Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive $$\mathcal {R}$$ R -sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.

PARABOLIC BOUNDARY VALUE PROBLEMS IN A PIECEWISE HOMOGENEOUS WEDGE-SHAPED SOLID CYLINDER

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous wedge-shaped solid cylinder were constructed at first time by the method of integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices). The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable $\varphi \in (0; \varphi_0)$, a Fourier integral transform on the Cartesian segment $(-l_1;l_2)$ by an applicative variable $z$ and a hybrid integral transform of the Hankel type of the first kind on a segment $(0;R)$ of the polar axis with $n$ points of conjugation by an radial variable $r$ were used to construct solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image. The structure of the solution of the problem in the case of setting the Neumann boundary conditions on the wedge edges is analyzed. Exact analytical formulas for the components of the main solutions are written and the theorem on the existence of a single bounded classical solution of the problem is formulated. The obtained solutions are algorithmic in nature and can be used (using numerical methods) in solving applied problems.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.