On Increased Summability of the Solution to the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift
DOI:
https://doi.org/10.52575/2687-0959-2024-56-2-124-135Keywords:
Higher Integrability of the Solution Gradient, Boyarsky – Meyers Inequality, Dirichlet Boundary Value ProblemAbstract
This paper is devoted to study of the Dirichlet problem in bounded Lipschitz domain for linear second order equation with drift (lower terms). Assuming that the coefficients of the first-order derivatives belong to the Lebesgue class, we prove Theorem of existence and uniqueness of solutions to this problem, i.e. we show the unique solvability of this problem. Cases of different dimensions of space are analyzed. We also prove the Boyarsky–Meyers inequality, i.e. we prove higher integrability of the gradient of solutions to the Dirichlet problem in bounded Lipschitz domain for the Laplacian with lower order terms.
The proof of main theorems is based on obtaining the inverse H¨older inequality for the gradient of the solution to the Dirichlet problem with subsequent application of the generalized Hering lemma. The Boyarsky – Meyers inequalities are useful for homogenization theory and in asymptotic methods.
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