**R. Duduchava and B. Silbermann**

##
Boundary Value Problems in Domains with Peaks

**abstract:**

We obtain criteria of solvability of the Dirichlet and
the Neumann boundary value problems (BVPs) for the Laplacian
in 2D domains with angular points and peaks on the boundary.
We start with the correct formulation of BVPs and modify it for
domains with outward peaks (classical conditions are incorrect).
Boundary integral equations (BIEs), obtained by the indirect potential
method, turn out to be equivalent to the corresponding
BVPs only when inward peaks are absent. BIEs on boundary curve
with angular points are investigated in different weighted function
spaces. If boundary curve has a cusp, corresponding to an inward or
an outward peak, equations are
non--Fredholm in usual spaces and we should impose restrictions
on the right--hand sides. The conditions are defined with the
Cesaro--type integrals. We consider also equivalent reduction
to boundary pseudo-differential equations (BPsDEs) of orders $\pm1$
by the direct potential method. Crucial role in our investigations
of BVPs and of corresponding BIEs, PsDOs belongs to the equivalent
reduction of BVPs to the Riemann--Hilbert problem for
analytic functions on the unit disk. The latter problem
can be investigated thoroughly, even when peaks are present and
equations have non--closed image by invoking results on convolution
equations with vanishing symbols.

**Mathematics Subject Classification:**
47A68, 35J25, 35J55.

**Key words and phrases:**
Boundary integral equation,
Convolution equation, Non-elliptic symbols, Logarithmic potentials,
Boundary value problems