On existence of boundary and initial values for degenerate parabolic equations in stellar domains

T. V. Kapitsyna

Matematičeskie zametki SVFU, Tome 25 (2018), pp. 15-33.

Résumé

We establish the necessary and sufficient conditions for the solution of the second-order parabolic equation in a stellar domain with a lateral boundary in the class $C^{1+\lambda}$, $\lambda>0$, degenerate on the boundary of the domain, to have an average limit on the lateral surface of the cylindrical domain and the limit in the mean on its lower base. Also, we study the unique solvability of the first mixed problem for such equations in the case when the boundary and initial functions belong to spaces of the type $L_2$. The closest to the questions under consideration are the theorems of Riesz and Littlewood and Paley, in which criteria are given for the limit values in $L_p,$ $p>1$, of functions analytic in the unit disk. Further development of this topic for uniformly elliptic equations was obtained in the works V. P. Mikhailov and A. K. Gushchin. The boundary smoothness condition ($\partial Q\in C^2$) can be weakened, as was shown by I. M. Petrushko. Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criteria for the existence of a boundary value were established in by A. K. Gushchin. In this case, all directions of the acceptance of boundary values for uniformly elliptic equations turn out to be equal, the solution has the property similar to the property of continuity with respect to the set of variables. In the case of degeneracy of the equation on the boundary of the domain, when the directions are not equal, the situation is more complicated. In this case, the formulation of the first boundary value problem is determined by the type of degeneracy. When the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are different from zero (Tricomi type degeneracy), the Dirichlet problem is well-posed and the properties of such degenerate equations are very close to the properties of uniformly elliptic equations. In particular, in this situation analogues of the Riesz and Littlewood–Paley theorems are valid.

Détail

Keywords: degenerate parabolic equation, function space, first mixed problem, boundary and initial values of solutions, a priori estimate.