V. G. Maz’ya

Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations

Introduction Singularities of solutions to equations of mathematical physics: Prerequisites on operator pencils Angle and conic singularities of harmonic functions The Dirichlet problem for the Lame system Other boundary value problems for the Lame system The Dirichlet problem for the Stokes system Other boundary value problems for the Stokes system in a cone The Dirichlet problem for the biharmonic and polyharmonic equations Singularities of solutions to general elliptic equations and systems: The Dirichlet problem for elliptic equations and systems in an angle Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle The Dirichlet problem for strongly elliptic systems in particular cones The Dirichlet problem in a cone The Neumann problem in a cone Bibliography Index List of symbols.

Linear Water Waves: A Mathematical Approach

Preface Part I. Time-Harmonic Waves: 1. Greens functions 2. Submerged obstacles 3. Semisubmerged bodies, I 4. Semisubmerged bodies, II 5. Horizontally-periodic trapped waves Part II. Ship Waves on Calm Water: 6. Greens functions 7. The Neumann-Kelvin problem 8. Two-dimensional problem Part III. Unsteady Waves: 9. Submerged obstacles: existence 10. Waves due to rapidly stabilizing and high-frequency disturbances Bibliography Name index Subject index.

Traces and Extensions of Multipliers in Pairs of Sobolev Spaces

It is well known that the space W p l−1/p (R n−1) with integer l is the space of traces on R n−1 of functions in the Sobolev space W p l (R + n ), where R + n = {(x, y): x ∈ R n−1, y > 0 }. We show that a similar result holds for spaces of pointwise multipliers acting in a pair of Sobolev spaces. Namely, we prove that the traces on.R n of functions in the multiplier space M(W p m (R + n ) → W p l (R + n )) form the space M(W p m−1/P (R n−1) → W> p l−1/P (R n−1)), and that there exists a linear continuous extension operator which maps M(W p m−1/P (R n−1) → W p l−1/P (R n−1)) to M(W p m (R + n ) → W p l (R + n )). We apply this result to the Dirichlet problem for the Laplace equation in the half-space.

On the solvability of the Neumann problem in domains with peak

The Neumann problem is considered for a quasilinear elliptic equation of second order in a multidimensional domain with the vertex of an isolated peak on the boundary. Under certain assumptions, the study of the solvability of this problem is reduced to a description of the dual to the Sobolev space W-p(1)(Omega) or (in the case of a homogeneous equation with nonhomogeneous boundary condition) to the boundary trace space TWp1(Omega). This description involves Sobolev classes with negative smoothness exponent on Lipschitz domains or Lipschitz surfaces, and also some weighted classes of functions on the interval (0,1) of the real line. The main results are proved on the basis of the known explicit description of the spaces TWp1(Omega) on a domain with an outward or inward cusp on the boundary.