer are given.