From Sue Brenner:
Are there upper bounds on the number of eigenvalues between 0 and 1 at a cross point of an interface problem, say in terms of the number of subdomains and the coefficients of the second order differential operator?
How do the constants in the elliptic regularity estimates of interface problems depend on the coefficients of the differential operators? For a fixed number of subdomains, would these constants blow up as the coefficients of the differential operators tend to infinity/zero?
Is there an elliptic regularity theory using Besov spaces that can recapture the shift theorem at the exceptional cases where the shift theorem for Sobolev spaces fails?
Are there extraction formulas for vertex-edge singularities?
From Atans Dimitrov:
Singularities for non linear elastic problems, in particular discussion of eigenvalue problems of the following type: [A(u)+lambda*B(u)+lambda^2*C(u)] u = 0
From Chris Schwab:
Are there shift theorems for elliptic operator -div(A grad), Dirichlet BCs, in (0,1)^d with d>3, in weighted spaces?
From Zohar Yosibash:
Denote by P_0 the 2-D problem having a sharp V-notch. Its asymptotic solution consisting of eigen-pairs and the corresponding coefficients. Let us assume a perturbed problem P_1 so that the V-notch tip is not sharp, but has an arc of radius rho at its tip. Is it possible to explicitly construct an asymptotic expansion for P_1 based on the P_0 eigen-pairs and coefficients (being dependent on rho)?
I am interested in a set of problems (linear elasticity and scalar elliptic) with available analytic solutions having edge, vertex and edge-vertex singularities. These can serve as benchmarks for numerical methods used to compute the singular solutions of 3-D problems.
It is known that 2-D scalar elliptic problems with homogeneous boundary conditions along a V-notch faces have real eigen-pairs. However, for a bi-material interface, complex eigen-pairs exist. Is there a general theory for 2-D scalar elliptic problems for any singular point which defines the conditions for complex eigen-pairs?