Geodesic
Smoothness of generalized solutions of the equation $\biggl(\lambda-\displaystyle\sum_{i,j}\nabla_ia_{ij}\nabla_j\biggr)u=f$ with continuous coefficients
Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 403-415 
Yu. A. Semenov. Smoothness of generalized solutions of the equation $\biggl(\lambda-\displaystyle\sum_{i,j}\nabla_ia_{ij}\nabla_j\biggr)u=f$ with continuous coefficients. Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 403-415. http://geodesic.mathdoc.fr/item/SM_1983_46_3_a5/
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     title = {Smoothness of generalized solutions of the equation $\biggl(\lambda-\displaystyle\sum_{i,j}\nabla_ia_{ij}\nabla_j\biggr)u=f$ with continuous coefficients},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[2] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[3] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[4] Morrey C. B., “Functions of Several variables and absolute continuity. II”, Duke Math. J., 6:1 (1940), 181–215 | DOI | MR

[5] Kovalenko V. F., Semenov Yu. A., “Nekotorye voprosy razlozheniya po obobschennym sobstvennym funktsiyam operatora Shredingera s silno singulyarnymi potentsialami”, UMN, 33:4 (1978), 107–140 | MR | Zbl