Volume 38, Issue 4

We consider an overdetermined problem associated to an inhomogeneous infinity-Laplace equation. More precisely, the domain of the problem is required to contain a given compact set K of positive reach, and the boundary of the domain must lie within the reach of K . We look for a solution vanishing at the boundary and such that the outer derivative depends only on the distance from K . We prove that if the boundary gradient grows fast enough with respect to such distance (faster than the distance raised to 13{\frac{1}{3}}), then the problem is solvable if and only if the domain is a tubular neighborhood of K , thus extending a previous result valid in the case when K is made up of a single point.

We prove an explicit formula for the tangential derivatives of the double layer heat potential. By exploiting such a formula, we prove the validity of a regularizing property for the integral operator associated to the double layer heat potential in spaces of functions with high-order derivatives in parabolic Hölder spaces defined on the boundary of parabolic cylinders which are unbounded in the time variable.

In this work, the interior spectral data is employed to study the inverse problem for a differential pencil with a discontinuity on the half line. By using a set of values of the eigenfunctions at some internal point and eigenvalues, we obtain the functions q0⁢(x){q_{0}(x)} and q1⁢(x){q_{1}(x)} applied in the diffusion operator.