The definition of multiple layer potential for the biharmonic equation in ${\mathbb{R}}^n$ is given. In order to represent the solution of Dirichlet problem by means of such a potential, a singular integral system, whose symbol determinant identically vanishes, is considered. The concept of bilateral reduction is introduced and employed for investigating such a system.

Let Ω be a bounded domain in ${ℝ}^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2\left(\Omega \right)$ into itself with compact inverse, with eigenvalues $-{\lambda }_i$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+b{u}^{+}-a{u}^{-}=f\left(x\right)$ in Ω, u=0 on ∂ Ω. We assume that $a<{\lambda }_1$, ${\lambda }_2<b<{\lambda }_3$ and f is generated by ${\varphi }_1$ and ${\varphi }_2$. We show a relation between the multiplicity of solutions and source terms in the equation....

We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.

We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with $F\in BUC(ℝ;L_{3/2,\infty }\left(D\right))$, we consider this problem in D × ℝ and prove that there exists a unique solution $u\in BUC(ℝ;L_{3,\infty }\left(D\right))$ when F and |ω| are sufficiently small. If, in particular, the external force for...

In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-\left(\right|x^{{}^{\text{'}}}\left(t\right){{|}^{p-2}{x}^{{}^{\text{'}}}\left(t\right))}^{{}^{\text{'}}}=f(t,x\left(t\right),x^{{}^{\text{'}}}\left(t\right))$, a.e. on $T$, $x^{{}^{\text{'}}}\left(0\right)=x^{{}^{\text{'}}}\left(b\right)=0$, $2\le p<\infty$ in the order interval $[\psi ,\varphi ]$, where $\psi$ and $\varphi$ are respectively a lower and an upper solution of the Neumann problem.

In the paper we study the equation $Lu=f$, where $L$ is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set $\Omega$. We prove existence and uniqueness of solutions in the space $H\left(\Omega \right)$ for the Neumann problem.

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set $G$ in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on $\partial G$. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on $G$ a necessary and sufficient condition for the solvability...