V. Maz’ya

Unique solvability of the integral equation for harmonic simple layer potential on the boundary of a domain with a peak

The problem of finding a solution of the Dirichlet problem for the Laplace equation in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the kind Vρ = f to solve for the density, where f are boundary Dirichlet data. It is shown that if S is the boundary of an n-dimensional domain (n > 2) with an outward peak on S, then the operator V−1, which acts on the smooth functions on S, admits a unique extension to an isomorphism between the spaces of traces on S of functions with finite Dirichlet integral over Rn and the dual space. Thereby the equation Vρ = f is uniquely solvable for the density ρ for every trace f = u|S of function u with finite Dirichlet integral over Rn. Using an explicit description of the space of the traces specified, we can enunciate the theorem on solvability of a boundary integral equation Vρ = f in terms of the function describing the peak cusp.

On the Solvability of the Neumann Problem for a Planar Domain with a Peak

The Neumann problem for second-order elliptic quasi-linear equations on a planar domain whose boundary contains the vertex of an outward or inward peak. Under certain conditions, the solvability problem for the Neumann problem is reduced to a description of the space dual to the boundary trace space TWp1 (Ω) for functions from the Sobolev class (Wp1 (Ω), where 1 < p < ∞. This dual space is characterized in terms of Sobolev classes on Lipschitz curves with negative smoothness exponents and in terms of function spaces on the interval (0, 1) of the real line. The proofs of the main results are essentially based on an explicit description of the space TWp1 (Ω) for a planar domain with a peak due to the author. Necessary and sufficient conditions for q to be such that the Neumann problem is solvable provided that the boundary function belongs to Lq(∂Ω) are given.

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω+ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω− = Rn\Ω+. Let Γ be the common boundary of the domains Ω±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over Rn, and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω− with a cusp of an inward peak may be represented as Vρ−, where ρ− ∈ Tr(Γ)* is uniquely determined for all Ψ− ∈ Tr(Γ)*. If Ω+ is a domain with an inward peak and if Ψ+ ∈ Tr(Γ)*, Ψ+ ⊥ 1, then the solution of the Neumann problem for Ω+ has the representation u+ = Vρ+ for some ρ+ ∈ Tr(Γ)* which is unique up to an additive constant ρ0, ρ0 = V−1(1). These results do not hold for domains with outward peak.

The -dissipativity of first order partial differential operators

We find necessary and sufficient conditions for the -dissipativity of the Dirichlet problem for systems of partial differential operators of the first order with complex locally integrable coefficients. As a by-product we obtain sufficient conditions for a certain class of systems of the second order.