Sufficient Condition of Existence for Quadrature Surfaces as Free Boundaries

This paper addresses the Quadrature Surfaces Free Boundary Problem, a significant challenge in the field of shape optimization and mathematical physics. We explore the existence of solutions by utilizing the shape derivative and applying the maximum principle, providing a novel sufficient condition on the data for resolving this complex problem. Our findings contribute to the understanding of free boundary problems and offer a comprehensive analytical framework for further investigations. The approach combines theoretical insights with mathematical rigor, potentially impacting related areas in applied mathematics and engineering.