The Dirichlet problem with BMO boundary data and almost-real coefficients
In a two-dimensional Lipschitz domain \(V\) with convex boundary, the author studies solutions properties for the following Dirichlet boundary problem: \[ \mathrm{div}A\nabla u=0\text{ in }V\text{ and } u=f,\leqno{(\ast)} \] such that \(A\) represents a square matrix satisfying the uniform ellipticity conditions such that its entries are complex-valued functions and \(t\)-independent, i.e., do not depend on the second variable, and \(f\) belonging to BMO(\(\partial V\)), the set of functions of bounded mean oscillation on \(\partial V\) (the boundary of \(V\)) and endowed with the norm \(\|\cdot\|_{\text{BMO}(\partial V)}\). Then, the author states the following result: \(i)\) if \(f\in \text{BMO}(\partial V)\) and the imaginary part of each coefficient of \(A\) is bounded, then there is \(u\), a solution for \((*)\) and satisfies \[ \frac{1}{\sigma(\partial V\cap B(x_0,r))}\int_{V\cap B(x_0,r)}|\nabla u(x)|^2\mathrm{dist}(x,\partial V)dx\leq C\|f\|^2_{\text{BMO}(\partial V)},\leqno{(\ast\ast)} \] \(C\) is a positive constant, \(B(x_0,r)\) is the Euclidean ball centered at \(x_0\in \partial V\) and of radius \(r>0\), \(\sigma(\cdot)\) stands for the real-valued Lebesgue measure on \(\partial V\), and \(\mathrm{dist}(\cdot,\partial V):=\inf_{\xi\in \partial V}\mathrm{dist}(\cdot,\xi)\). Moreover, the author states that \(u\) is unique whenever the left hand side of \((**)\) is bounded uniformly (Theorem 1.4). The proof is given in the third section (it is ended in the fourth section) and it is based on using a few known results stated in the second section, e.g., on the existence of a solution for \((*)\), the author has recourse to use the classical (double) layer potential associated with the operator \(\mathrm{div}A\nabla(\cdot)\) (Subsection 2.3). Theorem 1.4 also treats the converse of \(i)\) where the proof is provided in the fourth section. The second result states that if \(f\) is a bounded function on \(\partial V\), then the problem \((*)\) has a unique solution and bounded, up to a multiplicative constant, by the \(L_\infty(\partial V)\)-norm of \(f\) (Theorem 1.7), the proof is pinpointed in the fourth section.
scientific article; zbMATH DE number 6496479
scientific article