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On the smoothness of the solutions of multidimensional weakly singular integral equations
G. M. Vainikko
Abstract:
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
$\nu(-\infty<\nu<n)$ 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.
UDC:
517.96
MSC: Primary
45E99; Secondary
45B05,
45M99,
45L10 Received: 27.06.1987
Fulltext:
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