Geodesic
On weak solutions of boundary value problems for some general differential equations
Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 891-905

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We study general settings of the Dirichlet problem, the Neumann problem, and other boundary value problems for equations and systems of the form $\mathcal{L}^+ A\mathcal{L}u=f$ with general (matrix, generally speaking) differential operation $\mathcal{L}$ and some linear or non-linear operator $A$ acting in $L^k_2(\Omega)$-spaces. For these boundary value problems, results on well-posedness, existence and uniqueness of a weak solution are obtained. As an operator $A$, we consider Nemytskii and integral operators. The case of operators involving lower-order derivatives is also studied.
Keywords: partial differential equation, general theory of boundary value problems, boundary value problem, well-posedness, weak solution. 
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     author = {V. P. Burskii},
     title = {On weak solutions of boundary value problems for some general differential equations},
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V. P. Burskii. On weak solutions of boundary value problems for some general differential equations. Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 891-905. http://geodesic.mathdoc.fr/item/IM2_2023_87_5_a2/