On the smoothness of the solutions of multidimensional weakly singular integral equations

G. M. Vainikko

Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 585-600

Résumé

Estimates are given for the derivatives of solutions of the integral equation $$ u(x)=\int_GK(x,y)u(y)\,dy+f(x), \qquad x\in G, $$ where $G\subset R^n$ is an open bounded set, the kernel $K(x,y)$ has continuous derivatives up to order $m$ on $(G\times G)\setminus\{x=y\}$, and there exists a $\nu(-\infty\nu$ such that \begin{gather*} \biggl|\biggl(\frac\partial{\partial x_1}\biggr)^{\alpha_1}\dotsb\biggl(\frac\partial{\partial x_n}\biggr)^{\alpha_n}\biggl(\frac\partial{\partial x_1}+\frac\partial{\partial y_1}\biggr)^{\beta_1}\dotsb\biggl(\frac\partial{\partial x_n}+\frac\partial{\partial y_n}\biggr)^{\beta_n}K(x,y)\biggr| \\ \leqslant c \begin{cases} 1+|x-y|^{-\nu-|\alpha|},\nu+|\alpha|\ne0, \\ 1+|\ln|x-y||,\nu+|\alpha|=0, \end{cases} \qquad |\alpha|+|\beta|\leqslant m. \end{gather*} Two weighted function classes are distinguished such that if the free term $f$ is in one of them, so is the solution. The main qualitative consequence is that the tangential derivatives of a solution behave essentially better than the normal derivatives when $f$ is smooth. Figures: 4. Bibliography: 13 titles.