The two-phase obstacle problem (Q2773606)

The article is devoted to the study of the two-phase obstacle problem NEWLINE\[NEWLINE \Delta u = \lambda_+\chi_{\{ u > 0\}} - \lambda_-\chi_{\{ u < 0\}} \quad\text{in } D\subset\mathbb R^n, NEWLINE\]NEWLINE where \(\chi_{\Omega}\) is the characteristic function, \(\lambda_+\) and \(\lambda_-\) are nonnegative constants such that \(\lambda_+ + \lambda_- > 0\). The above equation is the Euler equation for the energy integral NEWLINE\[NEWLINE I(v) = \int_D\bigl(|\nabla v|^2 + 2\lambda_+\max\{v,0\} - 2\lambda_-\min\{v,0\}\bigr) dx. NEWLINE\]NEWLINE Using the Alt--Caffarelli--Friedman monotonicity formula, the author obtains a priori estimates for solutions of the problem and proves boundedness of the second partial derivatives of the solutions.