Regularity of solutions to the Dirichlet problem for degenerate elliptic equation
The author considers the following Dirichlet problem \[ \begin{cases} Lu = - (a_{ij}u_{x_i})_{x_j} = f, &\text{ in }\Omega, \\ u=0, &\text{ on } \partial \Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and \(a_{ij}\) is a symmetric, measurable matrix such that \[ \exists \;\nu > 0 \;: \;\nu^{-1} \omega(x) | \xi| ^2 \leq a_{ij} \xi_i\xi_j \leq \nu \omega(x) | \xi| ^2\,, \;x \in \Omega\,, \;\xi \in \mathbb{R}^n\,, \] where \(\omega\) is a weight that belongs to the Muckenhoupt class \(A_2\). The aim of the paper is to describe how achieve regularity of a class of generalized solutions in terms of a suitable assumption on the known term \(f\). Essentially the paper contains the extension of some results already known in the uniformly elliptic case to the degenerate one. The results obtained are the following: Assume that the function \(f\) belongs to a class of potentials (Schechter class), then the author obtains an extra-integrability result. Under further assumptions on the function \(f\), the author proves that the corresponding solution coincides with the variational one, showing that it is more and more regular as the known term belongs to some Stummel-Kato class and Morrey classes. After that, his aim is to show that the conditions just proved are indeed also necessary. First he starts proving that all the conditions are necessary, assuming that the function \(f\) is nonnegative. After that, he tries to show that the same results hold true in case the function \(f\) has variable sign. Unfortunately, the paper contains a serious mistake because the results stated are false in the case of functions \(f\) with variable sign. The mistake is the following: the author tries to prove the variable sign case by reduction to the constant sign one by splitting the solution \(u\) in its positive and negative part. Then he states that, denoting by \(u^+\) and \(u^-\) respectively the positive and negative part of the solution, and setting \(f^+ = Lu^+\), \(f^- = Lu^-\) the functions \(f^+\) and \(f^-\) are both non negative and the argument goes back to the case in which the known term is non negative. At this point, the result should be achieved also in the case of a variable sign known term \(f\). The point is that this statement is false as the following simple example shows. Consider the operator \(L=-\Delta\) in the ball centered at the origin with radius \(\sqrt n\) and the function \(f=2 e^{-| x\,| ^2} (n - 2| x\,| ^2)\). It is a triviality to recognize that the solution is the function \(u(x) = e^{-| x\,| ^2} - e^{-n}\). In this case, \(u^+ = u\) and \(u^- = 0\), then \(f^+ = Lu^+ = Lu = f\) and \(f^-= Lu^- = L 0 = 0\) but \(f^+\) has variable sign.
scientific article; zbMATH DE number 2052052