Smoothness of generalized solutions of the equation $\biggl(\lambda-\displaystyle\sum_{i,j}\nabla_ia_{ij}\nabla_j\biggr)u=f$ with continuous coefficients

Yu. A. Semenov

Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 403-415

Résumé

It is shown that if $u$ is a weak solution in $L^2(\mathbf R^l)$ of the equation $$ \biggl(\lambda-\sum_{i,j=1}^l\nabla_i a_{ij}\nabla_j\biggr)u=f, \qquad f\in L^1\cap L^\infty, \quad \lambda\geqslant0, $$ with continuous $a_{ij}(\,\cdot\,)$ and the matrix $(a_{ij})$ is real-valued, symmetric, and positive-definite, then $u\in\bigcap_{1$, where $L_k^p(\mathbf R^l)$ is the Sobolev space of functions whose derivatives through order $k$ are $p$-integrable. It is also proved that if $(a_{ij})=(k^2\delta_{ij})$, $\delta_{ij}$ the Kronecker symbol, $1\leqslant k$, and $\overrightarrow\nabla k\in L^4$, then for a certain extension $A\supset 1-\overrightarrow\nabla k^2\overrightarrow\nabla\upharpoonright C_0^\infty$ it is true that $A^{-1}[L^2\cap L^\infty]\subset L_2^2 \cap L_1^4$, and, moreover, $k^2\nabla_i\nabla_j u \in L^2$ and $k\nabla_i u\in L^4$ $\forall u\in A^{-1}[L^2\cap L^\infty]$. Bibliography: 5 titles.